Mobile Networks and Applications

Mobile Networks and Applications 10, 7–8, 2005 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Guest Editorial

Algorithmic Solutions for Wireless, Mobile, Ad Hoc and Sensor Networks

AMOTZ BAR-NOY, ALAN A. BERTOSSI, CRISTINA M. PINOTTI and CAULIGI S. RAGHAVENDRA

The field of wireless and mobile computing is an important and research challenging area of computing today. This has beenmade possible due to the tremendous and continued growth of wireless technology, creating the need of ubiquitous distributedservices: anywhere and anytime. In addition to wireless networks based on a pre-existing infrastructure, where wirelesscommunications take place only between the end-nodes and the access points, mobile ad hoc wireless networks and sensornetworks are emerging rapidly. Such networks do not need any insfrastructure to work, but they comprise mobile clientsas well as mobile servers, and they pose challenges in diverse areas such as network topology control, routing and security,resource placement, allocation and discovery, energy consumption, and media access. In this scenario, there is a great needof algorithmic solutions to realize and maintain high-speed, high-performance, cost-effective, energy-efficient and reliablewireless networks.

This special issue brings together contributions in discrete algorithms, optimization techniques, and performance evaluationmethods in the context of wireless, ad-hoc, and sensor networks.

More than 60 papers were submitted, out of which only the 16 papers below have been selected for publication. Thesepapers do not cover all aspects that belong to the scope of this special issue. However, they represent interesting researchefforts and subjects that definitely belong to the core research on algorithmic solutions for wireless, mobile, ad hoc, and sensornetworks.

The first three papers deal with the network topology control problem.The paper “Dynamic Coverage in Ad-Hoc Sensor Networks”, by H. Huang, A.W. Richa and M. Segal, dynamically main-

tains measures on the quality of the coverage of a sensor network.The paper “Algorithmic Aspects of Topology Control Problems for Ad Hoc Networks”, by E.L. Lloyd, R. Liu,

M.V. Marathe, R. Ramanathan and S.S. Ravi, shows how to assign power values in ad hoc networks to obtain a graph topologysatisfying some specified important properties.

The paper “Wireless ATM Layouts for Chain Networks”, by M. Flammini, G. Gambosi and A. Navarra, integrates thebenefits of the ATM technology with the wireless communication, and studies the existence of optimal layouts for specialnetwork topologies.

The next two papers consider the problem of routing in ad hoc networks.The paper “Ad Hoc Multicast Routing Algorithm with Swarm Intelligence”, by C.-C. Shen and C. Jaikaeo, proposes a novel

idea for multicast routing based on Swarm Intelligence that refers to complex behaviors that arise from very simple individualbehaviors and interactions.

The paper “Regional Gossip Routing for Wireless Ad Hoc Networks”, by X.-Y. Li, K. Moaveninejad and O. Frieder,develops a location based routing protocol and presents a detailed analysis of this routing protocol.

The next three papers study the issues of placement, allocation and discovery of resources in cellular and ad hoc networks.The paper “Comparison and Evaluation of Multiple Objective Genetic Algorithms for the Antenna Placement Problem”, by

L. Raisanen and R.M. Whitaker, evaluates the performance of a greedy algorithm to select and configure base station locationsusing genetic algorithms methods.

The paper “A Characterisation of Optimal Channel Assignments for Cellular and Square Grids Wireless Networks”, byM.V.S. Shashanka, A. Pati and A.M. Shende, proposes optimal channel assignment algorithms in wireless networks whosetopology can be represented by square and cellular grids.

The paper “CARD: A Contact-Based Architecture for Resource Discovery in Ad Hoc Networks”, by A. Helmy, S. Garg,P. Pamu and N. Nahata, proposes a resource discovery mechanism based on distributed directories which is suitable for largead hoc networks.

The next group of papers involves the energy consumption problem in sensor and ad hoc networks.The paper “Energy-Balanced Task Allocation for Collaborative Processing in Wireless Sensor Networks” by Y. Yu and

V.K. Prasanna, considers the problem of scheduling a real-time application onto a single-hop wireless sensor network takinginto account energy requirements for both computation and communication.

8 GUEST EDITORIAL

The paper “Efficient and Robust Protocols for Local Detection and Propagation in Smart Dust Networks”, by I. Chatzigian-nakis, S. Nikoletseas and P. Spirakis, presents various protocols for smart dust based sensor networks for local event detectionand propagation of reports.

The paper “Training a Wireless Sensor Network”, by A. Wadaa, S. Olariu, L. Wilson, M. Eltoweissy and K. Jones, proposesa protocol for training nodes in a sensor network. The protocol, partitioning nodes into clusters, obtains a scalable and energy-efficient routing from cluster to the sink.

The paper “Quorum-Based Asynchronous Power-Saving Protocols for IEEE 802.11 Ad Hoc Networks”, by J.-R. Jiang,Y.-C. Tseng, C.-S. Hsu and T.-H. Lai, addresses the asynchronous power management problem for an IEEE 802.11-basedMulti-Hop MANET, correlating it to the concept of quorum system.

The last group of four papers deals with the media access and transmission scheduling problems.The paper “CROMA – An Enhanced Slotted MAC Protocol for MANETs”, by M. Coupechoux, B. Baynat, C. Bonnet and

V. Kumar, presents a TDMA based MAC protocol with high utilization in synchronized mobile ad hoc networks. CROMAhandles both the hidden terminal and exposed terminal problems to achieve a high throughput.

The paper “Dynamic Bandwidth Management in Single-Hop Ad Hoc Wireless Networks”, by S.H. Shah, K. Chen andK. Nahrstedt, presents dynamic bandwidth management and call admission control in a single hop ad hoc network at theapplication level. This paper shows an elegant solution to this problem with simulation and experimental results.

The paper “High Speed Networking Security: Design and Implementation of Two New DDP-Based Ciphers”, byN. Sklavos, N.A. Moldovyan and O. Koufopavlou, proposes two new fast ciphers suitable for wireless communications,which set hard specifications in security implementations.

Finally, the paper “Media Synchronization and Qos Packet Scheduling Algorithms for Wireless Systems”, by A. Boukercheand H. Owens II, considers the QoS requirements and the scheduling transmission problems arising when multiple streams oftext, images, audio and video are sent to mobile clients through a combined wired and wireless network.

The guest editors wish to thank all the referees for their valuable comments and suggestions, and all the authors for theirhigh quality submissions. Special thanks go to the Editor-in-Chief of MONET for hosting this special issue.


Dynamic Coverage in Ad-Hoc Sensor Networks

HAI HUANG ∗ and ANDRÉA W. RICHA ∗,∗∗Department of Computer Science and Engineering, Arizona State University, Tempe, AZ 85287-8809, USA

MICHAEL SEGALCommunication Systems Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

Abstract. Ad-hoc networks of sensor nodes are in general semi-permanently deployed. However, the topology of such networks con-tinuously changes over time, due to the power of some sensors wearing out, to new sensors being inserted into the network, or even dueto designers moving sensors around during a network re-design phase (for example, in response to a change in the requirements of thenetwork). In this paper, we address the problem of how to dynamically maintain two important measures on the quality of the coverageof a sensor network: the best-case coverage and worst-case coverage distances. We assume that the ratio between upper and lower trans-mission power of sensors is bounded by a polynomial of n, where n is the number of sensors, and that the motion of mobile sensors canbe described as a low-degree polynomial function of time. We maintain a (1 + ε)-approximation on the best-case coverage distance anda (

√2 + ε)-approximation on the worst-case coverage distance of the network, for any fixed ε > 0. Our algorithms have amortized or

worst-case poly-logarithmic update costs. We are able to efficiently maintain the connectivity of the regions on the plane with respect to thesensor network, by extending the concatenable queue data structure to also serve as a priority queue. In addition, we present an algorithmthat finds the shortest maximum support path in time O(n log n).

Keywords: coverage, ad hoc sensor network, kinetic data structure

1. Introduction

Ad-hoc sensor networks are emerging as a new sensing par-adigm and have thus received massive research interest re-cently. Usually sensor nodes are semi-permanently deployed,since the sensors themselves barely have any moving capac-ity. However, the topology of such networks continuouslychanges over time due to a variety of reasons: For example,a sensor node may wear out due to its very limited batterypower; a new sensor node may be inserted into the network;or the layout of a sensor network may need to be changedin order to improve the quality of the network coverage inresponse to a change in the network requirements, which isaccomplished by changing the placement of current (or in-serting, deleting) sensors in network.

In this paper, we address the problem of how to dynami-cally maintain two important measures on the quality of thecoverage of a sensor network: the best-case coverage distanceand the worst-case coverage distance of the network. We alsoaddress a closely related problem, namely that of finding ashortest maximum support path.

In a sensor network, each sensor bears the ability to detectobjects around it. The coverage of a sensor is limited by itsenergy level. Assuming that a sensor’s detecting ability isomnidirectional, we can model the coverage of a sensor as adisk (under 2-norm on the Euclidean plane1) centered at thesensor. The radii of such disks are determined by the energy

∗ This work was supported in part by NSF CAREER Award CCR-9985284.∗∗ Corresponding author.

1 A disk of radius r centered at (x, y) under 2-norm in R2 is the set of points

(p, q) such that√

(p − x)2 + (q − y)2 � r .

level of the sensors. The coverage area (or simply coverage)of the sensor network is the union of all such disks.

A sensor network is often used to detect intruders. An in-truder may start at a point S, follow an arbitrary trajectory(path) on the plane, and stop at some other point T on theplane. In some applications, a sensor network may need tokeep track of the intruder at all times, as the intruder followsits trajectory; in some other applications, the network’s func-tion may be simply to detect the presence of an intruder, inwhich case the network only needs to cover some part of thetrajectory. Thus, given two points S and T , two relevant typesof trajectories on the plane are proposed [10]: the maximumbreach path and the maximum support path. (In [10], thesepaths are called maximal breach path and maximal supportpath, respectively.)

The maximum breach path measures the vulnerability of asensor network by, as the name suggests, completely avoid-ing the coverage area of the sensor network: It is a trajectorybetween the start point S and the stop point T that stays “asfar away” from the sensors as possible. On the other hand,the maximum support path measures the efficiency of the net-work coverage: This path is a trajectory between S and T

which stays “as close to the sensors” as possible. The distanceof a point P to the sensor network is defined as the smallestEuclidean distance from P to one of the sensor nodes. A max-imum breach path from S to T is a path from S to T such thatthe minimum distance from a point P in the path to the sen-sor network is maximized: this distance is called the worst-case coverage distance of the network. Similarly, a maximumsupport path from S to T is a path such that the maximumdistance of a point P in the path to the sensor network is min-

10 HUANG ET AL.

imized: this distance is called the best-case coverage distanceof the network.

When the topology of a sensor network changes, the qual-ity of its coverage most probably will be affected. We wouldlike to maintain an assessment on the quality of the networkcoverage – which, as explained above, can be done by main-taining the worst-case and best-case coverage distances – ef-ficiently at all times. This would give a clear indication onhow effective the network coverage is at any given point intime, possibly calling for the insertion of new nodes in thenetwork (e.g., when the coverage deteriorates due to nodefailures) or to a network re-design phase. Whenever neces-sary, the actual paths which give the best-case and worst-casecoverage distances can be retrieved. As we will see later, insections 4 and 5, our algorithms for maintaining the worst-case and best-case coverage distances have poly-logarithmicupdate and query costs, as defined later. To the best of ourknowledge, this is the first work which formalizes and ad-dresses this problem in a dynamic scenario.

For a moment, let us assume that all sensors have the sameenergy power and thus all disks have the same radius r . Wecall such a sensor network a uniform sensor network with cov-erage radius r . In a uniform sensor network, all of the pathswhose minimum distance of a point in the path to a sensoris larger than the coverage radius are equivalent, in the sensethat the sensors in the network will not be able to detect an in-truder using any such path. Similarly, all of the paths whosemaximum distance of a point in the path to a sensor is smallerthan the coverage radius are equivalent, in the sense that anysuch path is entirely contained in the coverage area of the net-work. The worst coverage radius (see [10]) is defined to bethe maximum coverage radius r such that there exists a trajec-tory P between given points S and T which does not intersectthe interior region of the area covered by the uniform sensornetwork (i.e., P may “touch” the coverage area, intersectingit at a discrete number of points only). We can think of theworst-coverage radius as being the maximum energy that canbe assigned to the sensor nodes which still would not preventan intruder from escaping from S to T without being detected(for simplicity, we assume that a sensor will not be able todetect an intruder who only touches its coverage area). Cor-respondingly, the best coverage radius (see [10]) is definedto be the minimum coverage radius r such that there existsa trajectory between S and T that is totally covered by theuniform sensor network.

We introduce uniform sensor networks as a merely concep-tual tool in order to facilitate the presentation of our approxi-mation algorithms and their analyses, following a similar ap-proach as Li et al. [9]. (The actual sensor network in consid-eration has nodes with arbitrary energy levels and thereforeis not assumed to be uniform.) In fact, if we think of a uni-form sensor network built on top of the placement of the sen-sor nodes currently deployed in the general sensor network inconsideration, the worst-coverage radius of the uniform net-work is indeed equal to the worst-case coverage distance ofthe general sensor network, and the best-coverage radius isindeed equal to the best-case coverage distance.

In order to dynamically maintain the best- and worst-casecoverage distance efficiently, we need to maintain some infor-mation on the current topology of the sensor network; whenthe network topology changes, we need to update this infor-mation. We also perform queries for the current best-caseand worst-case coverage distances, based on the informationmaintained. Hence, the cost (or running time) of our algo-rithms are measured in terms of their respective update cost– i.e., the cost to update the topology information, which ischarged per “relevant” topology change in the network – andthe query cost, which is the cost incurred when answeringa query for the current best-case or worst-case coverage dis-tance.

In sections 4 and 5, we formally define a “relevant topol-ogy change” – which will henceforth be called an event – forthe problems of maintaining the best-case and worst-case cov-erage distances, respectively.

The remainder of the paper is organized as follows. Sec-tion 1.1 states our results. In section 2, we present some re-lated work in the literature. Section 3 covers some prelim-inaries and sketches the basic framework of our solutions.We present the low constant approximation algorithms for thebest- and worst-case coverage distance in sections 4 and 5 re-spectively. In section 6 we address the closely related problemof efficiently finding a shortest maximum support path. Sec-tion 7 concludes the paper with some possible lines for futurework.

1.1. Our results

In this section, we summarize the main results of this paper.One of the main contributions of this work is to take into ac-count the dynamic nature of sensor networks, and to proposea framework which can be used to continuously monitor thequality of the network coverage. Let n denote the currentnumber of sensors in the network.

In the following sections, we present two algorithms tomaintain low constant approximations on the best-case andworst-case coverage distances. Both algorithms have lowupdate and query costs. Namely, our algorithms achieve a(1 + ε)-approximation on the best-case coverage distance,and a (

√2 + ε)-approximation on the worst-case cover-

age distance, for any fixed ε > 0. The amortized up-date cost per event of the best-case coverage distance algo-rithm is O(log3 n), and the respective query cost is worst-caseO(log n). For the worst-case coverage algorithm, the updatecost per event is worst-case O(log2 n) and the query cost isworst-case O(1). A formal definition of an event for each ofthe problems considered follows in sections 4 and 5, respec-tively.

As a byproduct of our algorithm for maintaining the worst-case coverage distance, we extend the concatenable queuedata structure to also serve as a priority queue. All the opera-tions on this extended data structure have worst-case O(log n)

running time.We also present an O(n log n) algorithm for computing

an exact shortest maximum support path between two given

DYNAMIC COVERAGE IN AD-HOC SENSOR NETWORKS 11

points S and T , improving on the best-known previous resultsby Li et al. [9]. A shortest maximum support path from S

to T is a maximum support path from S to T such that theEuclidean length of the trajectory followed in this path is min-imum. In [9], two algorithms are presented for computing themaximum support path: One algorithm computes an exactshortest maximum support path in O(n2 log n) time; the otheralgorithm provides a 2.5-approximation on the shortest max-imum support path in O(n log n) time. One should note thatthe algorithms presented by Li et al. can be implemented ina distributed fashion (we use the communication complexityas the time bound for the sequential versions of their algo-rithms), whereas the algorithms presented in this paper are allcentralized.

The update costs of our algorithms for approximatelymaintaining the best- and worst-case coverage distances aremuch cheaper than maintaining the best- or worst-case cov-erage distances using the best-known algorithms in the liter-ature prior to this work. In fact, the best previously knownalgorithm for maintaining the best-case (resp., worst-case)coverage distance maintains the exact distance by repeat-edly re-computing the maximum support path (resp., max-imum breach path) using the O(n log n) algorithm by Liet al. [9] (resp., the O(n2 log n) algorithm by Meguerdichianet al. [10]) each time an event occurs. To the best of ourknowledge, this is the first work that explicitly addressesthe problems of dynamically maintaining (approximations of)these two distances.

2. Related work

Meguerdichian et al. [10] considered the problems of findingthe maximum breach path and the maximum support path ona sensor network. They [10] present an O(n2 log �) runtimealgorithm for the maximum breach path problem, where n isthe number of sensors in the sensor network, and � is the dif-ference between the highest and the lowest weight of an edgein the Voronoi Diagram of the sensor network. Their algo-rithm for computing the maximum support path has the samerunning time as their maximum breach path algorithm. TheO(log �) factor can be easily converted into O(log n) in thealgorithm that solves the maximum breach path problem if weperform a binary search over a sorted list of the radii of sen-sors instead of using a linear search as in [10]. The algorithmspresented in [10] heavily rely on geometric structures such asthe Voronoi Diagram and Delaunay triangulation of the net-work, which cannot be constructed efficiently in a distributedmanner.

Li et al. [9] prove the correctness of the algorithms givenin [10]. They also show how to find a maximum supportpath in O(n log n) time using a centralized algorithm, or withO(n log n) communication complexity bits in a distributedfashion. In addition, Li et al. [9] present two algorithms forcomputing a shortest (with respect to the Euclidean length ofthe trajectory followed in this path) maximum support path:an algorithm that computes an exact shortest maximum sup-

port path with O(n2 log n) worst-case communication com-plexity, and an algorithm that computes a 2.5-approximationof a shortest maximum support path (i.e. the total length ofthe obtained path is at most 2.5 times the length of a short-est maximum support path) with O(n log n) communicationcomplexity.

Meguerdichian et al. [11] proposed an exposure-based for-mulation for analyzing the coverage of paths taken by polyg-onal objects: they define a path-dependent “integral”, whichconsists of the trajectories of all the points of a polygonal ob-ject (the polygonal object is able to rotate), and not only ofthe trajectory of the object’s center point.

Recently, Zhang and Hou [13] proved that if the commu-nication rage of a sensor is at least twice its sensing range,a complete coverage of a convex area implies connectivityamong the working set of nodes and derive optimality con-ditions under which a subset of working sensor nodes can bechosen for full coverage. Wang et al. [12] designed a Cov-erage Configuration Protocol (CCP) that can provide differ-ent degrees of connected coverage and present a geometricanalysis of the relationship between coverage and connectiv-ity. Huang and Tseng [8] present an algorithm with runtimeof O(n2 log n) that decides whether every point in a given ser-vice area is covered by at least one sensor.

3. Preliminaries

Before heading into the technical details of our algorithms,we introduce some basic concepts which will be used in bothsections 4 and 5. The first concept we introduce is that ofgrowing disks, which will help us translate our problems intograph connectivity problems.

The growing disks concept was previously proposed in [9].We restate it in terms of the coverage radius of a uniform sen-sor network as defined in section 1. (In section 1, we saw howthe coverage radius of a virtual uniform overlay sensor net-work directly relates to the worst-case and best-case coveragedistances of the actual network.) Assume we have a uniformsensor network with coverage disks centered at the sensors.Define U(r) to be the region on the plane composed of theunion of all of the coverage disks when the coverage radiusis r . Let U(r) be the complement of the region U(r). At thevery beginning, we set the coverage radius to be equal to 0.Then U(r) is the union of discrete singletons. As the cov-erage radius grows, the disks centered at the sensors becomelarger and might get connected into larger regions. Therefore,U(r) might get disconnected into separate regions. For anytwo given points S and T , the best coverage radius is the min-imum r such that S and T are in the same connected region ofU(r), while the worst coverage radius is the minimum r suchthat S and T belong to two disconnected regions in U(r).Hence, the best and worst coverage radius problems translateto connectivity problems on U(r) and U(r), respectively. Fig-ure 1 illustrates these ideas. We will further translate the bestand worst coverage radius problems into graph connectivityproblems.

12 HUANG ET AL.

(a)

(b)

Figure 1. Best and worst coverage radii. (a) Best-coverage radius: mini-mum r such that S and T are connected in U(r). (b) Worst-coverage radius:

minimum r such that S and T are disconnected in U(r).

We first show how to translate the best coverage radiusproblem into a graph connectivity problem. A uniform diskgraph is the intersection graph of disks with uniform radius r

(see [4]). In this graph, disks are vertices and there is an edgebetween two vertices if and only if the corresponding disksintersect.2 The connectivity of U(r) is naturally modeled bythat of a uniform disk graph of radius r , denoted by G(U(r)).The best coverage radius is the minimum r such that the ver-tex corresponding to the disk containing S is connected to thatcorresponding to the disk containing T in G(U(r)).

We also translate the worst coverage radius problem into agraph connectivity problem. However this case is rather moreinvolved and we delay its presentation to section 5.

When r is fixed, suppose that we have a poly-logarithmicrunning time query to check whether the region in either U(r)

or U(r) containing S is connected to that containing T . Thenwe can build an α-approximation algorithm, α > 1, for eitherthe best or the worst coverage radius problem, as we show inthe next paragraph.

For the best coverage radius, consider the sequence ofU(ri), such that ri = αri−1. Let i be such that S and T areconnected in U(ri) but not in U(ri−1). Since the best cover-age radius falls in the interval [ri−1, ri ] and since ri is at mostαri−1, we know that ri is an α-approximation on the best cov-erage radius. A similar argument on the sequence of U(ri)’sgives an α-approximation of the worst coverage radius.

Assume sensors occupy some space and cannot overlap.Then there is a constant lower bound on the coverage radius,denoted by rmin. Due to the limited battery power, we assumethat there is a constant upper bound on the coverage radius,

2 If we rescale one unit to be 2r , then a uniform disk graph is a unit-diskgraph.

denoted by rmax. Let R = rmax/rmin. We need to main-tain logα(R) copies of U(ri) or U(ri). If updating the rele-vant connectivity information for each U(ri) or U(ri) takestime g(n), the overall update time is logα(R) · g(n). The up-date time is poly-logarithmic on n provided that g(n) is poly-logarithmic on n, and that R is bounded by a polynomial on n.

4. Dynamic best-case coverage distance

In this section, we present our (1 + ε)-approximation algo-rithm to maintain the best-case coverage distance followingthe framework presented in section 3. Recall that, as shownin section 3, finding the best-case coverage distance for givenpoints S and T is equivalent to finding the minimum r suchthat S and T are connected in G(U(r)). Thus our main goalis to devise an approach to maintain the connectivity of theuniform disk graph G(U(r)) such that both the update costand the query cost are poly-logarithmic on n, where n is thenumber of sensors in the network.

Holm et al. [7] showed that the connectivity of a graphcan be maintained in amortized poly-logarithmic update cost,whereas each query takes worst-case O(log n/ log log n) time.Guibas et al. [5] used Holm et al.’s algorithm to maintain con-nectivity on a unit-disk graph. The update cost in [5,7] ischarged per edge insertion or deletion. In order to be ableto detect when uniform disks meet or separate on the plane(corresponding to an edge insertion or deletion on a unit-diskgraph, respectively), Guibas et al. [5] introduced a kinetic datastructure specially tailored to handle this scenario.

The kinetic data structure framework was first proposed byBasch et al. [2,3] to deal with dynamics. Their main contri-bution is a method to maintain an invariant of a set of movingobjects in a discrete manner. They introduce the idea of keep-ing certificates as triggers for updates. When an object movesand a certificate fails, the consistency of the kinetic data struc-ture is invalidated and an update is mandatory. Each failure ofa certificate incurs a setup of up to a constant number of newcertificates. Hence we are allowed to monitor the dynamicsof a set of objects discretely and efficiently. The kinetic datastructure requires that we know the flight plan (a specificationof the future motion) [2,5] of all disks, and that the trajectoryof each disk can be described by some low-degree algebraiccurve. We have the freedom to change the flight plan of a diskat any time. Basch [2] shows that kinetic data structures canefficiently support the dynamic operations of inserting anddeleting objects into the system, provided those operationsdo not occur too often. The details of kinetic data structuresare beyond the scope of this paper. Please refer to [2,3,5] formore information.

The kinetic data structure utilized in [5] can be viewed as adiscrete event monitor. The events we need to monitor in or-der to maintain accurate connectivity information on G(U(r))

are when two disks meet or separate. In [5], two types ofcertificates are set up and the data structure allows us to de-termine a priori the time when an event will occur. When anevent occurs, the topology of the uniform disk graph G(U(r))


changes and an update on the connectivity information is trig-gered. Hence the update cost is the cost to update the connec-tivity information of G(U(r)) per event. When a certificatefails and an event occurs, it takes constant time for the kineticdata structure to process the failure (due to the setup of at mosta constant number of new certificates). We do not explicitlytake this cost into account when computing the update cost ofthe maintenance of the connectivity information of G(U(r)),since it would not change the asymptotic bound on the updatecost.

We adapt the main theorem in [5, theorem 5.4], to betterserve our purposes. The uniform disk graph G(U(r)) corre-sponds to a unit-disk graph if we rescale one unit to be equalto 2r .

Lemma 1 (Adapted from [5, theorem 5.4]). In [5], an algo-rithm to dynamically maintain the connectivity of G(U(r)) ispresented. The update cost is amortized O(log2 n) per event.The query cost is worst-case O(log n/ log log n).

We still need to show how to determine which disks con-tain the given points S and T , at any given time. We sort allsensors according to their distances to the fixed point S. Wemaintain a binary heap on this ordering. Once the orderingchanges, we update the heap in O(log n) time. This intro-duces a new type of event – namely, when a sensor changesits location and needs to be re-inserted in this ordering – be-sides the other two events defined earlier. The update costfor this event is O(log n). To check which disk contains S,we find the closest sensor p to S. We check if the distancefrom p to S is larger than the coverage radius. If so, then S isnot contained in any disk. Otherwise, we know that the diskcentered at p contains the point S. This query takes constanttime. We maintain the ordering of the sensors with respectto T in a similar way.

Combining the result in this section with the algorithmicframework presented in section 3, we have our (1 + ε)-approximation algorithm (for any ε > 0) for the best-case coverage distance by maintaining log1+ε R copies ofG(U(r)), for r = 1, (1+ε), (1+ε)2, . . . . We perform a queryoperation by doing a binary search on the log1+ε R copies ofG(U(r)).

Theorem 1. Our algorithm dynamically maintains a (1 + ε)-approximation, for any ε > 0, of the best-case coveragedistance. The update cost of this algorithm is amortizedO(log2 n · log1+ε R) per event and the query cost is worst-case O((log n/ log log n) · log log1+ε R).

Corollary 1. If ε > 0 is fixed, then our algorithm has amor-tized O(log3 n) update cost per event, and worst-case O(log n)

query cost.

5. Dynamic worst-case coverage distance

In this section, we present our (√

2 + ε)-approximationalgorithm, for any ε > 0, to dynamically maintain the

worst-case coverage distance. We first present a (1 + ε)-approximation algorithm (for any ε > 0) for a simplified sen-sor network model, where the coverage disks are consideredunder infinity-norm. Since there is only a

√2 gap between

infinity-norm and 2-norm, a (1 + ε/√

2)-approximation fac-tor for infinity-norm dilates into a (

√2 + ε)-approximation

factor when applied to the 2-norm scenario, for any fixedε > 0. The infinity-norm of a vector v = (x1, . . . , xd) in ad-dimensional space is defined as ‖v‖∞ = max(|x1|,. . . , |xd |). Under infinity-norm, the distance between twopoints on the plane is the maximum of the difference oftheir x coordinates and the difference of their y coordinates.Hence the coverage region of a sensor is square shaped andits boundary is composed of four line segments. As we willsee later, this simple boundary shape allows for an efficientmaintenance scheme.

Recall the solution framework presented in section 3. Thecore of our algorithm is to check, for any two given points S

and T , whether the region in U(r) containing S is connectedto that containing T . If we can maintain some informationsuch that each query on connectivity of regions takes onlypoly-logarithmic time, the cost of update against mobility isalso poly-logarithmic.

In our algorithm, regions in U(r) are represented by theirboundaries. Only one region in U(r) may be infinite in area.We call such an unbounded region the outer face. All of theother (bounded) regions are called inner faces. Since we con-sider the infinity-norm, each disk is represented by a squareon the plane. Thus the boundary of any inner face is a sim-ple cycle composed of a sequence of line segments, while theboundary of the outer face comprises several simple cycles.To differentiate these cycles, we call a cycle that is the bound-

(a)

(b)

Figure 2. Representation in G(U(r)). (a) A square is represented by 8 ver-tices. (b) Dynamics of vertices and edges when squares overlap. Vertices C,C′, and E and E′ are relocated. Edges (C,C′) and (E,E′) are removed, and

edges (C,E′) and (E,C′) are inserted.

14 HUANG ET AL.

Figure 3. Outer- and inner-cycles. When the outer cycle in (a) breaks intotwo cycles, it can either break into an outer and an inner cycle, as shown

in (b); or it can break into two outer cycles, as shown in (c).

ary of an inner face an inner cycle, and a cycle on the bound-ary of the outer face an outer cycle. Figure 3 illustrates someof these concepts. The shaded areas in the figure define U(r),and the unshaded areas define U(r). In (b), U(r) is dividedinto two regions, the unbounded region is the outer face, thebounded region is the inner face. The boundary of the innerface is an inner cycle and that of the outer face is an outercycle. In (c), the boundary of the outer face consists of twodisjoint outer cycles.

Below we describe a method which translates the connec-tivity of regions in U(r) into a graph connectivity problem.The first step is to represent outer cycles and inner cycles bya graph. There are only vertical line segments and horizon-tal line segments in both outer and inner cycles, and thoseline segments only meet at their endpoints. Hence we candraw a graph such that the vertices are the endpoints and theedges are the line segments. We call this graph the connectiv-ity graph G(U(r)). (For convenience, the connectivity graphwill actually be implemented in a slightly different way, as weexplain in section 5.1.)

Every outer or inner cycle is a cycle in the graph and anytwo of them are disjoint, i.e., disconnected in the graph. Thiscoincides with the fact that any two distinct inner faces aredisconnected, and that any inner face is disconnected fromthe outer face.

The connectivity of G(U(r)) is thus analogous to that ofU(r): Two regions are connected in U(r) if and only if theirboundary cycles are connected in the graph, or they are bothpart of the outer face boundary. Thus we could apply the algo-rithm proposed by Holm et al. [7], which dynamically main-tains graph connectivity, to maintain the connectivity of theregions in U(r). The update cost per edge insertion or dele-

tion for each G(U(r)) in [7] is amortized O(log2 n) and thequery cost is O(log n/ log log n), implying an overall amor-tized update cost of O(log3 n) and worst-case query cost ofO(log n), with an approximation factor of (1 + ε), for anyfixed ε > 0. However, G(U(r)) is the union of simple dis-joint cycles, each uniquely defining a region in U(r) and thusit allows for a more efficient update and query scheme, at theexpense of a small degradation in the approximation factor.As we will show later, we can maintain the connectivity of allG(U(r)) in overall worst-case update cost of O(log2 n), withworst-case query cost of O(1), while maintaining a (

√2 + ε)-

approximation on the worst-case coverage distance, for anyfixed ε > 0.

In the remainder of this section, we first describe the dy-namics of the connectivity graph. Then we define three typesof events which mandate updates. Our update cost is chargedper event. Following that, we present a data structure, whichis an extension of concatenable queues [1], to maintain theconnectivity of the graph efficiently. Finally, we present ourmajor result on the worst-case coverage distance.

5.1. Dynamics of cycles

In this section, we first formally define the representation weuse for the connectivity graph G(U(r)). Second, we ad-dress the dynamics of the connectivity graph. And finally,we present an algorithm for maintaining the connectivity in-formation on the regions of U(r).

The boundary of a standalone square is the simplest cyclein G(U(r)). We represent a square by eight vertices and eightedges as shown in figure 2. For every corner X of a square,we introduce two vertices X and X′. Hence we have O(n)

vertices and edges in G(U(r)), where n always denotes thecurrent number of sensors in the network. The extra verticeshelp us to efficiently maintain the graph when squares start tomove and overlap on the plane (including when sensors areadded or removed from the network). In the following, wewill show that the dynamics of sensors will not change theO(n) bound on the number of vertices and edges.

When two squares meet, at most two pairs of line seg-ments of their boundaries intersect. Without loss of gener-ality, suppose a vertical edge B ′C intersects with a horizon-tal edge E′F at a point Z, and the new boundary comprisesedges B ′Z and ZF . Then we simply relocate vertices C andE′ to Z, insert an edge CE′ and remove edges CC′ and EE′from G(U(r)). Figure 2 illustrates this operation. Note thatwe do not introduce any new vertex or remove any old ver-tex. In fact, since G(U(r)) contains no information of thevertex’s location, we do not need to perform any “relocation”of a vertex when we operate on G(U(r)). The cases of a verti-cal edge intersecting with a vertical edge, and of a horizontaledge intersecting with a horizontal edge are analogous, andcan thus be also handled by at most two edge insertions andat most two edge deletions. Since we never change the num-ber of vertices in the graph, and since each vertex has degreeat most 2, the O(n) upper bound on number of vertices andedges in G(U(r)) always hold. The following fact follows:


Fact 1. When two squares meet or separate, up to four edgeinsertions and deletions are needed to update the connectivitygraph G(U(r)).

When the topology of the network changes, cycles inG(U(r)) may also undergo changes. A cycle may break intotwo smaller cycles; or two cycles may merge into a longercycle. Both these operations impose changes on the connec-tivity of G(U(r)). Cycles break or merge only when two sen-sors’ coverage disks meet or separate. Hence we need to de-tect the time when those happen in order to trigger an update.

When a cycle breaks, it could break into an outer cycle andan inner cycle (as shown in figure 3). We need to differentiateouter cycles from inner cycles since all outer cycles definethe same region, namely the outer face. In order to determinewhether a cycle is an outer cycle, one only needs to identifythe topmost edge of the cycle: If the topmost edge of the cycleis the top boundary of a square, then the cycle is an outercycle; otherwise, the topmost edge of a cycle is the bottomboundary of a square, and the cycle is an inner cycle. Hencewe need to maintain the topmost edge of each cycle as sensorsmove. The topmost edge of a cycle may change only whentwo horizontal line segments swap their y position. Thereforewe also need to monitor these line segment swaps.

Recall that the original problem we aim to solve is to checkwhether the region containing a given point S is connected tothat containing T . We need to determine which region con-tains a given point and also to update this information as sen-sors move. As described in section 4, we sort all sensors ac-cording to the distance from the fixed point S and maintain abinary heap on this ordering, with update cost O(log n) on theheap. In order to check which region S belongs to, we needto find the cycle representing the region. Again we find theclosest sensor p to S and check if the distance is smaller thanthe radius of the coverage disk of p. If so, then the point S

does not belong to any region of U(r). Otherwise, we checkthe eight vertices of the square representing the closest sensorto S, find the closest one of these vertices to S, and the cyclecontaining this closest vertex represents the region contain-ing S. This query takes constant time. We maintain a similardata structure for T . Thus we also need to monitor and de-tect the time when two sensors swap their relative position inthese orderings.

We summarize all of the above in the following threetypes of events, which we need to monitor in order to trig-ger mandatory updates, as sensors move on the plane:

(I) Two vertical line segments swap their x position,

(II) Two horizontal line segments swap their y position, and

(III) Two sensor swap their position in the orderings of thesensor’s distance to the given points S and T .

When events (I) or (II) occurs, we can check in constanttime whether two coverage disks meet or separate. If they do,we check whether the event leads to a cycle break or merge,and update the data structure accordingly. When event (II) oc-curs, we can check whether the two horizontal line segments

belong to the same cycle. If so, we may also need to updatethe topmost edge of the cycle. When event (III) occurs, weupdate the orderings with respect to distances to S and T .

We use the kinetic data structure as defined in [3] as ourevent monitor (unlike G(U(r)), G(U(r)) is not a unit-diskgraph and therefore the results in [5] do not apply). Eachevent can be detected and processed in constant time.

In the following, we present our update scheme. We willalso show that the update cost per event is O(log n). We storea cycle as a sequence of consecutive edges. In section 5.2we introduce a data structure which supports the followingoperations on sequences of edges:

INSERT – insert an edge into a sequenceDELETE – delete an edge from a sequenceCONCATENATE – concatenate a sequence to the end of an-

other sequenceSPLIT – split a sequence into two sequencesSWAP – swap the y position of two edgesMAX – return the topmost edge of a sequenceMEMBER – return the representative edge of a sequence

Each of these operations can be executed in worst-caserunning time O(log n), as stated in lemma 4.

The update per type (I) or (II) event is as follows. Whensquares move and the shape of a cycle changes, up to a con-stant number of INSERT and DELETE operations are neededto update the cycle per event. When two edges in a cycle ex-changes their y position, we execute SWAP to update the y

position per event. We can execute MAX to know whether acycle is an outer cycle or not. Recall that a cycle is an outercycle if and only if the topmost edge of the cycle is the topboundary line segment of a square. Cycle merges or breakscan be carried out by a constant number of CONCATENATEand SPLIT operations. Since only a constant number of IN-SERT, DELETE, CONCATENATE, SPLIT, SWAP and MAXoperations are executed per event, the update cost per event isworst-case O(log n). As we have explained earlier, the updatecost per type (III) event is also O(log n).

A data structure that supports the operations above can alsobe used for efficiently performing a connectivity check. As-sume that S and T are not covered by any sensor in U(r)

and therefore both belong to U(r). We can find the closestvertices u and v to points S and T , respectively, in constanttime. Then we check if u and v belong to the same cycle byperforming two MEMBER operations. If so, then S and T

belong to the same region in U(r). Otherwise, we need tocheck whether the closest cycles to S and T are both outercycles by two executions of the MAX operation. If both ofthem are outer cycles, then both S and T belong to the outerface, and hence are in the same region. Otherwise, S and T

belong to two disconnected regions. This procedure can beimplemented in O(log n) time.

We summarize all of the above in lemma 2.

Lemma 2. For any two given points S and T , we maintain adata structure with O(log n) update cost per event such that

16 HUANG ET AL.

the query to check whether the region in U(r) containing S isconnected to that containing T takes O(log n) time.

Combining lemma 2 with the algorithmic framework pre-sented in section 3, we have our (1 + ε)-approximation al-gorithm, for any ε > 0, for the worst-case coverage distanceunder infinity-norm, as stated in the lemma below. If everytime we perform an update operation, we keep track of thesmallest ri such that S and T are disconnected in G(U(ri)),then each query operation can be performed in O(1) time.

Lemma 3. Under infinity-norm, our algorithm dynamicallymaintains a (1 + ε)-approximation of the worst-case cover-age distance for any ε > 0. The update cost is worst-caseO(log n · log1+ε R) per event, and the query cost is worst-case O(1).

Hence the (√

2+ε)-approximation algorithm for the worst-case coverage distance under 2-norm follows:

Theorem 2. Our algorithm dynamically maintains a (√

2+ε)

-approximation of the worst-case coverage distance, for anyε > 0. The update cost is worst-case O(log n · log1+ε/

√2 R)

per event, and the query cost is worst-case O(1).

Corollary 2. If ε > 0 is fixed, then our algorithm has worst-case O(log2 n) update cost per event, and worst-case O(1)

query cost.

5.2. Extended concatenable queue

In this subsection we introduce a data structure that sup-ports the operations INSERT, DELETE, CONCATENATE,SPLIT, SWAP, MAX and MEMBER efficiently. The datastructure is an extension of the concatenable queue data struc-ture [1]. In [1], a concatenable queue is implemented by a2–3 tree (a Red–Black tree would also work, for example),and all the data is stored at the leaf nodes. A concaten-able queue supports the operations INSERT, DELETE, CON-CATENATE, SPLIT and MEMBER, and each operation takestime O(log n) in the worst case. In the following paragraphs,we will show how to also implement the SWAP and MAXoperations on a concatenable queue in O(log n) time.

We associate each edge’s y coordinate to the correspond-ing leaf node in the 2–3 tree. To each internal node t , we asso-ciate the maximum y coordinate of a leaf node in the subtreerooted at t . This is done by comparing all the y coordinatesassociated to t’s children in the tree, taking constant time perinternal node. When the y coordinate of an edge changes, anda SWAP operation is invoked, it takes at most O(log n) time toclimb up the tree and update all the internal nodes on the wayup. Starting from any given edge on a cycle, it takes O(log n)

time to reach the root of the 2–3 tree where we can find thetopmost edge of the cycle. Hence the O(log n) running timeof MAX follows.

We need also to justify that the above modification doesnot increase the running time of all other operations. Per each

INSERT or DELETE, it takes an additional O(log n) time toupdate the y coordinate of all internal nodes due to the edgeinsertion or deletion. Both CONCATENATE and SPLIT areimplemented by up to O(log n) joins or breaks of trees at theroot node. Since updating the y coordinate at the root nodetakes constant time (by comparing all the children of the root),we incur at most an additional O(log n) time per CONCATE-NATE or SPLIT. Thus the asymptotic running time of IN-SERT, DELETE, CONCATENATE, and SPLIT remains un-changed. The running time of MEMBER is not affected bySWAP or MAX operations.

Lemma 4. The extension of the concatenable queue datastructure supports the operations of INSERT, DELETE,CONCATENATE, SPLIT, SWAP, MAX and MEMBER.Each operation has worst-case running time of O(log n).

6. Exact shortest maximum support path

We consider the problem of finding a maximum support pathbetween S and T such that the Euclidean length of the tra-jectory followed by this path is minimum. Below we presentan O(n log n) runtime solution, thus improving on the best-known previous results by Li et al. [9]. One should note thatthe algorithms presented in [9] can be implemented in a dis-tributed fashion, whereas the algorithm we present in this sec-tion is intrinsically centralized.

We proceed as follows. First we compute the best coverageradius rbest using the algorithm of Li et al. [9] in O(n log n)

time. Next, we obtain a collection of uniform disks by settingthe radius of each sensor to be rbest. Let U denote the unionof all these uniform disks. Define the complement region ofthe union C = R

2 \ U .The problem of finding a shortest maximum support path

is equivalent to the problem of finding a shortest S, T -pathin R

2 avoiding C, since we are seeking for a maximum sup-port path and rbest is the best coverage radius. (Since rbest isthe best coverage radius, any maximum support path is con-tained in U ; in fact any path from S to T in U is a maximumsupport path.) A shortest maximum support path can onlycontain straight line segments as edges, otherwise the pathwould not be shortest. Therefore, we can replace each arc inC by a straight line segment. In such fashion we obtain a newset of obstacles C′ as a collection of polygonal objects withpossible “holes” that have a total O(n) number of vertices.We can remove these “holes” and obtain slightly larger num-ber of disjoint polygonal objects by cutting the existing ob-jects with segments that connect the vertices of the holes andthe external boundary in an arbitrary fashion. Note that thenumber of total vertices of the disjoint polygonal objects hasnot changed, i.e., it is still O(n). Thus, our problem translatesto that of finding a shortest path in R

2 that avoids the polyg-onal obstacles in C′. The idea now is to use an algorithm byHershberger and Suri [6], which finds a shortest path betweenS and T on the Euclidean plane that avoids polygonal obsta-cles in O(n log n) time. Hence the total running time of ouralgorithm is O(n log n).


As described in section 2, two algorithms are presentedfor computing the maximum support path by Li et al. in [9]:One algorithm computes an exact shortest maximum sup-port path in O(n2 log n) time; the other algorithm provides a2.5-approximation on the shortest maximum support path inO(n log n) time. Our algorithm improves on the running timeof the former and on the approximation factor of the latter al-gorithm. One should note, however, that the algorithms pre-sented by Li et al. can be implemented in a distributed fashion(we use the communication complexity as the time bound forthe sequential versions of their algorithms), whereas our al-gorithm is centralized in nature.

7. Future work

In this paper, we present poly-logarithmic dynamic algo-rithms to maintain approximations of two relevant measures– namely, the best- and worst-case coverage distances – of thequality of the network coverage in wireless sensor networks.An interesting open question is whether we can maintain ex-act best-case and worst-case coverage distances for Euclideanmetric with poly-logarithmic update time.

Acknowledgement

We express our thanks to Micha Sharir for his comments.

References

[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis ofComputer Algorithms (Addison-Wesley, Reading, MA, 1974).

[2] J. Basch, Kinetic data structures, Ph.D. dissertation, Stanford Univer-sity (1999).

[3] J. Basch, L.J. Guibas and J. Hershberger, Data structures for mobiledata, in: Proc. of 8th ACM–SIAM Symposium on Discrete Algorithms(1997) pp. 747–756.

[4] B.N. Clark and C.J. Colbourn, Unit disk graphs, Discrete Math. 86(1990) 165–177.

[5] L.J. Guibas, J. Hershberger, S. Suri and L. Zhang, Kinetic connectivityfor unit disks, Discrete Comput. Geom. 25 (2001) 591–610.

[6] J. Hershberger and S. Suri, An optimal algorithm for Euclidean shortestpaths in the plane, SIAM J. Comput. 28(6) (1999) 2215–2256.

[7] J. Holm, K. de Lichtenberg and M. Thorup, Poly-logarithmic deter-ministic fully-dynamic graph algorithms I: Connectivity and minimumspanning tree, Technical Report DIKU-TR-97/17, Department of Com-puter Science, University of Copenhagen (1997).

[8] C.-F. Huang and Y.-C. Tseng, The coverage problem in a wireless sen-sor networks, in: Proc. of the 2nd ACM Internat. Conf. on WirelessSensor Networks and Applications (2003) pp. 115–121.

[9] X.-Y. Li, P.-J. Wan and O. Frieder, Coverage in wireless ad-hoc sensornetworks, IEEE Trans. Comput. 52 (2003) 1–11.

[10] S. Meguerdichian, F. Koushanfar, M. Potkonjak and M.B. Srivastava,Coverage problems in wireless ad-hoc sensor networks, in: Proc. of the20th IEEE INFOCOM (2001) pp. 1380–1387.

[11] S. Meguerdichian, F. Koushanfar, G. Qu and M. Potkonjak, Exposure inwireless ad-hoc sensor networks, in: Proc. of the 7th ACM MOBICOM(2001) pp. 139–150.

[12] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless and C.D. Gill, Integratedcoverage and connectivity configuration in wireless sensor networks,in: Proc. of the 1st ACM Conf. on Embedded Networked Sensor Systems(2003).

[13] H. Zhang and J.C. Hou, Maintaining sensing coverage and connectivityin large sensor networks, Technical Report UIUCDCS-R-2003-2351,UIUC (2003).

Hai Huang is a Ph.D. student in the Department ofComputer Science and Engineering at Arizona StateUniversity. He has received a M.S. in the samedepartment under the supervision of Prof. AndreaW. Richa in 2003. He received a B.A. and a M.A.in the Mathematics Department at Tsinghua Univer-sity, P. R. China, in 1996 and 1999, respectively. Hiscurrent research work focus on clustering and rout-ing problems in mobile ad-hoc networks, and on thechordal graph completion problem with applications

to scientific computing.E-mail: [email protected]

Andréa W. Richa joined the Department of Com-puter Science and Engineering at Arizona State Uni-versity in 1998, where she is now an Associate Pro-fessor. She received her M.S. and Ph.D. degreesfrom the School of Computer Science at CarnegieMellon University, in 1995 and 1998, respectively.She also earned an M.S. degree in computer systemsfrom the Graduate School in Engineering (COPPE),and a B.S. degree in computer science, both at theFederal University of Rio de Janeiro, Brazil, in 1992

and 1990, respectively. Prof. Richa’s main area of research is in networkalgorithms. Some of the topics Dr. Richa has worked on include packetscheduling, distributed load balancing, packet routing, mobile network clus-tering and routing protocols, and distributed data tracking. Prof. Richa’s datatracking (or lookup) algorithm has been widely recognized as the first bench-mark algorithm for the development of distributed databases in peer-to-peernetworking, having received over 55 academic journal or conference publi-cations, and being implemented as part of two of the current leading pojectsin peer-to-peer networking. Dr. Richa’s was the recipient of an NSF CA-REER Award in 1999. For a selected list of her publications, CV, and currentresearch projects, please visit http://www.public.asu.edu/∼aricha.E-mail: [email protected]

Michael Segal was born at October 12, 1972 inUSSR. In 1991 he immigrated to Israel and startedto study computer science in Ben-Gurion Universityof the Negev. He finished his B.Sc., M.Sc. and Ph.D.degrees in 1994, 1997, and 1999, respectively. Dur-ing a period of 1999–2000 Dr. Michael Segal held aMITACS National Centre of Excellence PostdoctoralFellow position in University of British Columbia,Canada. Dr. Segal joined the Department of Com-munication Systems Engineering, Ben-Gurion Uni-

versity, Israel in 2002 where he holds now a position of senior lecturer andserves as department’s Deputy Chairman. His primary research is algorithms(sequential and distributed), data structures with applications to optimizationproblems, mobile wireless networks, communications and security.E-mail: [email protected]


Algorithmic Aspects of Topology Control Problems for Ad HocNetworks ∗

ERROL L. LLOYD ∗∗ and RUI LIU∗∗,Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716, USA

MADHAV V. MARATHE ∗∗∗Los Alamos National Laboratory, MS M997, P.O. Box 1663, Los Alamos, NM 87545, USA

RAM RAMANATHANInternetwork Research Department, BBN Technologies, Cambridge, MA, USA

S.S. RAVI ∗∗∗∗Department of Computer Science, University at Albany – SUNY, Albany, NY 12222, USA

Abstract. Topology control problems are concerned with the assignment of power values to the nodes of an ad hoc network so that the powerassignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimizationobjectives, including minimizing the maximum power and minimizing the total power. A general approach leading to a polynomial algorithmis presented for minimizing maximum power for a class of graph properties called monotone properties. The difficulty of generalizing theapproach to properties that are not monotone is discussed. Problems involving the minimization of total power are known to be NP-completeeven for simple graph properties. A general approach that leads to an approximation algorithm for minimizing the total power for somemonotone properties is presented. Using this approach, a new approximation algorithm for the problem of minimizing the total power forobtaining a 2-node-connected graph is developed. It is shown that this algorithm provides a constant performance guarantee. Experimentalresults from an implementation of the approximation algorithm are also presented.

Keywords: power control, approximation algorithms, topology

1. Introduction

1.1. Motivation

An ad hoc network consists of a collection of transceivers.All communication among these transceivers is based on ra-dio propagation. For each ordered pair (u, v) of transceivers,there is a transmission power threshold, denoted by p(u, v),with the following significance: A signal transmitted by thetransceiver u can be received by v only when the transmis-sion power of u is at least p(u, v). The transmission powerthreshold for a pair of transceivers depends on a number offactors including the distance between the transceivers, an-tenna gains at the sender and receiver, interference, noise,etc. [23].

∗ A preliminary version of this paper appeared in Proc. of the Third ACMInternational Symposium on Mobile Ad Hoc Networking and Comput-ing (MobiHoc 2002), Lusanne, Switzerland, June 2002, pp. 123–134.

∗∗ Prepared through collaborative participation in the Communicationsand Networks Consortium sponsored by the U.S. Army Research Lab-oratory under the Collaborative Technology Alliance Program, Coop-erative Agreement DAAD19-01-2-0011. The U.S. Government is au-thorized to reproduce and distribute reprints for Government purposesnot withstanding any copyright notation thereon.

∗∗∗ Research supported by the Department of Energy under ContractW-7405-ENG-36.

∗∗∗∗ Supported by NSF Grant CCR-97-34936.

Given the transmission powers of the transceivers, anad hoc network can be represented by a directed graph. Thenodes of this directed graph are in one-to-one correspondencewith the transceivers. A directed edge (u, v) is in this graphif and only if the transmission power of u is at least the trans-mission power threshold p(u, v).

The main goal of topology control is to assign transmissionpowers to transceivers so that the resulting directed graph sat-isfies some specified properties. Since the battery power ofeach transceiver is an expensive resource, it is important toachieve the goal while minimizing a given function of thetransmission powers assigned to the transceivers. Examplesof desirable graph properties are connectivity, small diameter,etc. Examples of minimization objectives considered in theliterature are the maximum power assigned to a transceiverand the total power of all transceivers (the latter objectiveis equivalent to minimizing the average power assigned to atransceiver).

As stated above, the primary motivation for studying topol-ogy control problems is to make efficient use of availablepower at each node. In addition, using a minimum amountof power at each node to achieve a given task is also likely todecrease the MAC layer interference between adjacent radios.We refer the reader to [20,22,23,26,28,31] for a thorough dis-cussion of the power control issues in ad hoc networks.

20 LLOYD ET AL.

1.2. Formulation of topology control problems

Topology control problems have been studied under twograph models. The discussion above corresponds to the di-rected graph model studied in [23]. The undirected graphmodel proposed in [16] represents the ad hoc network as anundirected graph in the following manner. First, the directedgraph model for the network is constructed. Then, for anypair of nodes u and v, whenever both the directed edges (u, v)

and (v, u) are present, this pair of directed edges is replacedby a single undirected edge {u, v}. All of the remaining di-rected edges are deleted. Under this model, the goal of atopology control problem is to assign transmission powers tonodes such that the resulting undirected graph has a specifiedproperty and a specified function of the powers assigned tonodes is minimized. Note that the directed graph model al-lows two-way communication between some pairs of nodesand one-way communication between other pairs of nodes.In contrast, every edge in the undirected graph model corre-sponds to a two-way communication.

In general, a topology control problem can be specifiedby a triple of the form 〈M, P, O〉. In such a specification,M ∈ {DIR, UNDIR} represents the graph model, P repre-sents the desired graph property and O represents the mini-mization objective. For the problems considered in this pa-per O ∈ {MAXP, TOTALP} (abbreviations of Max Power andTotal Power). For example, consider the 〈DIR, STRONGLY

CONNECTED, MAXP〉 problem. Here, powers must be as-signed to transceivers so that the resulting directed graph isstrongly connected and the maximum power assigned to atransceiver is minimized. Similarly, the 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉 problem seeks to assign powers tothe transceivers so that the resulting undirected graph has anode connectivity (see below for definition) of (at least) 2 andthe sum of the powers assigned to all transceivers is mini-mized.

2. Additional definitions

This section collects together the definitions of some graphtheoretic and algorithmic terms used throughout this paper.

Given an undirected graph G(V,E), an edge subgraphG′(V ,E′) of G has all of the nodes of G and the edge set E′is a subset of E. Further, if G is an edge weighted graph, thenthe weight of each edge in G′ is the same as it is in G.

The node (edge) connectivity of an undirected graph is thesmallest number of nodes (edges) that must be deleted fromthe graph so that the resulting graph is disconnected. Forexample, a tree has node and edge connectivities equal to 1while a simple cycle has node and edge connectivities equalto 2. When the node (edge) connectivity of a graph is greaterthan or equal to k, the graph is said to be k-node connected(k-edge connected). Given an undirected graph, polynomialalgorithms are known for finding its node and edge connec-tivities [30].

The main results of this paper use the following definition.

Definition 2.1. A property P of the (directed or undirected)graph associated with an ad hoc network is monotone if theproperty continues to hold even when the powers assigned tosome nodes are increased while the powers assigned to theother nodes remain unchanged.

Example. For any k � 1, the property k-NODE CONNECTED

for undirected graphs is monotone since increasing the pow-ers of some nodes while keeping the powers of other nodesunchanged may only add edges to the graph. However, prop-erties such as ACYCLIC or BIPARTITE are not monotone.

Some of the topology control problems considered in thispaper are NP-complete. For such problems, we study approx-imation algorithms. In this context, an approximation algo-rithm provides a performance guarantee of ρ if for every in-stance of the problem, the solution produced by the approxi-mation algorithm is within the multiplicative factor of ρ of theoptimal solution. A polynomial time approximation scheme(PTAS) is an approximation algorithm that, given a probleminstance and an accuracy requirement ε, produces a solutionthat is within a factor 1 + ε of the optimal solution.

3. Previous work and summary of results

3.1. Previous work

The form of topology control problems considered in thispaper was proposed by Ramanathan and Rosales-Hain [23].They presented efficient algorithms for two topology controlproblems, namely 〈UNDIR, 1-NODE CONNECTED, MAXP〉and 〈UNDIR, 2-NODE CONNECTED, MAXP〉. After deter-mining the minimum value for the objective, their algorithmsalso reduce the power assigned to each transceiver such thateach power level is minimal while maintaining the desiredgraph property. In addition, they presented efficient distrib-uted heuristics for these problems.

Several groups of researchers have studied the 〈DIR,

STRONGLY CONNECTED, TOTALP〉 problem [5,7,8,16].However, it is not difficult to see that their NP-hardness re-sults as well as approximation algorithms also hold for the〈UNDIR, 1-NODE CONNECTED, TOTALP〉 problem. The pa-per [5] proves that the problem is NP-hard and presentsan approximation algorithm with a performance guaranteeof 2. The other references consider a geometric version ofthe problem along with a symmetry assumption concerningtransmission power thresholds. More precisely, these refer-ences assume the following: (a) Each transceiver is locatedat some point of d-dimensional Euclidean space. (b) Forany pair of transceivers u and v, p(u, v) = p(v, u) =the Euclidean distance between the locations of u and v.For a justification of this model, see [16]. They showthat the 〈DIR, STRONGLY CONNECTED, TOTALP〉 problemis NP-hard when transceivers are located in 3-dimensionalspace. They also present an approximation algorithm witha performance guarantee of 2 for the problem in any met-ric space. In addition, they provide some results for the

ALGORITHMIC ASPECTS OF TOPOLOGY CONTROL PROBLEMS 21

1-dimensional version of the 〈DIR, STRONGLY CONNECTED,

TOTALP〉 problem where there is an additional constraint onthe diameter of the resulting undirected graph. Clementiet al. [7] show that the 2-dimensional version of the〈DIR, STRONGLYCONNECTED, TOTALP〉 problem remainsNP-hard. They also show that the 2-dimensional versionwith a diameter constraint can be efficiently approximated towithin some constant factor and that the 3-dimensional ver-sion does not have a polynomial time approximation scheme.Under a slightly different model, where there is an explicit re-lationship between the transmission power and distance, ref-erences [2,3] study topology control problems for connectiv-ity properties. The complexity of several problems under thismodel is established in [2]. A (1 + ln 2)-approximation algo-rithm for the problem is presented in [3]. The approximationratio is improved to 5/3 in a journal submission based on [3].

Additional related work may be found in [13,18–20,22,31].

3.2. Summary of main results

Throughout this paper, it is assumed that the power thresholdvalues are symmetric. The main results of this paper are thefollowing.

1. We show that for any monotone graph property P thatcan be tested in polynomial time for undirected (directed)graphs, the problem 〈UNDIR, P, MAXP〉 (〈DIR, P, MAXP〉)can be solved in polynomial time. This generalizessome of the results in [23] where efficient algorithmswere presented for two monotone properties, namely,1-NODE CONNECTED and 2-NODE CONNECTED. Ourpolynomial time algorithm can also be extended to graphproperties specified by proper functions1 [12].

2. We establish that there are nonmonotone and efficientlytestable properties (e.g., GRAPH IS A TREE) for whicheven determining whether there is a power assignmentthat can induce a graph with the specified property isNP-complete. This result shows that, in general, if themonotonicity condition is eliminated, then obtaining anefficient algorithm for minimizing maximum power maynot be possible.

3. As mentioned above, for any monotone and efficientlytestable property P, a solution that minimizes the maxi-mum power can be obtained in polynomial time. How-ever, if we introduce the additional requirement that thenumber of nodes that use the maximum power must alsobe minimized, we show that there are monotone propertiesfor which the resulting problem is NP-complete.

4. We present a general approach for developing approxima-tion algorithms for NP-hard topology control problems un-der the TOTAL POWER minimization objective. The ap-proximation results of [5,6,16] are special cases of this

1 Given a graph G(V,E), a function is f : 2V → {0, 1} is proper if it satisfiesthe following two conditions: (1) f (S) = f (V − S) for all S ⊆ V ; and(2) If A ∩ B = ∅, then f (A) = f (B) = 0 implies f (A ∪ B) = 0.

general approach. As an illustration of our general ap-proach, we present a constant factor approximation algo-rithm for the 〈UNDIR, 2-NODE CONNECTED, TOTALP〉problem. No approximation algorithm was previouslyknown for this problem. In analyzing this approxi-mation algorithm, we use some properties of critically2-node connected graphs [10,21,32]. By a minor modi-fication to this approximation algorithm, we also obtaina constant factor approximation algorithm for producing2-edge-connected graphs. As in the case of minimizingmaximum power, our general heuristic for approximatingtotal power is also applicable to graph properties specifiedby proper functions.

5. Finally, we present experimental results obtained froman implementation of the above approximation algorithmand compare its performance with an algorithm discussedin [23].

4. Results for minimizing maximum power

In this section, we present our results for the MAX POWER

objective. We begin with a general algorithm for the topologycontrol problem where the graph property is both monotoneand polynomial time testable. For a problem with n trans-ceivers, the algorithm uses O(log n) invocations of the algo-rithm to test the graph property. We also present a polynomialtime approximation scheme which can, under certain circum-stances, substantially reduce the number of invocations of theproperty testing algorithm. Next, we give an example of anonmonotone property for which the problem of minimizingthe maximum power is NP-hard. Finally, we show that the ad-ditional requirement of minimizing the number of nodes thatuse the maximum power also renders the problem NP-hard,even for certain monotone properties. Note that both of theNP-hardness results utilize arbitrary power thresholds. Thecomplexity of the problems in the geometric model (i.e., thepower threshold is a function of the Euclidean distance) re-mains open.

4.1. An algorithm for monotone and efficiently testableproperties

We begin with a simple lemma that points out the usefulnessof monotonicity.

Lemma 4.1. For any instance of 〈UNDIR, P, MAXP〉 and〈DIR, P, MAXP〉 where the graph property P is monotone,there is an optimal solution in which all of the nodes are as-signed the same power value.

Proof. Consider an optimal solution to the given instancewhere the nodes do not necessarily have the same power val-ues. Let Q denote the maximum power assigned to any node.Since the graph property is monotone, for any node whosepower value is less than Q, we can increase it to Q withoutdestroying the property. �

22 LLOYD ET AL.

Theorem 4.1. For any monotone and polynomial time tes-table graph property P, the problems 〈UNDIR, P, MAXP〉 and〈DIR, P, MAXP〉 can be solved in polynomial time.

Proof. We will present the proof for 〈DIR, P, MAXP〉. (Theproof for 〈UNDIR, P, MAXP〉 is virtually identical.)

Consider an instance of 〈DIR, P, MAXP〉. By lemma 4.1,there is an optimal solution in which every transceiver is as-signed the same power value. We can estimate the number ofcandidate optimal power values as follows. Let T denote theset of all transceivers in the system and let |T | = n. Considerany transceiver u ∈ T . The number of different power valuesthat need to be considered for u is at most n−1, since at mostone new power value is needed for each transceiver in T −{u}.Therefore, for all of the n transceivers, the total number ofcandidate power values to be considered is n(n−1) = O(n2).

For each candidate power value, the corresponding di-rected graph can be constructed in O(n2) time. Let FP(n)

denote the time needed to test whether property P holds fora directed graph with n nodes. Thus, the time needed totest whether property P holds for each candidate solutionvalue is O(n2 + FP(n)). An optimal solution can be ob-tained by sorting the O(n2) candidate solution values and us-ing binary search to determine the smallest value for whichproperty P holds. Since the number of candidate solu-tion values is O(n2), the time taken by the sorting step isO(n2 log n). The binary search would try O(log n) candidatesolution values and the time spent for testing each candidateis O(n2 + FP(n)). Thus, the total running time of this algo-rithm is O((n2 + FP(n)) log n). Since FP(n) is a polynomial,the algorithm runs in polynomial time. �

As an illustration of the above theorem, let P denote theproperty 2-NODE CONNECTED for undirected graphs. It isknown that this property can be tested in O(n2) time for agraph with n nodes [30]. For this property, the general al-gorithm outlined in the proof of theorem 4.1 yields an algo-rithm with a running time of O(n2 log n). This running timematches the time of the algorithm given in [23]. However, itshould be noted that the algorithm in [23] not only finds an op-timal solution but also reduces the power of each transceiverso that the power levels are minimal. There is no increase intheir asymptotic running time.

Instead of requiring the entire graph to be connected, onemay require connectivity only for a specified subset of thenodes. Such a requirement arises in the context of multicas-ting (see, for example, [27]), where the subset of nodes in-cludes the sender and all the intended receivers. Connect-edness of a specified subset of nodes can be seen to be amonotone property. Thus, the general approach presentedabove leads to a polynomial time algorithm for this prop-erty as well. In fact, the result extends to large class ofnetwork design problems that can be specified using properfunctions [1,12]. As noted in [12], the Steiner tree problemand the Steiner forest problem can be specified using this for-malism. Given a network and a proper function specifica-tion, it is easy to test in polynomial time if the network sat-

isfies the given proper function. Moreover, it is easy to seethat any graph property specified using a proper function isa monotone property. Thus, our results apply to this class ofnetwork design problems as well.

We now present a polynomial time approximation schemefor 〈UNDIR, P, MAXP〉 and 〈DIR, P, MAXP〉 problems. As acompensation for the slight deviation from the optimal value,this approach has the potential to reduce the running time sub-stantially.

Theorem 4.2. Let P be a monotone graph property that canbe tested for an n-node graph in time FP(n). For any fixedε > 0, the problems 〈UNDIR, P, MAXP〉 and 〈DIR, P, MAXP〉can be approximated to within the factor 1 + ε in O((n2 +FP(n)) log log(max / min)) time, where max and min are re-spectively the maximum and minimum power threshold val-ues in the given problem instance.

Proof. We will present the proof for 〈DIR, P, MAXP〉.Since the number of power threshold values is O(n2), the val-ues of min and max can be found in O(n2) time. Note thatfor any candidate power value (which is assigned to all thenodes), testing whether P holds for the induced graph can bedone in O(n2 + P(n)) time.

Let k be the smallest integer such that (1+ε)k min � max.Thus, k = O(log(max / min)). Consider the following set ofk + 1 power values: {min, (1 + ε) min, (1 + ε)2 min, . . . ,

(1 + ε)(k−1) min, max}. By doing a binary search on thisset, we can determine the smallest integer j such thatthe power value (1 + ε)j min causes the induced graph tohave the property P. The binary search uses O(log k) =O(log log(max / min)) calls to the algorithm for testing P.Thus, the running time of the algorithm is O((n2 +FP(n)) loglog(max / min)).

Further, since j is the smallest value for which the powervalue (1 + ε)j min causes the induced graph to have the prop-erty P, the optimal value must be at least (1 + ε)(j−1) min.Thus, the solution found by the algorithm is within a factor(1 + ε) of the optimal value. �

When the ratio max / min is substantially smaller than 2n,the above approximation scheme reduces the number of callsto the property testing algorithm to a value that is asymptoti-cally smaller than O(log n).

4.2. Difficulty of generalizing to nonmonotone properties

We now show that there is a natural nonmonotone graphproperty for which the problem of minimizing the maximumpower is NP-hard. As mentioned earlier, this result points outthat if the monotonicity requirement is omitted, then an effi-cient algorithm for minimizing maximum power may not bepossible.

The property that we use for this purpose is “G IS A

TREE”. Surprisingly, we show that this property makes thetopology control problem NP-complete even without anyminimization objective. The proof of lemma 4.2 utilizes a


reduction from Exact Cover by 3-Sets (X3C), which is knownto be NP-complete [11].

Lemma 4.2. To determine whether there is a power assign-ment such that the resulting undirected graph G is a tree isNP-complete.

Proof. See appendix A. �

Theorem 4.3. There is a nonmonotone property P for which〈UNDIR, P, MAXP〉 is NP-hard.

Proof. Let P denote the property “G IS A TREE”.The NP-hardness of 〈UNDIR, P, MAXP〉 follows fromlemma 4.2. �

4.3. Difficulty of minimizing the number of nodes ofmaximum power

An extension of 〈UNDIR, P, MAXP〉 for monotone graphproperties is explored in this section. While such problemscan be solved efficiently, our algorithm in section 4.1 assignsthe maximum power value to all of the nodes. From a prac-tical point of view, it is important to reduce the number ofnodes with maximum power without affecting the requiredproperty. In this section, we show that this additional require-ment renders the problem NP-hard even for certain monotonegraph properties. A formal statement of the decision versionof the problem is as follows.

Max-power users.

• Instance: A positive integer M , a positive number P (max-imum allowable power value), a node set V , a powerthreshold value p(u, v) for each pair (u, v) of transceiversand a graph property P.

• Question: Is there a power assignment where the powerassigned to each node is at most P and the number of thenodes that are assigned power P is at most M , such thatthe resulting undirected graph G satisfies P?

Theorem 4.4. There is a monotone and polynomial timetestable property P for which the problem Max-power usersis NP-complete.

Proof. See appendix B. �

5. A general approach for minimizing total power

5.1. Approximating minimum total power

Topology control problems in which the minimization objec-tive is total power tend to be computationally intractable. Forexample, the problem is NP-hard even for the (simple) prop-erty 1-NODE-CONNECTED [16]. A common way of copingwith such problems is to develop polynomial time approxima-tion algorithms for them. In this section, we present a general

outline for such an approximation algorithm for topology con-trol problems of the form 〈UNDIR, P, TOTALP〉. We observethat this general outline encompasses the approximation al-gorithm for 〈UNDIR, 1-NODE CONNECTED, TOTALP〉 pre-sented in [16]. Based on the general outline, we also developan approximation algorithm with a constant performanceguarantee for 〈UNDIR, 2-NODE CONNECTED, TOTALP〉.A slight modification of this approximation algorithm yieldsan approximation algorithm for the problem of obtaining a2-edge-connected graph while minimizing total power.

In presenting our general scheme, we assume (as done insection 4.1) that the property P to be satisfied by the graph ismonotone and that it can be tested in polynomial time. Wealso assume symmetric power thresholds as in [7,8,16]; thatis, for any pair of transceivers u and v, the power thresholdsp(u, v) and p(v, u) are equal.

An outline for our general approximation algorithm (calledHeuristic GEN-TOTAL-POWER) is shown in figure 1. Notethat steps 1 and 3 of the outline can be implemented in poly-nomial time. The time complexity of step 2 depends cru-cially on the property P. For some properties such as 1-NODE CONNECTED, step 2 can be done in polynomial time.For other properties such as 2-NODE CONNECTED, step 2cannot be done in polynomial time, unless P = NP [11].In such cases, an efficient algorithm that produces an ap-proximately minimum solution can be used in step 2. Thefollowing theorem proves the correctness of the general ap-proach and establishes its performance guarantee as a func-tion of some parameters that depend on the property P andthe approximation algorithm used in step 2 of the general out-line.

Theorem 5.1. Let I be an instance of 〈UNDIR, P, TOTALP〉where P is a monotone property. Let OPT(I) and GTP(I)

denote respectively the total power assigned to the nodes inan optimal solution and in a solution produced by HeuristicGEN-TOTAL-POWER for the instance I .

(i) The graph G′′ resulting from the power assignment pro-duced by the heuristic (i.e. step 3) satisfies property P.

(ii) Consider the complete graph Gc(V ,Ec) constructed instep 1 of the heuristic. Let H(V,EH) be an edge sub-graph of Gc with minimum total edge weight satisfyingproperty P and let W(H) denote the total edge weightof H . Let step 2 of the heuristic produce an edge sub-graph G′(V ,E′) of G with total edge weight W(G′).Suppose there are quantities α > 0 and β > 0 such that(a) W(H) � αOPT(I) and (b) W(G′) � βW(H). Then,GTP(I) � 2αβOPT(I). That is, Heuristic GEN-TOTAL-POWER provides a performance guarantee of 2αβ.

Before proceeding to the proof of this result, we illus-trate its use by discussing how the 2-approximation algorithmpresented in [16] for the 〈UNDIR, 1-NODE CONNECTED,

TOTALP〉 problem can be derived from the above general out-line. In step 2 they use an efficient algorithm for constructinga minimum spanning tree of Gc. They also show that the total

24 LLOYD ET AL.

power assigned by any optimal solution is at least the weightof a minimum spanning tree of Gc. Thus, using the notationof theorem 5.1, α = β = 1 for their approximation algo-rithm. Since 1-NODE-CONNECTED is a monotone property,it follows from theorem 5.1 that the performance guarantee oftheir algorithm is 2.

Proof of theorem 5.1. Part (i). The edge subgraph G′(V ,E′)constructed in step 2 of the heuristic satisfies property P. Weshow that every edge in E′ is also in the subgraph G′′ inducedby the power assignment produced in step 3. Then, even if G′′has other edges, the monotonicity of P allows us to concludethat G′′ satisfies P.

Consider an edge {u, v} with weight p(u, v) in E′. Recallthat p(u, v) is the minimum power threshold for the existenceof edge {u, v} and that the power thresholds are symmetric.Since step 3 assigns to each node the maximum of the weightsof edges incident on that node, we have π(u) � p(u, v) andπ(v) � p(u, v). Therefore, the graph G′′ induced by thepower assignment also contains the edge {u, v} and this com-pletes the proof of part (i).

Part (ii). By conditions (a) and (b) in the statement of thetheorem, we have W(G′) � αβOPT(I). We observe thatGTP(I) � 2 W(G′). This is because in step 3 of the heuris-tic, the weight of any edge is assigned to at most two nodes(namely, the endpoints of the edge). Combining the two in-equalities, we get GTP(I) � 2αβOPT(I), and this completesthe proof of theorem 5.1. �

5.2. New approximation algorithms

This section presents two new approximation algorithms de-rived from the general approach outlined in figure 1. Thesealgorithms are for the two monotone properties 2-NODE

CONNECTED and 2-EDGE CONNECTED respectively. Thecorresponding problems are denoted by 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉 and 〈UNDIR, 2-EDGE CONNECTED,

TOTALP〉.

5.2.1. An approximation algorithm for 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉This section presents an approximation algorithm for the〈UNDIR, 2-NODE CONNECTED, TOTALP〉 problem. TheNP-hardness of this problem is established in [4]. Ouralgorithm is derived from the general approach outlined infigure 1. The following notation is used throughout thissection. I denotes the given instance of 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉 with n transceivers. For each trans-ceiver u, π∗(u) denotes the power assigned to u in an optimalsolution. Further, OPT(I) denotes the sum of the powers as-signed to the nodes in an optimal solution.

We obtain an approximation algorithm for the 〈UNDIR,

2-NODE CONNECTED, TOTALP〉 problem from the outlineof figure 1 by using an approximation algorithm from [14] forthe minimum weight 2-NODE-CONNECTED subgraph prob-lem in step 2 of the outline. This approximation algorithmprovides a performance guarantee of (2 + 1/n). Using thenotation of theorem 5.1, we have β � (2 + 1/n).

We also show (see lemma 5.1 below) that for the com-plete edge weighted graph Gc(V ,Ec) constructed from theinstance I in step 1 of the outline, there is an edge sub-graph G1(V ,E1) such that G1 is 2-NODE-CONNECTED andthe total weight W(G1) of the edges in G1 is at most (2 −2/n)OPT(I). Using the notation of theorem 5.1, this resultimplies that α � (2 − 2/n).

Thus, once we establish lemma 5.1, it would followfrom theorem 5.1 that the performance guarantee of the re-sulting approximation algorithm for the 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉 problem is 2(2 − 2/n) (2 + 1/n),which approaches 8 asymptotically from below. The remain-der of this section is devoted to the formal statement and proofof lemma 5.1.

Lemma 5.1. Let I denote an instance of the 〈UNDIR,

2-NODE CONNECTED, TOTALP〉 problem with n transceiv-ers. Let OPT(I) denote the total power assigned to the trans-ceivers in an optimal solution to I . Let Gc(V ,Ec) denotethe complete graph constructed in step 1 of Heuristic GEN-TOTAL-POWER. There is an edge subgraph G1(V ,E1) of Gc

Input: An instance I of 〈UNDIR, P, TOTALP〉 where the property P is monotone and polynomial time testable.Output: A power value π(u) for each transceiver u such that the graph induced by the power assignment satisfies

property P and the total power assigned to all nodes is as small as possible.Steps:

1. From the given problem instance, construct the following undirected complete edge weighted graph Gc(V ,Ec).The node set V is in one-to-one correspondence with the set of transceivers. The weight of every edge {u, v} inEc is equal to the power threshold value p(u, v) (which is also equal to p(v, u) by the symmetry assumption).

2. Construct an edge subgraph G′(V ,E′) of Gc such that G′ satisfies property P and the total weight of the edgesin E′ is minimum among all edge subgraphs of Gc satisfying property P.

3. For each node (transceiver) u, assign a power value π(u) equal to the weight of the largest edge incident on u.

Figure 1. Outline of heuristic GEN-TOTAL-POWER for approximating total power.


Figure 2. A simple cycle 〈v1, v2, v3, v4, v5, v6, v1〉 with two chords {v1, v5}and {v3, v6}.

such that G1 is 2-NODE-CONNECTED and the total weightW(G1) of the edges in G1 is at most (2 − 2/n)OPT(I).

Our proof of lemma 5.1 begins with an optimal powerassignment to instance I and constructs a graph G1 satisfy-ing the properties mentioned in the above statement. Thisconstruction relies on several definitions and known resultsfrom graph theory. We begin with the necessary defini-tions.

Definition 5.1. Let G(V,E) be an undirected graph. Sup-pose the node sequence 〈v1, v2, v3, . . . , vk, v1〉 forms a sim-ple cycle C of length at least 4 in G. Any edge {vi, vj } of G

(1 � i �= j � k) which is not in C is a chord.

Figure 2 shows a simple cycle of length 6 with two chords.

Definition 5.2. An undirected graph G(V,E) is critically 2-NODE-CONNECTED if it satisfies both of the following con-ditions: (i) G is 2-NODE-CONNECTED. (ii) For every edgee ∈ E, the subgraph of G obtained by deleting the edge e isnot 2-NODE-CONNECTED.

For example, a simple cycle on three or more nodes is crit-ically 2-NODE-CONNECTED. This is because such a cycleis 2-NODE-CONNECTED, and deleting any edge of the cycleyields a simple path which is not 2-NODE-CONNECTED.

A number of properties of critically 2-NODE-CONNECTED

graphs have been established in the literature (see, for exam-ple, [10,21,32]). We use the following property in provinglemma 5.1.

Theorem 5.2. If a graph G is critically 2-NODE-CON-NECTED then no cycle of G has a chord.

For a proof of the above2 theorem, see [10,21]. We alsouse some terminology associated with Depth-First-Search(DFS) [9]. When DFS is carried out on a connected undi-rected graph G(V,E), a spanning tree T (V,ET ) is produced.Each edge in T , called a tree edge, joins a child to its parent.An ancestor of a node u in T is a node which is not the parentof u but which is encountered in the path from u to the rootof T . Each edge in E−ET , called a back edge, joins a node u

to an ancestor of u. The following lemma establishes a simpleproperty of back edges that arise when DFS is carried out ona critically 2-NODE-CONNECTED graph.

Lemma 5.2. Let G(V,E) be a critically 2-NODE-CON-NECTED graph and let T (V,ET ) be a spanning tree for G

produced using DFS. For any node u, there is at most oneback edge from u to an ancestor of u in T .

Proof. The proof is by contradiction. Suppose a node u hastwo or more back edges. Let v and w be two ancestors of u

in T such that both {u, v} and {u,w} are back edges. Note thatthese two edges are in G. Without loss of generality, let w beencountered before v in the path in T from the root to u. Thepath from w to u in T together with the edge {u,w} formsa cycle in G. By our choice of w, this cycle also includesthe node v. Therefore, the edge {u, v} is a chord in the cycle.This contradicts the assumption that G is critically 2-NODE-CONNECTED since by theorem 5.2, no cycle in G can have achord. The lemma follows. �

We now prove several additional lemmas that are used inour proof of lemma 5.1. Consider the given instance I ofthe 〈UNDIR, 2-NODE CONNECTED, TOTALP〉 problem andlet V denote the set of transceivers. Fix an optimal solution tothe instance I and let p∗ denote the maximum power value as-signed to a node in this optimal solution. Let the chosen opti-mal power assignment induce the graph G∗(V ,E∗). Note thatG∗ is 2-NODE-CONNECTED. Let G∗

1(V ,E∗1 ) be an edge sub-

graph of G∗ such that G∗1 is critically 2-NODE-CONNECTED.

(Such a subgraph can be obtained by starting with G∗ and re-peatedly removing edges until no further edge deletion is pos-sible without violating the 2-NODE-CONNECTED property.)For each edge {u, v} of G∗

1, we assign a weight w1(u, v) asfollows.

1. Let r be a node such that π∗(r) = p∗. Using r as theroot, perform a DFS of G∗

1. Let T (V,ET ) be the resultingspanning tree. Thus, each edge of G∗

1 is either a tree edgeor a back edge.

2. For each tree edge {u, v} where v is the parent of u, letw1(u, v) = π∗(u).

3. For each back edge {u, v} where v is an ancestor of u, letw1(u, v) = π∗(u).

2 It should be noted that the graph theoretic terminology used in [10,21] isdifferent from ours. The statement of theorem 5.2 given above is from [32].

26 LLOYD ET AL.

The following lemma bounds the total weight W1(G∗1) of

all the edges in G∗1 under the edge weight function w1 chosen

above.

Lemma 5.3. W1(G∗1) � (2 − 2/n)OPT(I).

Proof. As mentioned above, each edge of G∗1 is either a tree

edge or a back edge. Consider the tree edges first. For eachtree edge {u, v}, where v is the parent of u, w1(u, v) = π∗(u).Thus, the weight π∗(u) is assigned to at most one tree edge(namely, the edge that joins u to the parent of u if anyin T ). The power value of the root r in the optimal solution,namely p∗, is not assigned to any tree edge (since the roothas no parent). Thus, the total weight of all of the tree edgesunder the weight function w1 is bounded by OPT(I) − p∗.

Now consider the back edges. For each back edge {u, v},where v is an ancestor of u, w1(u, v) = π∗(u). Since G∗

1is critically 2-NODE-CONNECTED, by lemma 5.2, each nodehas at most one back edge to an ancestor. Thus, the weightπ∗(u) is assigned to at most one back edge. Again, thepower value p∗ of the root r in the optimal solution is notassigned to any back edge. Thus, the total weight of all of theback edges under the weight function w1 is also bounded byOPT(I) − p∗.

Therefore, the total weight W1(G∗1) of all of the edges

in G∗1 under the edge weight function w1 is at most 2OPT(I)−

2p∗. Since p∗ is the largest power value assigned to a nodein the optimal solution, p∗ is at least OPT(I)/n. Hence,W1(G

∗1) is bounded by (2 − 2/n)OPT(I) as required. �

The following lemma relates the weight w1(u, v) of anedge {u, v} to the power threshold p(u, v) needed for the ex-istence of the edge.

Lemma 5.4. For any edge {u, v} in G∗1, p(u, v) � w1(u, v).

Proof. Consider any edge {u, v} in G∗1. Since G∗

1 is an edgesubgraph of G∗ (the graph induced by the chosen optimalpower assignment), {u, v} is also an edge in G∗. Also, re-call that the minimum power threshold values are symmetric.Therefore, π∗(u) � p(u, v) and π∗(v) � p(u, v). Hencemin{π∗(u), π∗(v)} � p(u, v). The weight assigned to theedge {u, v} by the edge weight function w1 is either π∗(u) orπ∗(v). Therefore, w1(u, v) � min{π∗(u), π∗(v)}. It followsthat w1(u, v) � p(u, v). �

We are now ready to complete the proof of lemma 5.1.

Proof of lemma 5.1. Starting from an optimal power assign-ment to the instance I , construct the graph G∗

1(V ,E∗1 ) as de-

scribed above. Since the graph Gc constructed in step 1 ofthe heuristic (Figure 1) is a complete graph, every edge inG∗

1 is also in Gc. Consider the edge subgraph G1(V ,E1) ofGc where E1 = E∗

1 . Since G∗1 is 2-NODE-CONNECTED,

so is G1. By lemma 5.4, for each edge {u, v} in E1,p(u, v) � w1(u, v). Therefore, the total weight W(G1)

of all of the edges in G1 under the edge weight function p

is at most W1(G∗1). By lemma 5.3, W1(G

∗1) is bounded

by (2 − 2/n)OPT(I). Therefore, W(G1) is also boundedby (2 − 2/n)OPT(I). In other words, the edge subgraphG1(V ,E1) is 2-NODE-CONNECTED and the total weight ofall its edges is at most (2 − 2/n)OPT(I). This completes theproof of lemma 5.1. �

The following is a direct consequence of the above discus-sion.

Theorem 5.3. There is a polynomial time approximation al-gorithm with a performance guarantee of 2(2−2/n)(2+1/n)

(which approaches 8 asymptotically from below) for the〈UNDIR, 2-NODE CONNECTED, TOTALP〉 problem.

5.2.2. An approximation algorithm for 〈UNDIR, 2-EDGE

CONNECTED, TOTALP〉A result analogous to theorem 5.3 can also be obtained for〈UNDIR, 2-EDGE CONNECTED, TOTALP〉 where the goal isto induce a graph that has the monotone property 2-EDGE

CONNECTED. This problem has also been shown to be NP-complete in [4]. To obtain an approximation algorithm forthis problem from the general framework, we use an ap-proximation algorithm of Khuller and Vishkin [15]. Theirapproximation algorithm produces a 2-edge-connected sub-graph whose cost is at most twice that of a minimum 2-edgeconnected subgraph. In the notation of theorem 5.1, we haveβ � 2. Again using the notation of theorem 5.1, it is possibleto show that α � (2 − 1/n). The proof of this result is almostidentical to that for the 2-Node-Connected case, except thatwe need an analog of theorem 5.2. Before stating this ana-log, we have the following definition (which is analogous todefinition 5.2).

Definition 5.3. An undirected graph G(V,E) is critically2-EDGE-CONNECTED if it satisfies both of the followingconditions. (i) G is 2-EDGE-CONNECTED. (ii) For everyedge e ∈ E, the subgraph of G obtained by deleting the edge e

is not 2-EDGE-CONNECTED.

We can now state and prove the analog of theorem 5.2 forcritically 2-edge connected graphs.

Lemma 5.5. If a graph G is critically 2-EDGE-CONNECTED

then no cycle of G has a chord.

Proof. The proof is by contradiction. Suppose G is crit-ically 2-EDGE-CONNECTED but there is a cycle C=〈v1, v2, . . . , vr 〉, with r � 4, with a chord {vi, vj }. Con-sider the graph G′ obtained from G by deleting the chord{vi, vj }. We will show that G′ is 2-EDGE-CONNECTED, thuscontradicting the assumption that G is critically 2-EDGE-CONNECTED.

To show that G′ is 2-EDGE-CONNECTED, it suffices toshow that G′ cannot be disconnected by deleting any singleedge. Consider any edge {x, y} of G′, and let G′′ denote thegraph created by deleting {x, y} from G′. Since we deleted


only one edge from G′, all the nodes of the cycle C are inthe same connected component of G′′. Thus, if we create thegraph G1 by adding the chord {vi, vj } to G′′, the two graphsG1 and G′′ have the same number of connected components.However, G1 is also the graph obtained by deleting the edge{x, y} from G. Since G is 2-EDGE-CONNECTED, G1 is con-nected. Thus, G′′ is also connected. We therefore concludethat G′ is 2-EDGE-CONNECTED, and this contradiction com-pletes the proof of lemma 5.5. �

The remainder of the proof to show that α � (2 − 1/n)

is identical to that for the 2-Node-Connected case. Withα � (2 − 1/n) and β � 2, the following theorem is a di-rect consequence of theorem 5.1.

Theorem 5.4. There is a polynomial time approximationalgorithm with a performance guarantee of 8(1 − 1/n)

(which approaches 8 asymptotically from below) for the〈UNDIR, 2-EDGE CONNECTED, TOTALP〉 problem.

Our performance guarantee results are somewhat pes-simistic since they are derived from a general framework.Using a different method of analysis, Calinescu and Wan [4]have shown recently that both of our heuristics provide a per-formance guarantee of 4.

As in the case of minimizing maximum power, our generalframework for minimizing total power can also be used toobtain polynomial time approximation algorithms for topol-ogy control problems wherein the connectivity requirementsare specified using proper functions. To obtain this result,we use the general method outlined in [1,12] as the algo-rithm in step 2 of our general heuristic. The method of [1,12]gives a 2-approximation algorithm for network design prob-lems specified using proper functions. Using the notation oftheorem 5.1, β = 2. It is also straightforward to show that thecomplete graph constructed in step 1 of our heuristic has a re-quired subgraph of weight at most the optimal solution value.In other words, α � 1. Thus, we obtain a 4-approximationalgorithm for the general class of problems defined in [1,12].An important example of a problem in this class is the Steinervariant of connectivity, where the goal is to assign power lev-els so as to connect only a specified subset of nodes of a graphrather than all the nodes. An approximation algorithm with aperformance guarantee of (1+ln

√3 ) is known for the Steiner

tree problem in graphs [25]. Thus, using this approximationalgorithm, our approach yields a (2 + ln 3)-approximation forthe Steiner variant.

6. Experimental results

In the preceding section, we showed that our algorithm for〈UNDIR, 2-NODE CONNECTED, TOTALP〉 provides a con-stant factor approximation. In this section, we report onthe experimental performance of this algorithm. Since

there are no existing approximation algorithms3 specificallyfor 〈UNDIR, 2-NODE CONNECTED, TOTALP〉, in the exper-iments described here we compare the performance of ouralgorithm with Ramanathan and Rosales-Hain’s algorithmin [23]. Recall that their algorithm finds an optimum so-lution for 〈UNDIR, 2-NODE CONNECTED, MAXP〉 in whichthe power level of each node is minimal. Our experimentswere conducted using a customized implementation on bothrandomly generated networks and on networks derived fromrealistic data generated by the TRANSIMS project [29].

6.1. Randomly generated networks

6.1.1. Experimental environmentThe experimental setup used here is similar to the one de-scribed in [23]. The radio wave propagation model used isthe Log-distance Path Loss Model:

PL(d) = −10 log10

[GtGrλ

2

(4π)2d20

]+ 10η log10

[d

d0

],

where η is the path loss exponent, d0 is the close-in referencedistance, λ is the radio wavelength, Gt is the transmitter an-tenna gain, Gr is the receiver antenna gain, and d is the sepa-ration distance between transmitter and receiver (see [24] fordetailed descriptions of these parameters). All of the parame-ters are chosen to emulate a 2.4 GHz wireless radio, and if d

is less than a certain threshold, the transmission power is setto the minimum transmission power of 1 dBm.

The experiments are conducted by varying the density ofthe network and the spatial distribution of the nodes. In totalthere are 38 sets of experiments, and 10 trials are run on eachset. Each of the results we cite is the average over the 10 trials.

The node density varies from 0.625 node/sq mile to6.25 nodes/sq mile (10 nodes to 100 nodes) in a 4 mile by4 mile area. The experiments are conducted using two nodedistributions: one uniform and one skewed. Specifically, inthe uniformly distributed networks, all nodes are placed us-ing a random uniform distribution. In the networks with askewed distribution, the network area is equally divided intoa 2 by 2 grid, with 80% of the nodes uniformly distributedin two diagonal squares, and the other 20% of the nodes uni-formly distributed in the other two diagonal squares.

In each experiment, after generating a placement of thenodes, both our approximation algorithm (MIN TOTAL) andthe algorithm of [23] (MIN MAX) are run on the networkconsisting of those nodes. Each algorithm assigns powersto nodes such that the resulting network is 2-NODE CON-NECTED. For each algorithm we measure both the maximumand average power assigned, as well as the maximum and av-erage degrees of the nodes in the resulting network.

Prior to discussing the results, we first provide figure 3 thatshows the actual topologies for one simulated network with60 nodes. Figures 3(a) and (b) are respectively the topologies

3 While this paper was under review, an algorithm that provides a con-stant factor approximation for the geometric version of 〈UNDIR, 2-NODE

CONNECTED, TOTALP〉 was presented in [4].

28 LLOYD ET AL.

Figure 3. Examples of network topologies.

resulting from our approximation algorithm (MIN TOTAL)and Ramanathan and Rosales-Hain’s algorithm (MIN MAX).

6.1.2. Experimental results and discussionIn reporting our experimental results, we plot four differentquantities: (i) average power assigned to a node, (ii) maxi-mum power assigned to any node, (iii) average degree of anode and (iv) maximum degree of all of the nodes. The ex-perimental results on power and node degree are shown infigure 4.

• In figures 4(a) and (c), “Min Max” AVG (“Min Max”MAX) and “Min Total” AVG (“Min Total” MAX) arethe average (maximum) power using the MIN MAX andMIN TOTAL algorithms, respectively.

• In figures 4(b) and (d), “Min Max” AVG (“Min Max”MAX) and “Min Total” AVG (“Min Total” MAX) are theaverage (maximum) degrees using the MIN MAX and theMIN TOTAL algorithms, respectively.

Figure 4 illustrates the results on power and node de-gree. In the cases where nodes are uniformly distrib-uted, our MIN TOTAL algorithm consistently outperforms theMIN MAX algorithm in [23] in regard to average power by5%–19%. This improvement increases as the density of thenetwork increases. In contrast, the maximum power assignedby our algorithm is 14%–37% larger than that of [23]. Theaverage power is about 60%–83% of the maximum powerusing the MIN MAX algorithm, and about 39%–70% usingour algorithm. Those numbers decrease as the density of thenetwork increases, which implies that the average power de-creases faster than the maximum power, and a smaller per-centage of nodes have the maximum power as the networkdensity increases.

In skewed placements of nodes, our MIN TOTAL algorithmoutperforms the MIN MAX algorithm with respect to aver-age power by 6%–25%. We observe that the difference be-tween average power and maximum power is larger in skewedplacements than in uniform placements. The average power is

about 40%–76% of the maximum power using MIN MAX al-gorithm of [23], and about 25%–64% using our algorithm. Inother words, for a given average node density, the maximumpower in a skewed network is higher than that in a uniformlydistributed network, while the average power in the skewednetwork is lower. The reason is that in a skewed network thenode density varies significantly from region to region. Witha larger number of nodes in a smaller area, the average dis-tance between two nodes is less, hence the required powerlevels are, on the average, smaller.

As a general rule, smaller is better in regard to node de-grees in the network induced by the power assignments (e.g.,increases spatial spectrum reuse). In that context, in the casewhere nodes are uniformly distributed, the average (maxi-mum) degree of the network with power assigned by ourMIN TOTAL algorithm is consistently smaller than the aver-age (maximum) degree of the network with power assignedby the MIN MAX algorithm in [23]. When using either ofthe algorithms, the average degree does not vary much as thenetwork density changes. Specifically, the average degree isaround 2.73 using our algorithm, which is very close to thesmallest possible degree, since in a 2-node-connected graph,the degree of each node must be at least 2.

The results in regard to node degrees under the skewednode distribution are similar to those for the uniform case.

6.2. The TRANSIMS networks

In addition to the randomly generated networks, we also con-ducted an experimental study on a more realistic networkobtained from the TRANSIMS Portland Study by the LosAlamos National Laboratory [29]. This data set containslocations of 1716 nodes over a 3 km by 3 km area. Thelocations were generated by carrying out a detailed simula-tion of the traffic in the Portland, OR, metropolitan area us-ing the TRANSIMS simulation tool. Since the running timeof our algorithm would be prohibitively high if run on all1716 nodes, we selected two characteristic areas of this net-


Figure 4. Experimental results.

Table 1Experimental results in area 1.

Max range Average range Max degree Average degree

MIN MAX 158 m 67.75 m 12 4.80MIN TOTAL 193 m 55.07 m 5 2.72

work and conducted experiments on those two areas. Byso doing, the spatial effects of the network are preservedand the experimental results can be obtained in a reason-able time frame. Area 1 is a 1 km by 1 km square, where284 nodes are somewhat uniformly distributed. Area 2 is a600 meter by 1650 meter rectangle, where the majority of the271 nodes are concentrated along a curve and the others aresparsely distributed over the remaining area. Similar to ran-dom networks, for each area, we conducted two experiments:One uses our approximation algorithm (MIN TOTAL) for〈UNDIR, 2-NODE CONNECTED, TOTALP〉; the other usesRamanathan and Rosales-Hain’s algorithm (MIN MAX) for〈UNDIR, 2-NODE CONNECTED, MAXP〉. However, insteadof measuring transmission power, we measure the transmis-sion range. That is because the nodes in the TRANSIMS dataset are much more dense than our randomly generated net-

Table 2Experimental results in area 2.

Max range Average range Max degree Average degree

MIN MAX 153 m 73.59 m 28 7.94MIN TOTAL 222 m 51.95 m 7 2.73

works. So, if one utilizes the propagation model we used inprevious experiments, most nodes would use the minimumtransmission power of 1 dbm. The results are presented in thefollowing tables and figures.

• Tables 1 and 2 present the experimental results for area 1and area 2, respectively.

• Figure 5(a) shows the entire network of 1716 nodes giventhat every node has a 75 meter transmission range. Thetwo selected areas are highlighted.

• Figures 5(b) and (c) illustrate the topologies of area 1 andarea 2, respectively, after using our algorithm (MIN TO-TAL). Note that in figure 5(b), several nodes appear not tobe 2-node-connected (e.g., the node at the top middle partof the figure). The reason is that three nodes that are on or

30 LLOYD ET AL.

Figure 5. Topologies of the TRANSIMS network.

almost on a straight line, are connected to each other, andthe edges between them overlap in the figure.

Our experiments with TRANSIMS data show that topol-ogy control can significantly reduce the average transmis-sion power. In figure 5(a), where each node has a transmis-sion range of 75 meters, the induced graphs in areas 1 and 2are not even connected. After the application of MIN TO-TAL algorithm, the induced graphs in both areas are 2-node-connected and the average range for the two areas is reducedto 55.07 meters and 51.95 meters, respectively.

In area 1, the average range assigned by our MIN TOTAL

algorithm is 18.7% lower than that assigned by the MIN MAX

algorithm, while the maximum transmission range of our al-gorithm is 22.2% higher than the MIN MAX algorithm. Theinduced maximum and average degrees are always smallerusing the MIN TOTAL algorithm than using the MIN MAX

algorithm. For area 2, our MIN TOTAL algorithm assigns av-erage range 29.4% lower, but 45% higher maximum range.The contrast on the induced maximum and average degreesby using the two algorithms is even larger in area 2.

These results are consistent with the experimental resultson randomly generated networks. Our MIN TOTAL algorithmconstantly outperforms the MIN MAX algorithm on averagepower (transmission range), and the margin is larger whenthe network is more skewed.

7. Directions for future research

Our work provides several directions for future research.First, it will be of interest to investigate whether approxima-tion algorithms with performance guarantees better than 4 canbe developed for inducing 2-node connected and 2-edge con-nected graphs. Second, it will be useful to consider topologycontrol problems for other graph properties. In that direc-tion, some complexity and approximation results for proper-ties such as bounded diameter and lower bounds on node de-grees under the objective of minimizing total power are pre-sented in [17]. A third direction is to investigate the behav-ior of topology control problems under the asymmetric powerthreshold model. Some results in that direction are also pre-sented in [17]. Finally, it will be of interest to develop distrib-uted versions of algorithms for topology control problems.References [4,18,23] present some results along that direc-tion.

Acknowledgements

We thank the reviewers for a careful reading of the manuscriptand for providing valuable suggestions.


Appendix A. Proof of lemma 4.2

We first restate the lemma.

Lemma 4.2. To find a power assignment such that the result-ing undirected graph G is a tree is NP-complete.

By abuse of terminology, we use 〈UNDIR, TREE, ∗〉 to de-note this problem. The NP-hardness of this problem is es-tablished using a reduction from the X3C problem definedbelow.

Exact cover by 3-sets (X3C).

• Instance: A set S = {x1, x2, . . . , xn} of elements, wheren = 3r for some integer r; a collection C = {C1, C2, . . . ,

Cm} of subsets of S such that |Cj | = 3, 1 � j � m.

• Question: Does C contain an exact cover for S, that is, isthere a subcollection C′ of C such that the sets in C′ arepairwise disjoint and their union is equal to S?

Note that whenever there is a solution to an instance ofX3C, the number of sets in the solution is exactly r (i.e. n/3).

Proof of lemma 4.2. In the 〈UNDIR, TREE, ∗〉 problem, weare given a collection of nodes, and a (symmetric) powerthreshold p(u, v) for each pair of nodes. The question iswhether there exists a power assignment such that the graphinduced by the power assignment is a tree.

It is easy to see that 〈UNDIR, TREE, ∗〉 is in NP since onecan guess a power assignment and verify in polynomial timethat the resulting graph is a tree. We prove the NP-hardnessof the problem by a reduction from X3C (defined above).

Given an instance I of X3C consisting of a set S with n

elements and a collection C of m subsets, we construct aninstance I ′ of the 〈UNDIR, TREE, ∗〉 problem as follows. Thenode set V of I ′ contains a total of n + m + 1 nodes: There isone node (called an element node) ui corresponding to eachelement xi of S (thus, there are totally 3r element nodes),one node (called a set node) vj corresponding to each set Cj

of C (thus, there are totally m set nodes), and a special node(called the root node) denoted by R. The power thresholdsare chosen as follows. (The reader should bear in mind thatthe power thresholds are symmetric; that is, for any pair ofnodes u and v, p(u, v) = p(v, u).)

p(R, vj ) = 1, 1 � j � m,

p(ui, vj ) = 2, if xi ∈ Cj , 1 � i � n, 1 � j � m.

For all other pairs of nodes, the power thresholds are setto 3. This completes the construction of the instance I ′ of〈UNDIR, TREE, ∗〉. It is easy to verify that the constructioncan be carried out in polynomial time. We now argue thatthere is a solution to the 〈UNDIR, TREE, ∗〉 instance if andonly if there is a solution to the X3C instance.

If. Suppose the X3C instance has a solution C′. We choosethe following power assignment: p′(R) = 1, p′(ui) = 2 (1 �i � n), p′(vj ) = 2 if Cj is in C′ and p′(vj ) = 1 otherwise

(1 � j � m). It can be seen that the graph G resulting fromthis power assignment contains only the following edges:

(a) The edge {R, vj }, for each j , 1 � j � m.

(b) For each node vj whose corresponding set Cj is in C′,there are edges from vj to the three nodes correspondingto the elements in Cj .

By choosing R as the root and using the fact that C′ is anexact cover, it can be verified that G is a tree: the root node R

is adjacent to each of the set nodes; and, each element nodeappears as one of the three children of a set node correspond-ing to a subset in the collection C′.

Only if. Now, suppose the 〈UNDIR, TREE, ∗〉 instance hasa solution. Let p′(x) denote the power assigned to node x andlet G denote the graph induced by the power assignment.

We first observe that p′(R) � 1; otherwise, R would bean isolated node and thus G cannot be a tree. Similarly,p′(vj ) � 1 for every set node vj and p′(ui) � 2 for everyelement node ui . As a consequence, the root node R is adja-cent to each of the set nodes v1, v2, . . . , vm, and the maximumpower assigned is at least 2. Therefore, there are two cases toconsider:

Case 1. The maximum power assigned is 2. Let X ={vjk : p′(vjk ) = 2}. We claim that the collection C′ ={Cjk : vjk ∈ X} is an exact cover for S. We prove this by firstshowing that each element xi appears in some subset of C′.To see this, we note that the graph G is connected (since it isa tree). Thus, there is at least one edge from the element nodeui (corresponding to element xi) to some other node of G.Since the maximum power assigned to any node is 2 and thepower threshold for the element node ui to have an edge to R

or an edge to any other element node is 3, ui must be adjacentto a set node vj . Further, because the threshold values aresymmetric, p′(vj ) = 2. Thus, vj ∈ X and the correspond-ing subset Cj is in C′. Hence, each element appears in somesubset in the collection C′.

We now show that the subsets in the collection C′ are pair-wise disjoint. Suppose some pair of subsets Ca and Cb in C′have a common element xi . By our choice of C′, the powervalues assigned to the corresponding set nodes va and vb areboth 2. Further, the power assigned to node ui is also 2. Thus,in the graph G, ui is adjacent to both va and vb. As observedearlier, the root node R is adjacent to both va and vb . Now,the four edges {R, va}, {va, ui}, {ui, vb} and {vb, R} create acycle in G. This contradicts the assumption that G is a tree.So, the subsets in C′ are pairwise disjoint, and C′ is indeed anexact cover for S.

Case 2. The maximum power assigned is 3. First, notethat at most two nodes can have power 3, since if three nodeshave power 3, then they are mutually adjacent, and thus G isnot a tree.

Second, if the power assignment is as in the followingcases, we argue that there is an equivalent assignment inwhich the maximum power is 2. These cases are: only onenode has power 3; R and one set node vi have power 3; and,one element node ui and one set node vj have power 3 where

32 LLOYD ET AL.

xi ∈ Cj . In any of these cases, the resulting graph G hasno edge with power threshold 3, so an assignment with max-imum power 2 can be obtained by reducing the power levelof the nodes with power 3 while keeping the assignments toall of the other nodes unchanged. The induced graph does notchange. Thus, the new assignment is a solution with maxi-mum power 2 to the instance of 〈UNDIR, TREE, ∗〉. Follow-ing the argument in case 1, a solution to X3C can be con-structed.

Finally, we claim that there are no such valid power as-signments in the remaining cases (i.e. R and ui have power 3;vi and vj have power 3; ui and uj have power 3; or, ui and vj

have power 3 where xi /∈ Cj ). The reasons are the following:

1. If two set nodes vi and vj have power 3, then the edges{R, vi}, {R, vj } and {vi, vj } form a cycle.

2. If the root node R and one element node ui have po-wer 3, the edge {R, ui} is in G. Therefore, edge{ui, vj },1 � j � m, is not in G, otherwise R, ui , and vj form acycle. Recall that p′(ui) � 2 for every element node ui ,therefore each vj with power 2 is adjacent to exactly 3 el-ement nodes. No two set nodes can be adjacent to thesame element node, otherwise those three nodes and R

form a cycle. Hence, totally 3k (where k is the numberof set nodes with power 2) element nodes are adjacent tosome set node. Further, no two element nodes can be ad-jacent to each other since the power thresholds betweensuch nodes are 3. Thus, there are 3k + 1 element nodes.This is a contradiction since we know in this instance of〈UNDIR, TREE, ∗〉, the number of element nodes is a mul-tiple of 3.

3. If two element nodes ui and uj have power 3, the edge{ui, uj } is in G. Recall that all set nodes must be adjacentto R, so one and only one of ui and uj is adjacent to aset node. Suppose it is ui . We know from above that 3k

element nodes are adjacent to some set node. So, togetherwith uj , there are 3k + 1 element nodes – a contradiction.

4. If one element node ui and one set node vj have power 3,where xi /∈ Cj , then ui is adjacent to vj . Therefore, thereare 4 nodes adjacent to vj , which are ui and three elementnodes whose corresponding elements are in set Cj . Hence,there are totally 3k+1 element nodes – a contradiction.

This completes the proof of the case 2 as well as that oflemma 4.2. �

Appendix B. Proof of theorem 4.4

Proof. We use a reduction from SET COVERING (SC), awell-known NP-complete problem [11].

Set covering (Sc).

• Instance: A set S = {x1, x2, . . . , xn}, a collection C ={C1, C2, . . . , Cm}, where Ci is a subset of S (1 � i � m),and a positive integer K � m.

• Question: Does there exist a subcollection C′ ⊆ C, suchthat |C′| � K and the union of the sets in C′ is equal to S?

Let P be the property “THE DIAMETER OF G IS LESS

THAN OR EQUAL TO 6”. This property implies that in G,each node is at most 6 hops away from any other node. Ob-viously, P is monotone, and can be tested in O(n3) time byusing the Floyd–Warshall algorithm, where n is the numberof nodes in the graph [9]. Thus, Max-power users is in NP. Toprove the NP-hardness, we provide a reduction from SC.

Given an instance I of SC, an instance I ′ of Max-powerusers is constructed as follows: For each element xi of S,create a node ui in V and for each Ci of C, create a nodevi in V . Further, V also contains four special nodes: w, s1,s2, s3. The power threshold function p is defined as follows.(It should be noted that the power thresholds are symmetric.)

p(ui, vj ) = 1, if xi ∈ Cj ,

p(w, vj ) = P, 1 � j � m,

p(w, s1) = p(s1, s2) = p(s2, s3) = 1.

For all other pairs of nodes x and y, p(x, y) = P + 1.The value of M is set to K + 1. This completes the con-

struction of an instance I ′ of Max-power users. It is clear thatthe construction can be done in polynomial time. Now, weshow that there is a solution to the Max-power users instanceif and only if there is a solution to SC.

If. Suppose C′ is a solution to the instance of SC. Weconstruct a power assignment p′ as follows.

p′(w) = P,

p′(vi) = P, if Ci ∈ C′

(Note: there are at most K such nodes),

p′(x) = 1 for any other node x.

We now argue that p′ is a solution to the instance of Max-power users. Obviously, the maximum power assigned is P

and at most M (i.e., K + 1) nodes have power P . To establishthat the resulting graph G(V,E) satisfies P (i.e., the graph hasdiameter at most 6), we show that w is within 3 hops of everyother node. This follows from the following observations.

1. Nodes s1, s2, and s3 are respectively 1, 2, and 3 hops awayfrom w.

2. For any Ci ∈ C, if Ci ∈ C′, then the edge {vi, w} ∈ E.Hence, node vi is only one hop away from w.

3. For any xi ∈ S, node ui is 2 hops away from w, since ui

is adjacent to some node vi that has an edge to w. (Other-wise, C′ does not cover the element xi .)

4. For any Ci ∈ C, if Ci /∈ C′, then vi is 3 hops away from w,since vi is adjacent to some uj .

Only if. Suppose we have a power assignment p′ thatis a solution to the instance of Max-lpower users, and thatG(V,E) is the resulting graph. We construct a solution to theSC instance as follows. If there is an edge between w andvi in E (there are at most M − 1 such edges), then include


set Ci in C′. We claim that C′ is a solution to the instance ofSC. Since |C′| � M − 1 = K , we need only show that C′covers S. Since the diameter of G(V,E) � 6, s3 is at most6 hops away from any other node. It follows that w must bewithin 3 hops of every other node. For each vi , if edge {vi, w}∈ E, vi is one hop away from w. However, if edge {vi, w} /∈E, vi is at least 3 hops away from w. Now suppose there isan element xi ∈ S that is not in any set of C′. Then, ui is notadjacent to any node vj that is one hop away from w. Thus,ui must be adjacent to some node vj that is at least 3 hopsaway from w. Thus, node ui is at least 4 hops away from w –a contradiction. This completes the proof of theorem 4.4. �

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[13] L. Hu, Topology control for multi-hop packet radio networks, IEEETrans. Commun. 41(10) (1993) 1474–1481.

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[19] L. Li and J.Y. Halpern, Minimum energy mobile wireless networksrevisited, in: Proc. of the IEEE Conf. on Communications (ICC’01)(June 2001) pp. 278–283.

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[26] E.M. Royer, P. Melliar-Smith and L. Moser, An analysis of the opti-mum node density for ad hoc mobile networks, in: Proc. of the IEEEInternat. Conf. on Communication (ICC’01), Helsinki, Finland (June2001) pp. 857–861.

[27] E.M. Royer and C. Perkins, Transmission range effects on AODV mul-ticast communication, ACM Mobile Networks Appl. (Special Issue onMultipoint Communication in Wireless Networks) 7(6) (2002) 455–470.

[28] H. Takagi and L. Kleinrock, Optimal transmission ranges for randomlydistributed packet radio terminals, IEEE Trans. Commun. 32(3) (1984)246–257; also see in: Multiple Access Communications, Foundationsfor Emerging Technologies, ed. N. Abramson (IEEE Press, New York,1992) pp. 342–353.

[29] TRANSIMS, http://transims.tsasa.lanl.gov/.[30] J. van Leeuwen, Graph algorithms, in: Handbook of Theoretical Com-

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[32] D.B. West, Introduction to Graph Theory (Prentice-Hall, EnglewoodCliffs, NJ, 1996).

Errol L. Lloyd is a Professor of Computer and Infor-mation Sciences at the University of Delaware. Pre-viously he served as a faculty member at the Univer-sity of Pittsburgh and as Program Director for Com-puter and Computation Theory at the National Sci-ence Foundation. From 1994 to 1999 he was Chairof the Department of Computer and Information Sci-ences at the University of Delaware. Concurrently,from 1997 to 1999 he was Interim Director of theUniversity of Delaware Center for Applied Science

and Engineering in Rehabilitation. Professor Lloyd received undergraduatedegrees in both computer science and mathematics from Penn State Uni-

34 LLOYD ET AL.

versity, and a Ph.D. in computer science from the Massachusetts Instituteof Technology. His research expertise is in the design and analysis of al-gorithms, with a particular concentration on approximation algorithms forcomputationally difficult problems. He has published over thirty journal pa-pers and numerous conference papers. In 1989 Professor Lloyd received anNSF Outstanding Performance Award, and in 1994 he received the Univer-sity of Delaware Faculty Excellence in Teaching Award.E-mail: [email protected]

Rui Liu received the BS degree in mathematics fromPeking University, Beijing, China, in 1998; the MSdegree in applied mathematics from University ofDelaware in 2000. He is a doctoral candidate incomputer and information sciences at the Universityof Delaware. His research interests include designand analysis of algorithms for combinatorial opti-mization problems, parallel and distributed comput-ing and computer networks.E-mail: [email protected]

Madhav V. Marathe received his B. Tech in com-puter science from IIT Madras and his Ph.D. incomputer science from the University at Albany –SUNY in 1994. Since that time, he has been withLos Alamos National Laboratory, where he currentlyleads the Mathematics and Computer Science teamin the Basic and Applied Simulation Science Group.His research interests include mobile computing,simulation of large socio-technical systems such astransportation, telecommunication and markets and

design and analysis of algorithms.E-mail: [email protected]

Ram Ramanathan is a Division Scientist at BBNTechnologies. His research interests are in the areaof wireless and ad hoc networks, in particular, rout-ing, medium access control and directional anten-nas. He is currently the principal investigator for aproject on architecture and protocols for opportunis-tic access of spectrum using cognitive radios. Re-cently, he was one of one of two principal inves-tigators for the DARPA project UDAAN (UtilizingDirectional Antennas for Ad hoc Networking), and

the co-investigator on NASA’s Distributed Spacecraft Network project. Ramis actively involved in the evolution of mobile ad hoc networking, and hasrecently served on the program committees of the ACM MobiHoc sympo-sium, and ACM Mobicom. He is on the editorial board of Ad Hoc Networksjournal. He has won three best paper awards at international conferences –at ACM SIGCOMM 92, at IEEE INFOCOM 96, and IEEE MILCOM 02.Dr. Ramanathan holds a B.Tech from the Indian Institute of Technology,Madras, and an M.S. and a Ph.D. from the University of Delaware. He isa senior member of the IEEE.E-mail: [email protected]

S.S. Ravi received his Ph.D. in computer sciencefrom the University of Pittsburgh in 1984. Since thattime, he has been with the Department of ComputerScience, University at Albany – State University ofNew York, where he is currently a Professor. His re-search interests include design and analysis of algo-rithms, mobile computing, fault-tolerance and VLSIdesign and testing.E-mail: [email protected]


Wireless ATM Layouts for Chain Networks ∗,∗∗

MICHELE FLAMMINIDipartimento di Informatica, University of L’Aquila, Via Vetoio loc. Coppito, I-67100 L’Aquila, Italy

GIORGIO GAMBOSIDipartimento di Matematica, University of Rome “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Rome, Italy

ALFREDO NAVARRA ∗∗∗Dipartimento di Informatica, University of L’Aquila, Via Vetoio loc. Coppito, I-67100 L’Aquila, Italy, and

MASCOTTE project, I3S-CNRS, INRIA, Université de Nice, Sophia Antipolis, route des Lucioles, B.P. 93 F-06902, Sophia Antipolis Cedex, France

Abstract. In this paper we consider the problem of constructing ATM layouts for wireless networks in which mobile users can move alonga chain of base stations. We first show that deciding the existence of a layout with maximum hop count h, load l and channel distance d

is NP-complete for every fixed value of d greater or equal to 1. We then provide optimal layout constructions for the case d � 2. Finally,optimal layout constructions are obtained also for any d within the class of the so-called canonic layouts, that so far have always been shownto be the optimal ones.

Keywords: capacity planning, ATM networks, wireless networks, mobile users, chains

1. Introduction

The Asynchronous Transfer Mode (ATM for short) is themost popular networking paradigm for Broadband ISDN[18,19,24]. It transfers data in the form of small fixed-sizecells, and in order to achieve the stringent transfer rate re-quirements, is based on two types of predetermined routes inthe network: virtual paths or VPs, constituted by a sequenceof successive edges or physical links, and virtual channels orVCs, each given by the concatenation of a proper sequenceof VPs. Routing in virtual paths can be performed very effi-ciently by dedicated hardware, while a cell passing from onevirtual path to another one requires more complex and slowerelaboration.

Given a network and a set of connections to be estab-lished, to provide the performance required by B-ISDN ap-plications it is important that routing is performed in a hard-ware fashion in most of the nodes a cell traverses, at the sametime limiting the number of paths sharing a same physicallink [1,4,15,25,26].

A graph theoretical model related to this ATM design prob-lem has been first proposed in [7,15]. In such a framework,the VP layouts determined by the VPs constructed on the net-work are evaluated mainly with respect to two different costmeasures: the hop count, that is the maximum number of

∗ Work supported by the IST Programme of the EU under contract num-ber IST-1999-14186 (ALCOM-FT), by the EU RTN project ARACNE,by the Italian project REAL-WINE, partially funded by the Italian Min-istry of Education, University and Research, by the French MASCOTTEproject I3S-CNRS/INRIA/Univ. Nice, Sophia Antipolis and by the Ital-ian CNR project CNRG003EF8 – “Algoritmi per Wireless Networks”(AL-WINE).

∗∗ Preliminary version of this paper appeared in [11].∗∗∗ Corresponding author.

VPs belonging to a VC, which represents the number of VPchanges of messages along their route to the destination, andthe load, given by the maximum number of virtual paths shar-ing an edge, that determines the size of the VP routing tables(see, e.g., [8]). For further details and technical justificationsof the model for ATM networks see for instance [1,15].

While the problem of determining VP layouts withbounded hop count and load is NP-hard under different as-sumptions [10,15], many optimal and near optimal construc-tions have been given for various interconnection networkssuch as chain, trees, grids and so forth [3,7,9,13,14,21,29](see [30] for a survey).

The integration of wireless and ATM networks is emergingas one of the most promising approaches able to support usersmobility while maintaining the quality of service offered bythe classical ATM. This combination occurs at different lev-els and yields different scenarios, such as End-to-End WATMand WATM Interworking, applied respectively to create newwireless networks with ATM virtual channels extending untilthe mobile terminals and at a more external level for intercon-necting different existing wireless subnets [16]. In both sce-narios, the mobility facility requires the efficient solution ofseveral problems, such as handover (users movement), rout-ing, location management, connection control and so forth.A detailed discussion of these and other related issues can befound in [2,5,6,16,23,27].

An extension of the basic ATM model of [7,15] able tocombine quality of service and mobility aspects in wirelessATM networks has been proposed in [12]. In this model asubset of the nodes of the network represents the base sta-tions and users are allowed to move between them accordingto an adjacency graph expressing their adjacencies in the ge-ographic space. Such a graph, in general, can differ from

36 FLAMMINI ET AL.

the physical topology of the infrastructured network. For in-stance, in nowadays cellular systems like GSM, the physicalgraph G is a tree, stations correspond to its leaves and the ad-jacency graph is an hexagonal grid (see, for instance, [22]).Standard ATM layouts must be constructed in order to estab-lish a different VC for each station, but their performance isevaluated by means of a further parameter, the virtual channeldistance, that measures the time needed to reconstruct virtualchannels during handover phases, that is when mobile termi-nals switch between adjacent stations. More precisely, thedistance between the virtual channels of two adjacent nodesis equal to the number of VPs that must be deleted and addedto one VC in order to obtain the other one. In order to makethe rerouting phase imperceptible to users and thus to obtaina sufficient quality of service, the maximum distance betweentwo virtual channels must be maintained as low as possible.Therefore, a natural combinatorial problem arises in whichsuitable trade-offs must be determined between the differentperformance measures.

In [12] it has been shown that the layout construction prob-lem is intractable, that is NP-hard. Moreover, optimal lay-out constructions are given when the physical and adjacencygraphs are coincident and correspond to basic interconnectionnetworks, such as chains and rings. Such results hold underthe assumption that all the VCs induce shortest paths in theunderlying network.

In this paper we consider the determination of WATM lay-outs for chains in the non-shortest path case in which thelengths of the paths induced by the VPs is not constrained.We first show that deciding the existence of a layout withmaximum hop count h, load l = 1 and distance d = 1 isNP-complete even when the adjacency graph is a chain ofbase stations with the source coinciding with one of its end-points. Moreover, such a hardness result is extended to everyfixed value of d . We then consider the case in which the phys-ical and adjacency graph coincide with chains and provideoptimal layout constructions for d � 2. Finally, optimal lay-out constructions are obtained also for any d within the classof the so-called canonic layouts, that so far have been alwaysshown to be the optimal ones.

The paper is organized as follows. In the next section weintroduce the model, the notation and the necessary defini-tions. In section 3 we provide the above mentioned hardnessresults for the layout construction problem. In section 4 weprovide the optimal layouts for chains when d = 2 and insection 5 the optimal canonic ones for any d . Finally, in sec-tion 6, we give some conclusive remarks and discuss someopen questions.

2. The WATM model

We model the network as an undirected graph G = (V ,E),where nodes in V represent switches and edges in E are point-to-point communication links. In G there exists a subset ofnodes U ⊆ V constituted by base stations, i.e., switchesadapted to support mobility and having the additional capa-

bility of establishing wireless connections with mobile termi-nals. A distinguished source node s ∈ V provides high speedservices to the users moving along the network. We observethat, according to the wireless nature of the system, duringthe handover phase mobile terminals do not necessarily haveto move along the network G, but they can switch directlyfrom one station to another, provided that they are adjacent inthe physical space. It is thus possible to define a (connected)adjacency graph A = (U, F ), whose edges in F representadjacencies between stations.

A layout � for G = (V ,E) with source s ∈ V is a col-lection of simple paths in G, termed virtual paths (VPs forshort), and a mapping that defines, for each station u ∈ U ,a unique virtual channel VC(u) connecting s to u, i.e., a sim-ple path from s to u in the virtual topology defined by the VPsof � . In other words, VC(u) is a collection of VPs whoseconcatenation forms a path in G from s to u.

Definition 2.1 [15]. The hop count h(u) of a node u ∈ U ina layout � is the number of VPs contained in VC(u), thatis |VC(u)|. The maximal hop count of � is Hmax(�) ≡maxu∈U {h(u)}.

Definition 2.2 [15]. The load l(e) of an edge e ∈ E in alayout � is the number of VPs ψ ∈ � that include e. Themaximal load Lmax(�) of � is maxe∈E{l(e)}.

As already observed, when passing from a station u ∈ U

to an adjacent one v ∈ U , the virtual channel VC(v) mustbe reconstructed from VC(u) changing only a limited numberof VPs. Once fixed VC(u) and VC(v), denoted as VC(u, v)

the set of VPs in the subchannel corresponding to the longestcommon prefix of VC(u) and VC(v), this requires the deletionof all the VPs of VC(u) that occur after VC(u, v), plus theaddition of all the VPs of VC(v) after VC(u, v). The numberof removed and added VPs, denoted as D(VC(u), VC(v)), iscalled the distance of VC(u) and VC(v) and naturally definesa channel distance measure d between pairs of adjacent nodesin A.

Definition 2.3 [12]. The channel distance of two nodes u

and v, such that, {u, v} ∈ F (i.e., adjacent in A) is d(u, v) =D(VC(u), VC(v)) = h(u)+h(v)−2|VC(u, v)|. The maximaldistance of � is Dmax(�) ≡ max{u,v}∈F {d(u, v)}.

It is now possible to give the following definition concern-ing layouts for WATM networks.

Definition 2.4. A layout � with Hmax(�) � h, Lmax(�)

� l and Dmax(�) � d is a 〈h, l, d〉-layout for G, s and A.

In the following we will always assume that all the VPsof � are contained in at least one VC. In fact, if such prop-erty does not hold, the not used VPs can be simply removedwithout increasing the performance measures h, l and d .

Before concluding the section, let us remark that for prac-tical purposes and quality of services guarantees, it makes

WIRELESS ATM LAYOUTS FOR CHAIN NETWORKS 37

sense to consider the case where d � h. In fact, while alittle communication delay proportional to the hop count ingeneral can be tolerated, connections gaps due to rerouting ofvirtual channels must not be appreciated by mobile users. Onthe other hand, when d � 2h, our model coincides with theclassical one presented in [15] for standard ATM networks,since the difference between any two virtual channels is al-ways at most equal to 2h.

3. Hardness and approximation results

In this section we show that constructing optimal dynamiclayouts is in general an NP-hard problem, even when l = 1,d = 1 and the adjacency graph is a chain of stations with thesource being one of its endpoints.

Before proving our results, let us briefly outline the basiccharacteristics of a layout with maximum delay d = 1.

Given any two stations u1, u2 ∈ U adjacent in A =(U, F ), during an handover from u1 to u2 if d = 1 by de-finition only one VP can be modified. This means that eitherVC(u2) is a prefix of VC(u2) and thus VC(u2) is obtainedfrom VC(u1) adding a new VP from u1 to u2, or vice versa.In any case, a VP between u1 and u2 must be contained in thelayout. As a direct consequence, the virtual topology definedby the VPs of � coincides with the adjacency graph A. More-over, A must be acyclic. In fact, moving from a station in onedirection along a cycle it is not possible to rebuild the virtualchannel of the station itself when it is reached twice. Finally,if the source coincides with a base station, the maximum hopcount of � is the eccentricity of s in A, that is, the maximumdistance in A between s and the other stations.

We are now ready to prove our first hardness result.

Theorem 3.1. Given a network G = (V ,E), a source s ∈ V ,a chain adjacency graph A = (U, F ) and a positive inte-ger h, deciding the existence of a 〈h, 1, 1〉-layout for G withsource s is an NP-complete problem.

Proof. First of all, observe that, for any h, l, d , the prob-lem of deciding the existence of a 〈h, l, d〉-layout is in NP, asgiven G = (V ,E), s ∈ V , A = (U, F ) and a layout � , it ispossible to check in polynomial time whether Hmax(�) � h,Lmax(�) � l and Dmax(�) � d .

We prove the claim by providing a polynomial time re-duction from Disjoint Paths problem (DP), known to beNP-complete [20]. An instance of this problem is consti-tuted by a graph G = (V ,E) and a collection of node pairs{(s1, t1), . . . , (sk, tk)}. We want to determine whether thereexist k edge-disjoint paths in G, each connecting a differentpair (s1, t1), 1 � i � k.

Without loss of generality, it is possible to assume thatall the pairs (si, ti ), 1 � i � k, are disjoint, i.e., allnodes s1, . . . , sk, t1, . . . , tk are different. In fact, any instancenot satisfying this property can be trivially modified into anequivalent one in which every node v occurring in k′ � k

pairs is connected in G to k′ new nodes v1, . . . , vk′ and thek′ pairs contain in the order v1, . . . , vk′ instead of v.

Starting from an instance of DP, we construct a networkG′ = (V ′, E′), a source s ∈ V ′ and a chain adjacency graphA = (U, F ) that admit a 〈h, 1, 1〉-layout with h = 2k − 1 ifand only if there exist the requested k edge-disjoint paths inthe instance of DP.

Let G′ = (V ′, E′) be such that, given k − 1 nodesw1, . . . , wk−1 not contained in the initial graph G, V ′ = V ∪{w1, . . . , wk−1} and E′ = E∪{{ti , wi}{wi, si+1} | 1 � i < k}.Concerning A = (U, F ), let U = {s1, . . . , sk, t1, . . . , tk} andF = {{si , ti} | 1 � i � k} ∪ {{ti , si+1} | 1 � i < k}. Finally,the source s = s1.

Assume first that there is a 〈2k − 1, 1, 1〉-layout � forG′ = (V ′, E′), s and A = (U, F ). By the considerationsat the beginning of this section, for each e ∈ F , a VP in �

must exist connecting the two endpoints of e. We can assumethat for each i, 1 � i < k, the VP connecting ti to si+1 is〈ti , wi, si+1〉, i.e., it is constituted by the new added path inG′ that goes from ti to si+1 through the new node wi . Infact, if this does not hold, it is possible to add to � the newVP 〈ti , wi, si+1〉, deleting the old one and then, in order tokeep l = 1, if there is another VP stepping through wi , it ismodified in such a way that its subpath between ti and si+1coincides with the old deleted VP.

Therefore, since l = 1 and for all i, 1 � i � k, the VPbetween si and ti does not step through any of the nodesw1, . . . , wk−1, there must exist k edge-disjoint paths in G

connecting the pairs (s1, t1), . . . , (sk, tk).Vice versa, if there are k edge-disjoint paths in G connect-

ing the pairs (s1, t1), . . . , (sk, tk), a 〈2k−1, 1, 1〉-layout � forG′ = (V ′, E′), s and A = (U, F ) can be constructed as fol-lows. For each i, 1 � i � k, the VP between si and ti is givenby the corresponding path in G, edge-disjoint with all the oth-ers. The VP between ti and si+1, 1 � i < k, is 〈ti , wi, si+1〉.Since s = s1 and the eccentricity in A = (U, F ) of the sta-tion s1 is 2k − 1, the layout � thus constructed gives directlya 〈h, 1, 1〉-layout with h = 2k − 1. �

Notice that in the above construction the source s corre-sponds to an endpoint of the chain A = (U, F ), so as alreadyremarked the NP-completeness holds also under this restric-tion.

The above result can generalized to any fixed d > 0 asfollows.

Theorem 3.2. For any fixed integer d > 0, given a networkG = (V ,E), a source s ∈ V , a chain adjacency graphA = (U, F ) and a positive integer h, deciding the existenceof a 〈h, 1, d〉-layout for G with source s is an NP-completeproblem.

Proof. Given G, a source s and an adjacency graph A =(U, F ), it is sufficient to construct in polynomial time G′, s′and A′ = (U ′, F ′), such that, G, s,A admit a 〈h, 1, 1〉-layoutif and only if G′, s′, A′ admit a 〈h′, 1, d〉-layout for a suitableh′ > 0.

By theorem 3.1, it is possible to assume that A = (U, F )

is a chain of the nodes u1, . . . , uk and s = u1.

38 FLAMMINI ET AL.

G′ = (V ′, E′) is obtained from G = (V ,E) by addingfor each ui , 1 � i < k, d − 1 other stations ui,1, . . . , ui,d−1connected by edges {ui, ui,1} and {ui,j , ui,j+1}, 1 � j <

d − 1, in such a way that ui, ui,1, . . . , ui,d−1 form a chain ofd nodes. s′ = u1 and the new adjacency graph A′ = (U ′, F ′)is, such that, U ′ = U ∪ {ui,1, . . . , ui,d−1 | 1 � i � k} andF ′ = {{ui, ui,1 | 1 � i � d − 1} ∪ {{ui,j , ui,j+1 | 1 � i < k,1 � j < d − 1} ∪ {{ui,d−1, ui+1 | 1 � i < k}. Hence,A = (U, F ) is a chain.

Since l = 1, in any layout � ′ for G′, s′, A′, all edges edges{ui, ui,1} and {ui,j , ui,j+1}, 1 � i < k and 1 � j < d − 1,must be VPs, as they are the only simple paths connectingthe respective endpoint stations. Then, during the handoverfrom a station ui,d−1 to ui+1, the d − 1 VPs {ui, ui,1} and{ui,j , ui,j+1}, 1 � j < d − 1, must be deleted and then asingle VP must be added from ui (the last station in the com-mon prefix of the virtual channels VC(ui,d−1) and VC(ui+1))to ui+1.

Since s′ = u1 and u1 has eccentricity h′ = max{h, k −2+d − 2} in A′ = (U ′, F ′), then G, s,A admit a 〈h, 1, 1〉-layoutif and only if G′, s′, A′ admit a 〈h′, 1, d〉-layout, hence thetheorem holds. �

Again, the NP-completeness still holds if the source s is anendpoint of the chain adjacency graph.

Before concluding the section, let us, finally, show that ford = 1 a stronger hardness result holds. To this aim observefirst that as remarked at the beginning of this section, the vir-tual topology induced by any 〈h, l, 1〉-layout coincides withthe adjacency graph. Moreover, if the source coincides with abase station, h is equal to the eccentricity in A of s, otherwiseconnecting s by a VP to a node of minimum eccentricity in A

it is possible to obtain a layout with a maximum hop countequal to its eccentricity increasing the load at most of one.Therefore, as far as approximation results are concerned, theinteresting parameter to be approximated remains the maxi-mum load.

The problem of minimizing the maximum load is equiv-alent from an approximation point of view to the optimiza-tion version of the decision problem DP in which we want todetermine k paths connecting the k source–destination pairs(s1, t1), . . . , (sk, tk) in such a way as to minimize the maxi-mum number of paths sharing a same edge.

In fact, any r-approximation algorithm A for the layoutproblem yields directly a O(r)-approximation algorithm ADPfor DP. Informally, ADP simply consists in running A on theinstance of the layout problem obtained by adding a newsource s, connecting s to each si , 1 � i � k, and lettingA = (U, F ) be, such that, U = {s, s1, . . . , sk, t1, . . . , tk} andF = {{s, si} | 1 � i � k} ∪ {{si, ti} | 1 � i � k}. Thek VPs connecting each si to ti , 1 � i � k, in the layoutreturned by A correspond to an O(r)-approximate solutionfor DP. A reverse reduction can be determined by observingthat an r-approximation algorithm ADP for DP yields directlyan O(r)-approximation algorithm A for the layout construc-tion problem that consists in running ADP on the instance ob-tained by associating a source–destination (s, t) to each edge

{s, t} ∈ F . The paths returned by ADP plus an eventual pathconnecting s to the node with minimum eccentricity in A ifs is not a base station form the VPs of an O(r)-approximatesolution for the layout problem.

To the best of our knowledge, the best general algorithmfor DP has an approximation ratio r = O(

√|E| log |V |),while r = O(polylog|V |) [28]. Therefore, an O(

√|E| log |V |)-approximation algorithm exists for the maximum load min-imization in layout with d = 1, while any algorithm withan asymptotic better approximation ratio would improveupon [28].

4. Optimal chain layouts for d � 2

Starting from the hardness results shown in the previous sec-tion, we now focus on specific topologies and provide optimallayouts for chain networks when the maximum channel dis-tance d is at most 2. More precisely, we consider the casein which the physical graph is a chain Cn of n nodes, that isV = {1, 2, . . . , n}, E = {{v, v + 1} | 1 � v � n − 1} andthe adjacency graph A coincides with Cn. Moreover, withoutloss of generality, we take the leftmost node of the chain asthe source, i.e., s = 1, as otherwise we can split the layoutconstruction problem into two equivalent independent sub-problems for the left- and the right-hand sides of the source,respectively.

Given fixed h,l,d and a 〈h, l, d〉-layout � for a chain Cn,we say that � is optimal if no 〈h, l, d〉-layout exists for anychain Cm with m > n.

By the considerations of the previous section for d = 1, thevirtual topology induced by the VPs of any 〈h, l, 1〉-layout �

coincides with the adjacency graph A and thus with Cn. As aconsequence, the largest chain admitting a 〈h, l, 1〉-layout is,such that, n = h + 1. Therefore, in the remaining part of thissection we focus on the case d = 2.

In the following we denote by 〈u, v〉 the unique VP cor-responding to the simple path from u to v in Cn and by〈〈s, v1〉〈v1, v2〉 . . . 〈vk, v〉〉 or simply 〈s, v1, v2, . . . , vk, v〉 thevirtual channel VC(v) of v given by the concatenation of theVPs 〈s, v1〉, 〈v1, v2〉, . . . , 〈vk, v〉.

The following lemma establishes that, when moving in onedirection along a chain, some VPs are “accumulated”, that isthey cannot be removed from the VCs of the successive nodesencountered along the same direction.

Lemma 4.1. Given a 〈h, l, 2〉-layout � for a chain networkand a node v, if VC(v) = 〈s, v1, v2, . . . , vk, v〉 and in VC(v)

there exist two consecutive VPs 〈vi−1, vi〉, 〈vi , vi+1〉 withvi−1 < v and vi < v (resp. vi−1 > v and vi > v), thenfor every u � v (resp. u � v), 〈s, v1, v2, . . . , vi−1, vi〉 is aprefix of VC(u).

Proof. Assume first that vi−1 < v and vi < v and letu > v be the first node, such that, 〈vi, vi+1〉 /∈ VC(u). Sinced(u − 1, u) � 2, to reach u we can only add 〈vi, u〉. Thisprocess can be iterated to every node w with w > u, hence the


claim holds. An analogous argument applies when vi−1 > v

and vi > v. �

Another useful property of 〈h, l, 2〉-layouts is that the pre-fixes of a VC are the VCs of their final nodes.

Lemma 4.2. There exists an optimal 〈h, l, 2〉-layout � fora chain network, such that, for every node v with VC(v) =〈s, v1, v2, . . . , vk−1, vk, v〉, VC(vi) = 〈s, v1, v2, . . . , vi〉 forevery i � k.

Proof. Let � be any optimal 〈h, l, 2〉-layout. We now provethat, if for a given node v each prefix of every virtual channelVC(u) with u < v is the virtual channel of the correspondingfinal node, then � can be modified in such a way that sucha property is satisfied also by VC(v). This clearly proves thelemma.

Trivially the property is satisfied by the virtual channel ofsource s, since it is empty. Therefore, let v � 2 be any node,such that, the property is true for all the VCs of the previousnodes and let VC(v) = 〈s, v1, v2, . . . , vk−1, vk, v〉. Recallingthat d(v − 1, v) � 2, it is possible to distinguish among thefollowing cases:

1. VC(v − 1) = 〈s, v1, v2, . . . , vk, v − 1〉,2. VC(v − 1) = 〈s, v1, v2, . . . , vk, v, v − 1〉,3. VC(v − 1) = 〈s, v1, v2, . . . , vk〉, that is vk = v − 1,

4. VC(v − 1) = 〈s, v1, v2, . . . , vk, vk+1, v〉, and

5. VC(v − 1) = 〈s, v1, v2, . . . , vk−1〉, that is vk−1 = v − 1.

Since by the hypothesis the claim is true for VC(v − 1)

and 〈s, v1, v2, . . . , vk〉 is a prefix of VC(v − 1), VC(vi) =〈s, v1, v2, . . . , vi〉 for each vi with 1 � i � k. Therefore,every prefix of VC(v) is a VC.

If VC(v − 1) = 〈s, v1, v2, . . . , vk−1〉, that is vk−1 = v − 1,we further distinguish the following two subcases.

(I) vk > v. In this case the VP 〈v, vk〉 must bedeleted in the VC of a node u with v < u � vk . Ifu = vk VC(vk) = 〈s, v1, v2, . . . , vk〉, otherwise, sinced(u − 1, u) � 2, VC(u) = 〈s, v1, v2, . . . , vk, u〉 and iteratingthe same argument to the VP 〈u, vk〉 we finally have that againVC(vk) = 〈s, v1, v2, . . . , vk〉. Therefore, since for each vi

with i < k 〈s, v1, v2, . . . , vi〉 is a prefix of VC(v−1), we haveVC(vi) = 〈s, v1, v2, . . . , vi〉 for every vi with 1 � i � k.

(II) vk < v − 1. If the VP 〈vk, v − 1〉 is contained in theVC of a node u < v − 1, then starting from the source s, inVC(u) 〈vk, v − 1〉 is not traversed from vk to v − 1, otherwiseby hypothesis 〈vk, v − 1〉 would be contained in VC(v − 1)

and thus it could not be added to VC(v − 1) with 〈vk, v〉 toobtain VC(v). Therefore, 〈vk, v−1〉 is traversed from v−1 tovk and again by hypothesis the prefix of VC(u) till v−1 coin-cides with VC(v−1) and VC(vk) = 〈s, v1, v2, . . . , vk−1, vk〉.As in the previous subcase, since for each vi with i < k

〈s, v1, v2, . . . , vi〉 is a prefix of VC(v−1), we have VC(vi) =〈s, v1, v2, . . . , vi〉 for every vi with 1 � i � k.

Assume then that the VP 〈vk, v − 1〉 is not contained inVC(u) for every u < v − 1. In this case also 〈vk, v〉 is notcontained in any VC(u) with u < v − 1, because otherwise,with no matter of the sense in which 〈vk, v〉 is traversed inVC(u), VC(v) could not contain 〈vk, v − 1〉, as by hypotheses〈vk, v − 1〉 does not belong to VC(u).

If 〈vk, v〉 is contained in VC(v + 1), then by lemma 4.1VC(v) is a prefix of all the VCs VC(u) with u � v. Therefore,the layout obtained by deleting the VPs 〈vk, v−1〉 and 〈vk, v〉,adding 〈v−1, v〉 and modifying each VC(u) = 〈s, v1, v2, . . . ,

v − 1, vk, v, . . . , u〉 with u � v as VC(u) = 〈s, v1, v2, . . . ,

v − 1, v, . . . , u〉 does not increase the hop count of any node,the load of any edge and the channel distance of the ad-jacent nodes. Therefore, since in the new layout the vir-tual channels of the nodes before v are not modified andVC(v) = 〈s, v1, v2, . . . , v − 1, v〉 = 〈s, v1, v2, . . . , vk−1, v〉,〈s, v1, v2, . . . , vi 〉 is a prefix of VC(v − 1) for every vi with1 � i < k and thus VC(vi) = 〈s, v1, v2, . . . , vi〉.

If 〈vk, v〉 is not contained in VC(v +1), then by lemma 4.1〈s, v1, v2, . . . , v −1, vk〉 is a prefix of all the VCs VC(u) withu � v. Moreover, all the VPs starting at vk contained inthe VC of some node u > v are not contained in any VCVC(w) with w � v − 1, as otherwise by hypothesis VC(u)

would not contain 〈vk, v − 1〉. Notice also that the otherendpoint of each such VP is greater than v, as otherwise bythe maximum channel distance it cannot be used in the VCsof the nodes after v. It is thus possible to modify the lay-out � as follows. The VPs 〈vk, v − 1〉 and 〈vk, v〉 are deleted,〈v − 1, v〉 is added, the VPs 〈vk,w〉 contained in the VC ofsome node u > v are substituted with 〈v,w〉 and, finally, eachVC(u) = 〈s, v1, v2, . . . , v − 1, vk,w, . . . , u〉 with u � v ismodified as VC(u) = 〈s, v1, v2, . . . , v − 1, v,w, . . . , u〉. Bythe above considerations, the new layout does not increase thehop count of any node, the load of any edge and the channeldistance of the adjacent nodes. Moreover, it does not modifythe virtual channels of the nodes before v. Therefore, againVC(v) = 〈s, v1, v2, . . . , v − 1, v〉 = 〈s, v1, v2, . . . , vk−1, v〉,〈s, v1, v2, . . . , vi 〉 is a prefix of VC(v − 1) for every vi with1 � i < k and thus VC(vi) = 〈s, v1, v2, . . . , vi〉. �

Motivated by the previous lemma, even if not explicitlystated, in the remaining part of this section, we restrict ourattention to layouts in which all the prefixes of each VC arethe VCs of the corresponding final nodes. In fact, this doesnot affect the correctness of our results, since the optimalityis preserved under such assumption.

The following corollary is a direct consequence of the pre-vious lemma.

Corollary 4.3. Every VP of a 〈h, l, 2〉-layout � for a chainnetwork is the final VP of exactly one of its two endpoints.

A last useful property that allows to suitably bound themaximum size of a chain admitting a 〈h, l, 2〉-layout is estab-lished in the following lemma.

Lemma 4.4. Given a 〈h, l, 2〉-layout � for a chain networkand any j � h, let u and v be the last nodes with hop count

40 FLAMMINI ET AL.

h(u) = j − 1 and h(v) = j , respectively. Then the last VPsof all the VCs reaching the nodes from u+1 to v share a samephysical link.

Proof. Let us first prove the claim for j = 1, and thus withu corresponding to the source s = 1.

In order to show that the lemma holds for j = 1, it issufficient to prove that no two edge-disjoint VPs exist in theVCs of the nodes from u + 1 to v.

Assume by contradiction that such property does not holdand let 〈x, y〉,〈w, z〉 be a pair of closest edge-disjoint VPs insuch VCs with x < y � w < z � v. Then y = w ory = w − 1, otherwise any other VP used to reach a nodebetween y and w would be disjoint from 〈x, y〉 or 〈w, z〉 andclosest to 〈x, y〉 or 〈w, z〉, thus contradicting the hypothesis.

If y = w − 1 then 〈x, y〉 is not used to reach y. Infact, 〈x, y〉 cannot be contained in VC(w), as by lemma 4.1it would be contained also in VC(v) against the hypothe-sis h(v) = 1. Thus, if 〈x, y〉 is used to reach y, sinced(y,w) � 2, when moving from y to w 〈x, y〉 must be re-moved and the VP 〈x,w〉 must be added to reconstruct VC(w).But then 〈x,w〉 would be a VP closer to 〈w, z〉 than 〈x, y〉,again contradicting the hypothesis.

By corollary 4.3, 〈x, y〉 is then used to reach x. If y isreached by a VP 〈q, y〉 with q < y, then by corollary 4.3〈w, z〉 is used to reach z, and thus by lemma 4.1 〈q, y〉 is con-tained in VC(v) again contradicting the hypothesis h(v) = 1.If q > y 〈x, y〉 and 〈y, q〉 would be closer than 〈x, y〉 and〈w, z〉: a contradiction.

Therefore, y = w − 1 cannot hold and it must be y = w.Recalling that by corollary 4.3 every VP is used to reach

exactly one of its endpoints, i.e., it is the final VP of exactlyone of the VCs of its endpoints, it is possible to distinguishthe following cases:

• 〈x, y〉 reaches x and 〈y, z〉 reaches z.By corollary 4.3 y is reached by another VP 〈q, y〉 withq �= x and q �= z. If q > y, then by lemma 4.1 〈q, y〉 iscontained in all the VCs of the nodes before x, x included,thus contradicting h(s) = 0. Similarly, if q < y, 〈q, y〉 iscontained in all the VCs of the nodes after z, z included,thus contradicting h(v) = 1. Therefore, this case cannothold.

• 〈x, y〉 reaches y and 〈y, z〉 reaches z.Since by lemma 4.2 every prefix of VC(z) corresponds tothe VC of its final node, 〈x, y〉 is contained in VC(z) andby lemma 4.1 〈x, y〉 belongs to VC(v), thus contradictingh(v) = 1.

• 〈x, y〉 reaches x and 〈y, z〉 reaches y.This case is symmetric to the previous one and completely

analogous considerations show that it would contradicth(s) = 0.

• 〈x, y〉 reaches y and 〈y, z〉 reaches y. This case is clearlyimpossible, since only one of the two VPs 〈x, y〉 and by〈y, z〉 can be the last one of VC(y).

In conclusion, neither y = w can hold and therefore notwo edge-disjoint VPs exist before v.

In order to extend the proof to every j � h, observe thatsince u is the last node with hop count h(u) = j − 1 andd = 2, all the VCs of the nodes w � u have VC(u) as prefix.Thus an identical proof shows that all the VPs after u in theVCs of the nodes w with u < w � v share a same physicaledge. �

It is thus possible to prove the following lemma.

Lemma 4.5. Given any 〈h, l, 2〉-layout � for a chain net-work and j � h, the last node v with h(v) = j is, suchthat, v � j l + 1.

Proof. The claim trivially holds for j = 0, since only thesource s = 1 has hop count h(s) = 0.

Assume by induction that the corollary holds for a givenj , such that, 1 � j < h and let u � j l + 1 the last nodewith hop count h(u) = j and v the last node with hop counth(v) = j + 1. Since by lemma 4.4 the last VPs of all the VCsreaching the nodes from u+1 to v share a same physical link,there cannot be more than l nodes from u + 1 to v, otherwisethe shared link would have load greater than l. Therefore,v � u + l � (j + 1)l + 1. �

In conclusion, the following theorem holds.

Theorem 4.6. For every h, l � 0, a 〈h, l, 2〉-layout for achain network Cn is optimal if and only if n = hl + 1.

Proof. By lemma 4.5, n � hl + 1 holds for any 〈h, l, 2〉-layout for a chain Cn. A layout attaining n = hl + 1 is de-picted in figure 1. �

Before concluding the section, we finally observe that thelayout in figure 1 is not the only optimal one. Another exam-ple with h = 4, l = 4 and d = 2 is shown in figure 2.

5. Optimal canonic layouts for chain networks

In this section we provide 〈h, l, d〉-layouts for chain networksthat are optimal within the class of the canonic layouts. In

Figure 1. Optimal 〈h, l, 2〉-layout for a chain network.


Figure 2. An alternative optimal 〈4, 4, 2〉-layout.

Figure 3. The recursive definition of T (h, l, d) for h > 0 and l > 0.

fact, such layouts have been shown to be the optimal onesunder different assumptions (see, for instance, [12,15]). In-formally speaking, a layout � is canonic if it does not containintersecting VPs and it induces a tree. More precisely, wehave the following definitions.

Definition 5.1. Two VPs 〈u, v〉 and 〈w, z〉 are crossing if u <

w < v < z. A layout � is crossing-free if it does not containany pair of crossing VPs.

Definition 5.2. A layout � is canonic if it is crossing-freeand the virtual topology induced by its VPs is a tree.

Let us say that a rooted tree is ordered if a total orderis defined on its nodes with the root being the lowest ordernode. Then there is a one-to-one corresponds between lay-outs for chains and ordered trees. Namely, each node of thetree corresponds to a node of the chain, the root correspondsto the source s, each edge to a VP of � and finally the to-tal order of the nodes of the tree is given by the order ofthe nodes along the chain. Clearly, not all the ordered treesyield canonic layouts, as their induced VPs might be cross-ing. However, the one-to-one correspondence between or-dered trees and canonic layouts is maintained if we restrictto ordered trees in which every subtree contains a subset ofnodes that forms an interval according to the node ordering.In other words, each subtree corresponds to a segment of thechain not touched by the other subtrees.

Given any ordered tree T , let the reverse tree T r be thesymmetric ordered tree obtained from T by inverting the or-der of the nodes (hence the root becomes the highest or-der node). We now introduce a new class of ordered trees�(h, l, d) that allows to completely define the structure of anoptimal 〈h, l, d〉-layout.

The definition of �(h, l, d) is recursive and the solutionof the associated recurrence gives the exact number of nodesreached by an optimal canonic 〈h, l, d〉-layout. Before intro-ducing �(h, l, d), let us define another ordered subtree that isexploited in its definition.

Definition 5.3. Given any h, l, d , T (h, l, d) is an ordered treerecursively defined as follows.

• If h = 0 or l = 0 T (h, l, d) consists of a single node.

• If h > 0 and l > 0 T (h, l, d) contains at least two nodesand the lowest order node u, that is the root, is connectedby an edge to the highest order node v.Moreover, a chain of min{h, �d/2 } trees T (h−j, l−1, d)

with 0 � j � min{h, �d/2 }− 1 is attached to u in such away that the lowest order node of T (h, l − 1, d) coincideswith u and the lowest order node of each T (h−j, l−1, d)

with 1 � j � min{h, �d/2 }−1 coincides with the highestorder node of T (h − j + 1, l − 1, d).Finally, a chain of min{h − 1, �(d − 1)/2 } reverse treesT r(h− j, l − 1, d) with 1 � j � min{h − 1, �(d − 1)/2 }is attached to v in such a way that the highest order nodeof T r(h − 1, l, d) coincides with v and the highest ordernode of each T r(h − j, l − 1, d) with 2 � j � min{h − j,

�(d − 1)/2 } coincides with the lowest order node ofT r(h − j + 1, l − 1, d).

An example of T (h, l, d) is depicted in figure 3. Infor-mally speaking, a T (h, l, d) corresponds to the sublayout ofa canonic layout � induced by all the VPs occurring undera given VP, with the lowest order node being closer to thesource. Thus, T (h, l, d) is the subtree induced by all the VPswhose endpoints occur from the first endpoint of the given VPuntil the second endpoint.

Directly from the definition, it follows that all the nodes inT (h, l, d) are at distance at most h from u, and thus at mosth additional hops from the node corresponding to u in � aresufficient to reach the other nodes corresponding to T (h, l, d)

in the chain. Moreover, the load yielded by T (h, l, d) on itssegment of the chain is bounded by l. Finally, the channeldistance between two consecutive nodes belonging to the sub-chain of T (h, l, d) is always at most equal to d . In fact, itis given by the maximum distance in T (h, l, d) between twonodes adjacent in the ordering. Therefore, assuming by induc-tion that such property holds inside the subtrees T (j, l −1, d)

(and thus T r(j, l − 1, d)), in order to show that it holds alsoin T (h, l, d) it is sufficient to prove that the final node of thechain of subtrees attached to u and the final node of the otherreverse chain attached to v, that is the only not yet consideredadjacent pair of nodes, is at distance at most d . But such nodesare at distance min{h, �d/2 }+1+min{h−1, �(d−1)/2 } ��d/2 + 1 + �(d − 1)/2 = d . Therefore, also the channel

42 FLAMMINI ET AL.

Figure 4. �(h, l, d) in terms of trees of type T (a) and the alternative recursive definition (b).

distance within the subchain of T (h, l, d) is bounded by d .Clearly, symmetric considerations hold for each T r(h, l, d).

We are now ready to define the final tree �(h, l, d).

Definition 5.4. The ordered tree �(h, l, d) is formed by thechain of h trees T (j, l, d), 1 � j � h, such that, the lowestorder node of T (j, l, d) coincides the highest order node ofT (j + 1, l, d) for 1 � j < h (see figure 4).

Notice that, if h = 0 or l = 0, �(h, l, d) consists of just asingle node. Moreover, an alternative recursive definition of�(h, l, d) is given by a T (h, l, d) attached to a �(h − 1, l, d)

tree (again see figure 4).Let Tn(h, l, d) denote the number of nodes of T (h, l, d)

(and thus of T r(h, l, d)) minus one. Then, directly fromdefinition 5.3, Tn(h, l, d) = 0 if h = 0 or l = 0, other-wise Tn(h, l, d) = 1 + ∑min{h,�d/2 }−1

j=0 Tn(h − j, l − 1, d) +∑min{h−1,�(d−1)/2 }

j=1 Tn(h − j, l − 1, d).Moreover, by definition 5.4, denoted as �n(h, l, d) the

number of nodes in �(h, l, d), �n(h, l, d) = 1 +∑hk=1 Tn(k, l, d).Clearly, by the above observations, �(h, l, d) corresponds

to a canonic 〈h, l, d〉-layout for a chain network. Actually, astronger result holds.

Lemma 5.5. The layout induced by �(h, l, d) is optimalwithin the class of the canonic 〈h, l, d〉-layouts for chain net-works.

Proof. Let � be any canonic 〈h, l, d〉-layout for a chain Cn.It is sufficient to show that n � �n(h, l, d).

Let VC(n) = 〈v1, . . . , vk〉 with v1 = s, vk = n and k �h + 1 the VC of the last node of the chain in � . We provethat vi − vi−1 � Tn(h − i + 2, l, d) for every i, such that,2 � i � k. In fact, this implies n = vk = v1 + ∑k

i=2(vi −vi−1) � 1 + ∑k

i=2 Tn(h − i + 2, l, d) = 1 + ∑k−2i=0 Tn(h −

i, l, d) � 1 + ∑h−1i=0 Tn(h − i, l, d) = 1 + ∑h

i=1 Tn(i, l, d) =�n(h, l, d).

In order to show that vi − vi−1 � Tn(h − i + 2, l, d) forevery i, such that, 2 � i � k, it suffices to prove that, givenany VP 〈u, v〉 of a canonic 〈h, l, d〉-layout, such that, h(u) =h − h′, h(v) = h − h′ + 1 or vice versa and there exist l − l′VPs over it, that is of the form 〈w, z〉 with w � u and z > v

or w < u and z � v, it is v − u � Tn(h′, l′, d). If l′ = 1,

it must be v = u + 1, otherwise the nodes between u and v

could not be reached from the source without exceeding themaximum load l. Recalling definition 5.3, v − u = 1 =Tn(h

′, 1, d).

Assume then that the claim is true for l′ − 1, that is, for allthe VPs of a canonic 〈h, l, d〉-layout � with l − l′ + 1 VPsover them and let 〈u, v〉 be a VP of � with h(u) = h − h′,h(v) = h − h′ + 1 and l − l′ VPs over it. Let w be thelast node with u � w � v (that is under 〈u, v〉) reachedby a VC stepping through u and not from v, and considerthe subchain of d1 VPs 〈w1, . . . , wd1+1〉 with w1 = u andwd1+1 = w connecting u to w. Similarly, let 〈zd2+1, . . . , z1〉with z1 = w + 1 and zd2+1 = v the subchain of d2 VPsconnecting v to w + 1. Since w and w + 1 are adjacent andthe maximum channel distance is d , it must be d1+d2+1 � d .Moreover, since h(w) � h and h(w + 1) � h, d1 � h′ andd2 � h′−1. Therefore, since such subchains and with all theirVPs occur under 〈u, v〉, by applying the inductive assumptionit follows that

v − u = (w − u) + (v − w)

=d1+1∑

i=2

(wi − wi−1) +(

1 +d2+1∑

i=2

(zi − zi−1)

)

� 1 +d1+1∑

i=2

Tn

(h′ − i + 2, l′ − 1, d

)

+d2+1∑

i=2

Tn

(h′ − d2 + i − 2, l′ − 1, d

)

= 1 +d1−1∑

i=0

Tn

(h′ − i, l′ − 1, d

)

+d2∑

i=1

Tn

(h′ − i, l′ − 1, d

)

� 1 +min{h′, d/2}−1∑

i=0

Tn

(h′ − i, l′ − 1, d

)

+min{h′−1, (d−1)/2}∑

i=1

Tn

(h′ − i, l′ − 1, d

)

= Tn

(h′, l′, d

).

A completely symmetric proof shows that v−u � Tn(h′, l′, d)

for every VP 〈u, v〉 of � with h(u) = h−h′+1, h(v) = h−h′and l − l′ VPs over it. �

Starting from lemma 5.5, in order to determine the largestchain admitting a canonic 〈h, l, d〉-layout, it is sufficient toestimate the number of nodes contained in the tree �(h, l, d),that is, �n(h, l, d).


Before solving the recurrence on Tn(h, l, d) and conse-quently estimate �n(h, l, d), we recall that given n + 1 pos-itive integers m, k1, . . . , kn, such that, m = k1 + · · · + kn,the multinomial coefficient

(m

k1,...,kn

)is defined as m!/(k1! ·

k2! · · · kn!) (see, for instance, [17]).

Lemma 5.6. For every h > 0, l > 0 and d > 1, if d is even

Tn(h, l, d)

=l∑

i=1

h−1∑

j=0

∑

0�kd/2−1�kd/2−2�···�k2�k1�i

k1+k2+···+kd/2−1=j

2k1

×(

i

i − k1, k1 − k2, . . . , kd/2−2 − kd/2−1, kd/2−1

),

while if d is odd

Tn(h, l, d)

=l∑

i=1

h−1∑

j=0

∑

0�k(d−1)/2�k(d−1)/2−1�···�k2�k1�i

k1+k2+···+k(d−1)/2=j

2k1−k(d−1)/2

×(

i

i − k1, k1 − k2, . . . , k(d−1)/2−1 − k(d−1)/2, k(d−1)/2

).

Proof. Let M be the matrix defined as follows:

Mi,j =

1 if i = 0 and j = 0,

0 if i = 0 and j > 0,j∑

t=max{0,j−�d/2 +1}Mi−1,t

+j∑

t=max{1,j−�(d−1)/2 }Mi−1,t otherwise.

Note that a generic element Mi,j represents the numberof subtrees T (h − j, l − i, d) and T r(h − j, l − i, d) thatoccur in T (h, l, d) or analogously in the expansion of the re-cursive definition of T (h, l, d) until obtaining only trees ofload l − i. Moreover, by the recurrence of Tn, it results that∑l

i=1∑h−1

j=0 Mi,j is exactly the number of nodes in T (h, l, d)

minus one, that is the value Tn(h, l, d).In order to determine the sum of the first h columns and

the l rows without the first of M , we observe that each row i

of M corresponds to the coefficients of the ith power of thepolynomial ((x�d/2 −1+x�d/2 −2+· · ·+x+1)+(x�(d−1)/2 +x�(d−1)/2 −1 + · · · + x))i . More precisely, a generic elementMi,j is equal to the coefficient of xj in the expansion of thepolynomial ((x�d/2 −1+x�d/2 −2+· · ·+x+1)+(x�(d−1)/2 +x�(d−1)/2 −1 + · · · + x))i .

If d is even, by applying d/2 − 1 times the well-knownequality (a+b)i = ∑i

k=0

(ik

)akbi−k to (2xd/2−1+2xd/2−2+

· · ·+2x2 +2x +1)i with a = 2xd/2−1 +2xd/2−2 +· · ·+2x2

+ 2x and b = 1 and iterating the same argument, we obtain

(2xd/2−1 + 2xd/2−2 + · · · + 2x2 + 2x + 1

)i

=i∑

k1=0

(i

k1

)(2xd/2−1 + 2xd/2−2 + · · · + 2x2 + 2x

)k1

=i∑

k1=0

2k1

(i

k1

)(xd/2−2 + xd/2−3 + · · · + x + 1

)k1xk1

=i∑

k1=0

2k1

(i

k1

) k1∑

k2=0

(k1

k2

)(xd/2−3 + xd/2−4 + · · ·

+ x + 1)k2xk1+k2 = · · ·

=i∑

k1=0

2k1

(i

k1

) k1∑

k2=0

(k1

k2

)· · ·

×kd/2−2∑

kd/2−1=0

(kd/2−2

kd/2−1

)xk1+k2+···+kd/2−1

=i∑

k1=0

k1∑

k2=0

· · ·kd/2−2∑

kd/2−1=0

2k1

(i

k1

)(k1

k2

)· · ·

×(

kd/2−2

kd/2−1

)xk1+k2+···+kd/2−1,

that can be rewritten as∑

0�kd/2−1�kd/2−2�···�k2�k1�i

2k1

×(

i

k1

)(k1

k2

)· · ·

(kd/2−2

kd/2−1

)xk1+k2+···+kd/2−1

=i(d/2−1)∑

j=0

∑

0�kd/2−1�kd/2−2�···�k2�k1�i

k1+k2+···+kd/2−1=j

2k1

×(

i

k1

)(k1

k2

)· · ·

(kd/2−2

kd/2−1

)xj .

Therefore, recalling the definition of multinomial coefficientand that Mi,j is the coefficient of xj in (2xd/2−1 + 2xd/2−2 +· · · + 2x2 + 2x + 1)i , it follows that

Mi,j =∑

0�kd/2−1�kd/2−2�···�k2�k1�i

k1+k2+···+kd/2−1=j

2k1

×(

i

i − k1, k1 − k2, . . . , kd/2−2 − kd/2−1, kd/2−1

).

For the case of odd d we obtain(x(d−1)/2 + 2x(d−1)/2−1 + · · · + 2x2 + 2x + 1

)i

=i∑

k1=0

(i

k1

)(x(d−1)/2 + 2x(d−1)/2−1 + · · · + 2x2 + 2x

)k1

=i∑

k1=0

(i

k1

)(x(d−1)/2−1

+ 2x(d−1)/2−2 + · · · + 2x + 2)k1xk1

44 FLAMMINI ET AL.

=i∑

k1=0

(i

k1

) k1∑

k2=0

(k1

k2

)(x(d−1)/2−2 + 2x(d−1)/2−3 + · · ·

+ 2x + 1)k22k1−k2xk1+k2

=i∑

k1=0

(i

k1

) k1∑

k2=0

(k1

k2

) k2∑

k3=0

(k2

k3

)(x(d−1)/2−3

+ 2x(d−1)/2−4 + · · · + 2x + 1)k3 2k1−k3xk1+k2+k3

= · · ·

=i∑

k1=0

k1∑

k2=0

· · ·k(d−1)/2−1∑

k(d−1)/2=0

2k1−k(d−1)/2

(i

k1

)(k1

k2

)· · ·

×(

k(d−1)/2−1

k(d−1)/2

)xk1+k2+···+k(d−1)/2,

that can be rewritten as∑

0�k(d−1)/2�k(d−1)/2−1�···�k2�k1�i

2k1−k(d−1)/2

×(

i

k1

)(k1

k2

)· · ·

(k(d−1)/2−1

k(d−1)/2

)xk1+k2+···+k(d−1)/2

=i((d−1)/2)∑

j=0

∑

0�k(d−1)/2�k(d−1)/2−1�···�k2�k1�i

k1+k2+···+k(d−1)/2=j

2k1−k(d−1)/2

×(

i

k1

)(k1

k2

)· · ·

(k(d−1)/2−1

k(d−1)/2

)xj .

Therefore,

Mi,j =∑

0�k(d−1)/2�k(d−1)/2−1�···�k2�k1�i

k1+k2+···+k(d−1)/2=j

2k1−k(d−1)/2

×(

i

i − k1, k1 − k2, . . . , k(d−1)/2−1 − k(d−1)/2, k(d−1)/2

).

In every case, the claim follows by recalling that Tn(h, l, d) =∑li=1

∑h−1j=0 Mi,j . �

Theorem 5.7. For every h > 0, l > 0 and d > 1, the max-imum number of nodes reachable in a chain network by acanonic 〈h, l, d〉-layout is

�n(h, l, d)

= 1 +h∑

k=1

Tn(k, l, d)

= 1 +h∑

k=1

l∑

i=1

h−1∑

j=0

∑

0�kd/2−1�kd/2−2�···�k2�k1�i

k1+k2+···+kd/2−1=j

2k1

×(

i

i − k1, k1 − k2, . . . , kd/2−2 − kd/2−1, kd/2−1

),

if d is even, and

�n(h, l, d)

= 1 +h∑

k=1

l∑

i=1

h−1∑

j=0

∑

0�k(d−1)/2�k(d−1)/2−1�···�k2�k1�i

k1+k2+···+k(d−1)/2=j

2k1−k(d−1)/2

×(

i

i − k1, k1 − k2, . . . k(d−1)/2−1 − k(d−1)/2, k(d−1)/2

),

if d is odd.

Unfortunately, �n(h, l, d) in general cannot be expressedby means of a more compact closed formula. However, insome cases it can be significantly simplified. For instance,

• d = 2: �n(h, l, 2) = h · l + 1.In fact, by the definition of the matrix M in the proof oflemma 5.6, the only non null elements of M belong to thefirst column and their value is always equal to one. Hence,the number of the nodes of every Tn(k, l, 2) is l and

�n(h, l, 2) = 1 +h∑

k=1

Tn(k, l − 1, 2) = 1 + h · l.

This coincides with the result obtained in the previous sec-tion, and in fact �(h, l, 2) coincides with the layout con-struction depicted in figure 1.

• d � 2h: �n(h, l, d) = ∑min{h,l}i=0 2i−1

(hi

)(li

) + 12 .

In fact, in this case, our model and constructions coincidewith the ones in [9].

6. Conclusion

The main question left open in the paper is if the family ofthe canonic layouts contains optimal layouts for d > 2. Evenif not claimed explicitly, our constructions show that this istrue for d � 2 and the previous results shown in the literatureseem to confirm this conjecture.

Moreover, it would nice to extend our results to more gen-eral topologies and to the case in which the physical and ad-jacency graphs are not coincident.

Another worth investigating issue is the extension to othercommunication patterns like multicast and all-to-all.

Finally, it would be worth to investigate the approximabil-ity of the layout construction problem for d > 1.

References

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[2] B.A. Akyol and D.C. Cox. Rerouting for handoff in a wireless ATMnetwork, in: Proc. of the IEEE Internat. Conf. on Universal PersonalCommunications (1996).

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[4] J. Burgin and D. Dorman, Broadband ISDN resource management: Therole of virtual paths, IEEE Communicatons Magazine 29 (1991).

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[7] I. Cidon, O. Gerstel and S. Zaks, A scalable approach to routing in ATMnetworks, in: Proc. of the 8th Internat. Workshop on Distributed Algo-rithms, eds. G. Tel and P.M.B. Vitányi, Terschelling, The Netherlands(October 1994), Lecture Notes in Computer Sience, Vol. 857 (Springer,New York, 1994) pp. 209–222; submitted for publication in IEEE/ACMTransactions on Networking.

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[9] Y. Dinitz, M. Feighelstein and S. Zaks, On optimal graphs embeddedinto path and rings, with analysis using l1-spheres, in: Proc. of the 23rdInternat. Workshop on Graph-Theoretic Concepts in Computer Science(WG), Lecture Notes in Computer Science, Vol. 1335 (Springer, NewYork, 1997) pp. 171–183.

[10] T. Eilam, M. Flammini and S. Zaks, A complete characterizationof the path layout construction problem for ATM networks withgiven hop count and load, in: Proc. of the 24th Internat. Collo-quium on Automata, Languages and Programming (ICALP), LectureNotes in Computer Science, Vol. 1256 (Springer, New York, 1997)pp. 527–537.

[11] M. Flammini, G. Gambosi and A. Navarra, Wireless ATM layouts forchain networks, in: Proc. of the 17th Internat. Parallel and DistributedProcessing Symposium (IPDPS), 3rd Workshop on Wireless, Mobileand Ad Hoc Networks (WMAN), IEEE Computer Society (2003)p. 220.1.

[12] M. Flammini, A. Gasparini, G. Gambosi and A. Navarra, Dy-namic layouts for wireless ATM, in: Proc. of the 9th Internat.Conf. on Parallel and Distributed Computing (Euro-Par), LectureNotes in Computer Science, Vol. 2790 (Springer, New York, 2003)pp. 1056–1063.

[13] O. Gerstel, I. Cidon and S. Zaks, The layout of virtual paths inATM networks, IEEE/ACM Transactions on Networking 4(6) (1996)873–884.

[14] O. Gerstel, A. Wool and S. Zaks, Optimal layouts on a chain ATMnetwork, in: 3rd Annual European Symposium on Algorithms (ESA),Corfu, Greece (September 1995), Lecture Notes in Computer Science,Vol. 979 (Springer, New York, 1995) pp. 508–522; to appear inDiscrete Applied Mathematics.

[15] O. Gerstel and S. Zaks, The virtual path layout problem in fastnetworks, in: Proc. of the 13th ACM Symposium on Principles of Dis-tributed Computing, Los Angeles, USA (August 1994) pp. 235–243.

[16] J.D. Gibson, The Mobile Communications Handbook, 2nd ed. (CRCPress/IEEE Press, 1999).

[17] R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics(Addison-Wesley, Reading, MA, 1989).

[18] R. Händler and M.N. Huber, Integrated Broadband Networks: AnIntroduction to ATM-Based Networks (Addison-Wesley, Reading, MA,1991).

[19] ITU recommendation, I series, Blue Book (November 1990).[20] R.M. Karp, On the computational complexity of combinatorial

problems, Networks 5 (1975) 45–68.[21] E. Kranakis, D. Krizanc and A. Pelc, Hop-congestion tradeoffs in

ATM networks, in: Proc. of the 9th IEEE Symposium on Parallel andDistributed Processing (1995) pp. 662–668.

[22] M. Mouly and M.B. Pautet, The GSM System for Mobile Communica-tions (Cell & Sys, 1993).

[23] G. Parry, Wireless ATM MAC protocols – a literature survey, WARPProject – URL, http://vera.ee.und.ac.za/coe/warp(1999).

[24] C. Partridge, Gigabit Networking (Addison-Wesley, Reading, MA,1994).

[25] K.I. Sato, S. Ohta and I. Tokizawa, Broad-band ATM network archi-tecture based on virtual paths, IEEE Transactions on Communications38(8) (1990) 1212–1222.

[26] Y. Sato and K.I. Sato, Virtual path and link capacity design for ATMnetworks, IEEE Journal on Selected Areas in Communications 9(1991).

[27] D. Sobirk and J.M. Karlsson, A survey of wireless ATM MACprotocols, in: Proc. of the Internat. Conf. on the Performanceand Management of Complex Communication Networks (PMCCN)(Chapman & Hall, London, 1997).

[28] A. Srinivasan, Improved approximations for edge-disjoint paths,unsplittable flow, and related routing problems, in: Proc. of the 38thAnnual IEEE Symposium on Foundations of Computer Science (FOCS)(IEEE Computer Society Press, Los Alamitos, CA, 1997) pp. 416–425.

[29] L. Stacho and I. Vrto, Virtual path layouts for some bounded degreenetworks, in: Proc. of the 3rd Colloquium on Structural Informationand Communication Complexity (SIROCCO) (Carleton Univ. Press,1996) pp. 269–278.

[30] S. Zaks, Path layouts in ATM networks, in: Proc. of the SOFSEMConf., Lecture Notes in Computer Science, Vol. 1338 (Springer, NewYork, 1997) pp. 144–160.

Michele Flammini received the degree in computerscience at the University of L’Aquila in 1990 and thePh.D. degree in computer science at the University ofRome “La Sapienza” in 1995. He is associate profes-sor at the computer science Department of the Uni-versity of L’Aquila since 2000. His research inter-ests include algorithms and computational complex-ity, communication problems in interconnection net-works and routing. He has authored and co-authoredmore than 50 papers in his fields of interest published

in the most reputed international conferences and journals.E-mail: [email protected]

Giorgio Gambosi received the degree in electronicengineering at the University of Rome “La Sapienza”in 1980. He is full professor at the Department ofMathematics of the University of Rome “Tor Ver-gata”. His research interests include distributed sys-tems, network management algorithms and routing.He has coauthored about 70 research papers in rele-vant international conferences and journals.E-mail: [email protected]

Alfredo Navarra received the degree in computerscience at the University of L’Aquila in 2000. Hespent one year at the research institute INRIA ofSophia Antipolis (France) collaborating with theMASCOTTE project group and now he is ending hisPh.D. degree in computer science at the University ofRome “La Sapienza”. His research interests includealgorithms and computational complexity, ATM, op-tical and wireless communication networks.E-mail: [email protected];

[email protected]


Ad Hoc Multicast Routing Algorithm with Swarm Intelligence ∗

CHIEN-CHUNG SHEN ∗∗Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716, USA

CHAIPORN JAIKAEODepartment of Computer Engineering, Faculty of Engineering, Kasetsart University, 50 Phahonyothin Rd., Lardyaw, JatuJak, Bangkok 10900, Thailand

Abstract. Swarm intelligence refers to complex behaviors that arise from very simple individual behaviors and interactions, which isoften observed in nature, especially among social insects such as ants. Although each individual (an ant) has little intelligence and simplyfollows basic rules using local information obtained from the environment, such as ant’s pheromone trail laying and following behavior,globally optimized behaviors, such as finding a shortest path, emerge when they work collectively as a group. In this paper, we applythis biologically inspired metaphor to the multicast routing problem in mobile ad hoc networks. Our proposed multicast protocol adaptsa core-based approach which establishes multicast connectivity among members through a designated node (core). An initial multicastconnection can be rapidly setup by having the core flood the network with an announcement so that nodes on the reverse paths to the corewill be requested by group members to serve as forwarding nodes. In addition, each member who is not the core periodically deploys a smallpacket that behaves like an ant to opportunistically explore different paths to the core. This exploration mechanism enables the protocol todiscover new forwarding nodes that yield lower total forwarding costs, where cost is abstract and can be used to represent any metric to suitthe application. Simulations have been conducted to demonstrate the performance of the proposed approach and to compare it with certainexisting multicast protocols.

Keywords: ad hoc networks, multicast routing, swarm intelligence

1. Introduction

Mobile wireless ad hoc networks consist of mobile nodesthat autonomously establish connectivity via multihop wire-less communications. Without relying on any existing, pre-configured network infrastructure or centralized control, theyare useful in many situations where impromptu communi-cation facilities are required, such as battlefield communi-cations and disaster relief missions. In many applications,nodes are likely to collaborate to achieve common goals andare expected to communicate as a group rather than as pairsof individuals (point-to-point). For instance, soldiers roam-ing in the battlefield may need to keep listening to theirgroup commander (point-to-multipoint), or a group of com-manders exchange current mission scenarios with one another(multipoint-to-multipoint). Therefore, multicast communica-tion serves as one critical operation to support these applica-tions.

Many different multicast protocols have been proposed forad hoc networks. Some protocols are based on constructinga tree spanning all the group members. A node then acceptspackets only when they are coming from another node withwhich a tree branch has been established. However, sincethere is only a single path between a pair of sender and re-ceiver, the scheme is vulnerable to network dynamics. Con-sequently, several protocols aim to construct a mesh that al-lows data packets to be transmitted over more than one path

∗ This work is supported in part by National Science Foundation under grantANI-0240398.

∗∗ Corresponding author.

from a sender to a receiver to increase robustness at the priceof redundancy in data transmission. Multicast protocols canalso be classified by how multicast connectivity is establishedand maintained. In a source-based approach, a tree or a meshis constructed per multicast sender, where the constructionprocess is often initiated by the sender. While in a group-shared tree/mesh approach, a single multicast connection isshared by all senders of the same group. One common tech-nique used in this approach is to assign a node, known as therendezvous point or the core [2], to accept join requests frommembers. The multicast connection then consists of shortestpaths from the core to each of the members.

In this paper, we propose a novel multicast routing pro-tocol for mobile ad hoc networks that adopts swarm intelli-gence to reduce the number of nodes used to establish mul-ticast connectivity. We name the protocol Multicast for AdHoc Networks with Swarm Intelligence or MANSI for short.Swarm intelligence refers to complex behaviors that arisefrom very simple individual behaviors and interactions, whichis often observed in nature, especially among social insectssuch as ants and honeybees. Although each individual (for in-stance, an ant) has little intelligence and simply follows basicrules using local information obtained from the environment,global optimization objectives1 emerge when they work col-lectively as a group. Similarly, MANSI utilizes small con-trol packets equivalent to ants in the physical world. Thesepackets, traveling like biological ants, deposit control infor-

1 An example of these is that ants often find a shortest path from their nestto the food source.

48 SHEN AND JAIKAEO

mation at nodes they visit, similar to the way ants layingpheromone trails. This information, in turn, affects the be-havior of other ant packets. With this form of indirect com-munication (known as stigmergy), the deployment of ant-likepackets resembles an adaptive distributed control system thatevolves itself to a more efficient state, accommodating thecurrent condition of the environment.

For each multicast group, MANSI determines a set of in-termediate nodes, forming a forwarding set, that connect allthe group members together and are shared among all thegroup senders. By adopting a core-based approach, the for-warding set is initially formed by nodes that are on the short-est paths between the core and the other group members,where the core may be one of the group members or senders.In addition, during the lifetime of the multicast session (i.e.,when there is at least one active sender), the forwarding setwill evolve, by means of swarm intelligence, over time intostates that yield lower cost, which is expressed in terms oftotal cost of all the nodes in the forwarding set. This evolv-ing, including exploring and learning, mechanism differen-tiates MANSI from other existing ad hoc multicast routingprotocols. Since a node’s cost is abstract and may be definedto represent different metrics, MANSI can be applied to manyvariations of multicast routing problems for ad hoc networkssuch as load balancing, secure routing, and energy conserva-tion.

The remainder of the paper is organized as follows. Wefirst describe the motivation and overview of the MANSI pro-tocol in the next section. Section 3 explains the protocol indetails. Simulation results are then presented and discussedin section 4. Related works are reviewed in section 5. Andsection 6 concludes the paper with future research efforts.

2. Overview of MANSI

MANSI is an on-demand multicast routing protocol that cre-ates a multicast connection among group members by deter-mining a set of intermediate nodes that serve as forwardingnodes. This set, called a forwarding set, is shared among allthe senders of the group. The protocol exploits a core-basedtechnique where each member joins the group via the corenode in order to establish a connection with the other groupmembers. Unlike the core-based tree (CBT) protocol [2],however, the core of each group is not statically assigned to aparticular node in the network and is not known in advanceby the members. Instead, the first member who becomesan active source (i.e., starts sending data to the group) takesthe role of the core and announces its existence to the othersby flooding the network with a CORE ANNOUNCE packet.Each member node then relies on this announcement toreactively establish initial connectivity by sending a JOIN

REQUEST back to the core via the reverse path. Nodes whoreceive a JOIN REQUEST addressed to themselves becomeforwarding nodes of the group and are responsible for accept-ing and rebroadcasting non-duplicated data packets, regard-less of which node the packets were received from. There-fore, MANSI does not rely on any unicast routing protocol.

Figure 1. Examples of multicast connectivity among three group members:(a) a forwarding set of six nodes formed by shortest paths from the core to theother two members, and (b) another forwarding set when A partially sharesthe same path to the core with B, which results in more efficient data packet

forwarding.

To maintain connectivity and allow new members to join, thecore floods CORE ANNOUNCE periodically as long as thereare more data to be sent. As a result, these forwarding nodesform a mesh structure that connects the group members to-gether, while the core serves as a focal point for forwardingset creation and maintenance. Since this process is performedonly when there is an active source sending data to the group,we do not waste valuable network bandwidth to unnecessarilymaintain group connectivity in such dynamic environments.

Similar to other core-based protocols, this process createsa forwarding set consisting of all the intermediate nodes onthe paths on which CORE ANNOUNCEs are accepted and for-warded from the core to the other members, which are oftenshortest paths, as illustrated in figure 1(a). However, groupconnectivity can be made more efficient by having A chooseanother path that is partially shared by B to reduce the sizeof the forwarding set, as shown in figure 1(b), which low-ers the total cost of forwarding data packets. Note that thecost is considered on a per-node basis, not per-link, due to thefact that wireless communication is broadcast in nature (i.e.,a single data packet broadcast by a node is expected to arriveat all of its immediate neighbors in one transmission). In gen-eral, the cost of the forwarding set does not always reflect thenumber of nodes in the set. Instead, the cost associated witheach node can represent different measurements, dependingon the desired properties of the forwarding set. For instance,if we aim to reduce the number of nodes in the forwardingset for efficient data forwarding, the cost associated with eachnode could be one. Table 1 lists a few more examples of whatnode cost would represent when MANSI is applied to othervariations of the multicast routing problem in wireless ad hocnetworks.

We adopt the swarm intelligence metaphor to allow nodesto learn a better multicast connection that yields a lower (to-tal) forwarding cost. Each member who is not the core pe-riodically deploys a small packet, called a FORWARD ANT,that opportunistically explores different, and hopefully betterpaths toward the core. This exploring process is illustrated infigure 2. If a FORWARD ANT arrives at a node who is cur-rently serving as a forwarding node for the group (node D inthis case), it turns itself into a BACKWARD ANT and travels

AD HOC MULTICAST ROUTING ALGORITHM 49

Table 1A few variations of the multicast routing problem and how each node would

compute its cost in MANSI.

Problem Cost calculation per node

Load balancing Current traffic load or the current queue sizePower-aware routing Node’s transmission powerEnergy conservation Inverse of the remaining energy of the nodeSecure routing Security risk of the area the node is located in

Figure 2. Behavior of forward and backward ants: (1) a FORWARD ANT

deployed from the member A choosing C as the next hop and encounter-ing a forwarding node D, and (2) at D, the FORWARD ANT becoming aBACKWARD ANT and following the reverse path back to A while depositing

pheromone along the way.

back to its originator via the reverse path. When the BACK-WARD ANT arrives at each intermediate node, it estimates thecost of having the node it is currently at join the forwardingset via the forwarding node it previously found. The com-puted cost, as well as a pheromone amount that is inverselyproportional to the cost, are updated on the node’s local datastructure. These pheromone amounts are then used by sub-sequent FORWARD ANTs that arrive at this node to make adecision which node they will travel to next, similar to howpheromone is used by biological ants. Let us consider thesame example shown in figure 2, when the BACKWARD ANT

leaves D and arrives at C, the cost of having C join the for-warding set via D is zero since D is already a forwardingnode and is directly connected to C. When the ant comesback to A, the cost of having A join the forwarding set viaD is the same as the cost associated with C because C wouldbe required to become a forwarding node to allow A to jointhe group via D. If A sees that the pheromone amount on thelink to C becomes the highest among links to all neighboringnodes, it will switch to join the group via C by sending a JOIN

REQUEST to C. Consequently, C will become a forwardingnode, while E, F and G will remove themselves from theforwarding set (since they no longer hear requests from A),which is similar to the connectivity shown in figure 1(b).

To prevent the race condition where members attempt toestablish group connectivity via one another’s forwardingpath and nobody remains connected to the core, each for-warding node is associated with a height which is identicalto the highest ID of the nodes that use it to connect to thecore. In addition, the core has its height set to infinity. Fig-ure 3 shows an example illustrating how heights are assigned

Figure 3. An example illustrating how heights are assigned to forwardingnodes used by the members with IDs 3, 6 and 8.

to forwarding nodes. A FORWARD ANT must stop and turninto a BACKWARD ANT only when it encounters a forwardingnode whose height is higher than the ID of the member whooriginated the ant. That means a member is allowed to con-nect to the core via an existing path that belongs to anothermember with a higher ID, but not vice versa, to assure thatthe core, whose height is always the highest, will eventuallybe connected to all the other members.

By following these simple rules, a majority of FORWARD

ANTs from each member will choose a path that connects toan existing forwarding node with a smaller total path cost.Nodes on this path are then used to forward multicast datapackets, resulting in a lower data forwarding cost. This ex-ploring and learning mechanism enables MANSI to learn abetter forwarding set for each group, depending on how nodecost is defined, as well as differentiates MANSI from otherexisting ad hoc multicast routing protocols. Note that, by do-ing so, MANSI attempts to evolve multicast connectivity intostates that yield lower cost. It, however, does not guaranteethat minimum-cost connectivity can be achieved.

3. MANSI protocol description

This section explains the operations of MANSI in details.

3.1. Local data structures

Each node in the network is assigned a unique ID. A nodewith a unique ID i maintains a list of neighboring nodes,ntab(i), obtained via a neighbor discovery protocol such asperiodic hello messaging. The node cost associated withi is denoted by cost(i), where cost(i) � 0, which shouldbe appropriately defined to reflect the performance metricsubject to minimization. In addition, for each multicastgroup g, MANSI maintains the following data structures ateach node i.

• Join table: maintains a list of nodes that have requested tojoin a multicast group via node i. The join table of node i

for multicast group g is denoted by joing(i). This table isupdated when i hears a JOIN REQUEST packet intended toitself. Each entry in joing(i) is of the form 〈r, hr 〉, where

50 SHEN AND JAIKAEO

Figure 4. Sample network snapshots illustrating the operations of MANSI: (a) network setup with three members: nodes 1 (lower-left), 47 (upper-right), and50 (upper-left), where node 1 is the core, (b) dissemination of CORE ANNOUNCE indicated by arrows, (c) initial multicast connectivity using reverse paths to

the core, resulting in a forwarding set of ten nodes (shown in gray), and (d) forwarding set of four nodes learned by ants later in time.

r is a requesting node’s ID and hr is its height (as de-scribed in section 2) that it has sent along with its JOIN

REQUEST. The join table is initially empty for each node.The node i becomes a forwarding node of the group g aslong as joing(i) �= ∅. When a neighbor j is removed fromntab(i) due to a link failure, i will remove all the corre-sponding entries 〈j, hj 〉 from all the join tables.

• Core ID: denoted by coreg(i) to indicate the current coreof group g. coreg(i) is initially set to INVALID_AD-DRESS.

• Core sequence number: keeps track of the latest CORE

ANNOUNCE’s sequence number, denoted by seqNog(i),and initially set to zero.

• Height: represents the height of i if it is currently a mem-ber or a forwarding node of the group g, defined as:

heightg(i) =

∞ if i = coreg(i),

max{i, max{hr | 〈r, hr 〉 ∈ joing(i)}

}

if i is a member of group g,

max{hr | 〈r, hr 〉 ∈ joing(i)

}

otherwise.(1)

As described in section 2, the height of i is the highest IDof the nodes, including i itself if it is a member, that areusing i to connect to the core, and the core has an infiniteheight.


• Pheromone table: maps neighboring nodes and heightsto pheromone intensities. For node i, the pheromone in-tensity associated to the height h of the link (i, j) forthe multicast group g is denoted by τg(i, j, h), where0 � τg(i, j, h) � 1. This table is initially empty. Similarto maintaining the join table, if a neighbor j is removedfrom ntab(i), all entries τg(i, j, h), ∀g, h are removed aswell. The maximum pheromone intensity of one is de-fined to prevent pheromone trails from being overly inten-sified, which, therefore, gives ants enough probabilities toexplore different paths in a timely manner.

• Best cost table: keeps track of how close node i thinks itis to forwarding nodes of certain heights in terms of pathcosts. The cost of the best path to any forwarding node ofheight h for group g that i has seen so far is representedby bestCostg(i, h). This best cost information is used todetermine whether a BACKWARD ANT has returned froma good path or a bad path. Initially, this table is also empty.

3.2. Forwarding set initialization

Since MANSI is a reactive protocol, it does not send anycontrol packet out (except hello packets for neighbor dis-covery) when there is no active source of multicast traf-fic. When a member c of a group g has data to send andit sees that the core does not exist for the group yet (i.e.,coreg(c) = INVALID_ADDRESS), it sets coreg(c) to itsown ID and floods the network with a CORE ANNOUNCE

packet to announce that it is becoming the core. The CORE

ANNOUNCE contains the node ID, c, the multicast group ID,g, a sequence number, and a cost which is initially set tozero, as shown in figure 5. Upon receiving this CORE AN-NOUNCE, each node i discards the packet if it has seen anannouncement from the same node with the same sequencenumber before, or if coreg(i) > c. This is to assure that du-plicate CORE ANNOUNCEs will not be processed, and onlyone CORE ANNOUNCE is allowed to be flooded if more thanone node are attempting to become the core and flooding theirCORE ANNOUNCEs simultaneously. Algorithm 1 presentsthe pseudo code of how node i processes a CORE ANNOUNCE

packet. If the conditions are satisfied, i sets its coreg(i) toc, increases the packet’s cost field by its own cost, then re-broadcasts the packet. In addition, i updates the best costtable, as well as the pheromone amount corresponding tothe height ∞ (i.e., the core’s height) and the neighbor fromwhich the CORE ANNOUNCE was received, by invoking theprocedure UpdatePheromoneAndCost shown in algorithm 2.The operations of algorithm 2 will be explained later in de-tails.

For every node i, coreg(i) is reset back to INVALID_ADDRESS if it has not heard any CORE ANNOUNCE withinthe ANNOUNCE_INTERVAL time period. The core nodec also keeps sending out an announcement packet for groupg every ANNOUNCE_INTERVAL time period as long ascoreg(c) = c and it had at least one data packet for the groupto send within the last ANNOUNCE_INTERVAL time pe-riod.

Figure 5. CORE ANNOUNCE packet format.

Figure 6. JOIN REQUEST packet format.

Algorithm 1. Node i processing a CORE ANNOUNCE packet

1: Input:2: announce ← incoming CORE ANNOUNCE

3: lastHop ← the node from which announce was received4: Begin:5: g ← announce.group6: if coreIdg(i) = INVALID_ADDRESS

OR coreIdg(i) � announce.coreOR seqNog(i) � announce.seqNo

7: Update local information:coreIdg(i) ← announce.coreIdseqNog(i) ← announce.seqNo

8: Invoke UpdatePheromoneAndCostg(lastHop,∞, announce.cost, TRUE)

9: Update cost in the announcement packet:announce.cost ← announce.cost + cost(i)

10: Rebroadcast announce11: end if

As long as a member or a current forwarding node i ofgroup g keeps hearing CORE ANNOUNCE from the core node,i.e., coreg(i) �= INVALID_ADDRESS, it periodically broad-casts a JOIN REQUEST packet to its neighbors. The JOIN

REQUEST packet contains an entry 〈g, k, heightg(i)〉, wherek is defined as:

k = arg maxn∈ntab(i)

∑

h>heightg(i)

τg(i, n, h)

bestCostg(i, h) + 1. (2)

The above formula implies that node i who is willing to joina group should send a request to a neighbor whose good-ness was recently confirmed by BACKWARD ANTs (i.e., hav-ing high pheromone intensity) and also potentially yields thelowest joining cost. In addition, node i only takes into ac-count the best cost information and pheromone intensities ofheights greater than its own height since it is not allowed toconnect to an existing forwarding node of a smaller height, asdiscussed in section 2. At this moment, however, no actualant packets are involved and each node has only one entry,whose height is ∞ (i.e., the core’s height), in each of its bestcost table and pheromone table. In other words, each nodehas just enough information to establish a connection directlyto the core via the reverse path. As a result, the initial for-warding set generally consists of all the nodes that are on the(often times, shortest) paths on which the CORE ANNOUNCE

are forwarded to the members. Figures 4(a)–(c) illustrate theforwarding set initialization process.

52 SHEN AND JAIKAEO

Algorithm 2. Procedure UpdatePheromoneAndCostg(next, height, cost, detFlag) executed by node i

1: Parameters:

2: next ← neighbor ID indicating

which pheromone table entry to be updated

3: height ← height associated with this update

4: cost ← cost of joining the group g at a forwarding

node of height height via next

5: detFlag ← flag indicating whether this

update is deterministic

6: Begin:

7: if τg(i, next, height) is not defined then

8: τg(i, next, height) ← 0

9: end if

10: if detFlag = TRUE then

11: bestCostg(i, height) ← cost

12: τg(i, next, height) ← τg(i, next, height)

+1/(2(1 + cost))

13: else

14: if bestCostg(i, height) is not defined OR

cost < bestCostg(i, height) then

15: bestCostg(i, height) ← cost

16: τg(i, next, height) ← 1 /* set intensity to max */

17: else

18: τg(i, next, height) ← τg(i, next, height)

+1/(1 + cost)

19: end if

20: end if

21: τg(i, next, height) ← min{τg(i, next, height), 1)}/* pheromone intensity is at most one */

If i is a member or a forwarding node belonging to morethan one group, it can combine multiple join entries into asingle JOIN REQUEST packet, as shown in figure 6. When anode j receives a JOIN REQUEST from i and sees that its IDis in the packet, it realizes that it should become a forward-ing node for the group g. It then inserts the sender’s ID andheight in its join table joing(j) and broadcasts its own JOIN

REQUEST containing the ID of the next hop obtained by thesame formula above. Therefore, requests made by memberswill eventually be propagated to the core, thus creating multi-cast connectivity among all the members. On the other hand,if node j hears a JOIN REQUEST from i again without its ID,or i is removed from ntab(j) by neighbor discovery due toa link failure, it removes i from its join table. Each node i

remains to serve as a forwarding node for group g as long asjoing(i) is not empty.

Figure 7. Ant packet format used by both FORWARD ANT and BACKWARD

ANT.

3.3. Forwarding set evolution

Once the initial forwarding set is formed, each group memberwho is not the core attempts to learn a better connection to thecore, in order to minimize the overall cost of the forwardingset, by deploying a FORWARD ANT every ANT_INTERVALtime period. A FORWARD ANT packet deployed by member i

for multicast group g, whose format is shown in figure 7, con-tains the following fields:

• group: multicast group ID.

• height: height of the forwarding node found by this ant.This field is used only after this ant has been turned to aBACKWARD ANT.

• f : forwarding flag indicating whether this ant is a FOR-WARD ANT or a BACKWARD ANT (since they share thesame structure). Since i is deploying a FORWARD ANT,this flag is set to TRUE.

• exLimit: the number of times the ant is allowed to proba-bilistically pick a next hop that is not the current best onein order to prevent it from aimlessly traversing the net-work. This field is initially set to EXPLORE_LIMIT anddecrements every time the ant makes a decision on a nexthop probabilistically, instead of deterministically choosingthe next hop given by (2).

• d: deterministic flag indicating whether the ant should al-ways follow the current best path in order to obtain theactual current cost for the best cost table. The reason forusing deterministic ants is that costs in the best cost tablemay no longer reflect the actual costs due to node mobility,dynamics of nodes’ costs, or dynamics of the forwardingset itself. If this flag is set, the exLimit field is alwaysignored. Every other ant deployed by each member is de-terministic.

• cost: the total cost of the nodes this ant has visited, initiallyset to zero.

• costLimit: the cost limit of the path that the ant is allowedto traverses after leaving its originator. This field is usedin conjunction with the cost field to prevent the ant fromtraversing forward after the accumulated cost exceeds thelimit. Usually this limit is set to minh>i bestCostg(i, h),the lowest known cost to a current forwarding node ofgroup g that i is allowed to connect to, plus some thresh-old. By this way, the ant can stop proceeding once it iscertain that it will not find any better path than what itsoriginator currently has. This cost limit is ignored if theant is deterministic since its goal is not to find a bettercost, but to find the actual current best cost.

• visitedNodes: the set of nodes visited by the ant, initiallyset to {i}.


Algorithm 3. Procedure ReleaseForwardAntg(fant)executed by node i.

1: Parameter:

2: fant ← a FORWARD ANT to be released

3: Begin:

4: Compute a desirability, dn, for node n, n ∈ ntab(i) fromsummations of only entries whose heights are higher thanfant.height in the main pheromone table:

dn =

0 if n ∈ fant.visitedNodes,

1 +∑

h>fant.height

τg(i, n, h)

bestCostg(i, h) + 1

if τg(i, n, h) exists,

1 otherwise.

(3)

5: if ∀n, dn = 0 then

6: return /* ant has no place to go */

7: end if

8: If fant is not deterministic (fant.d = FALSE) and it isallowed to explore (fant.exLimit > 0), with probability0.5, fant decides to randomly choose a next hop n, wherethe probably of choosing n depends on its desirability asfollows:

Prob(n) = dn∑k∈ntab(i) dk

(4)

fant.exLimit ← fant.exLimit − 1

/* ant just performs one more exploration */

9: Otherwise, the next hop is set to the one whose

desirability is maximum:

n ← arg maxk∈ntab(i)dk

10: append n to fant.visitedNodes and broadcast fant

A node deploying a FORWARD ANT invokes the procedureReleaseForwardAnt described in algorithm 3 to find the nexthop that the ant will travel to. A desirability, defined in (3),is computed for each the neighboring nodes by giving highervalues to neighbors that have higher pheromone intensitiesand potentially yield lower costs to connect to an existing for-warding node. On the other hand, zero desirability is given toall the nodes that have been visited before. If the ant is notdeterministic and is still allowed to explore, these desirabil-ities are then normalized to obtain a probability of choosingeach of the neighboring nodes. Otherwise the neighbor nodethat gives the maximum desirability is chosen, which has thesame effect as using (2) except that it excludes all the nodesin the visitedNodes field. Once a next hop is chosen, its IDis appended to the end of visitedNodes and the ant is broad-cast.

Algorithm 4. Node i processing a FORWARD ANT packet.

1: Input:

2: fant ← incoming FORWARD ANT

3: Begin:

4: if i = last entry in fant.visitedNodes then

5: g ← fant.group

6: if joing(i) �= φ AND fant.visitedNodes[0]< heightg(i) then

7: Convert fant to a BACKWARD ANT

fant.cost ← 0

fant.height ← heightg(i)

fant.f ← FALSE

8: Remove last entry from fant.visitedNodes

and broadcast fant

9: else

10: fant.cost ← fant.cost + cost(i)

11: if fant.cost < fant.costLimit OR

fant.d = TRUE then

12: Invoke ReleaseForwardAntg(fant)

13: end if

14: end if

15: end if

When a node j receives a FORWARD ANT, it checks if itsID matches the ID at the end of the ant’s visitedNodes field.If not, the ant is discarded. Otherwise, j knows that this antis intended to itself and accepts it. Algorithm 4 shows howa FORWARD ANT is processed. First, j checks if it is cur-rently a forwarding node of the group and its height is higherthan the ID of the ant’s originator. If so, j realizes that themember who deployed the ant is eligible to join the group viaj itself. This ant is then turned into a BACKWARD ANT byresetting its f flag. Its cost is then reset to zero in order tostart computing the total cost on the way back, and its heightfield is set to j ’s height. The last entry of its visitedNodesis removed in order to send this ant back to the previoushop.

If the condition is not satisfied to convert the ant to aBACKWARD ANT, j increases the ant’s cost field by its owncost cost(j). It then invokes the procedure ReleaseForwardAnt to forward the ant to a next hop, if the updated cost doesnot exceed the limit or the ant is deterministic.

When a node k hears a BACKWARD ANT from j , it in-vokes the procedure UpdatePheromoneAndCost, described inalgorithm 2, which updates the entries in k’s pheromone and

54 SHEN AND JAIKAEO

best cost tables in accordance with j and the height field. Ifthe ant is deterministic, the cost that it carries back is the ac-tual cost of the path its originator is currently using to join thegroup. Therefore, the best cost corresponding to the heightfield is updated to this value. If the ant is not deterministic,however, the best cost is updated only when it is higher thanthe returned cost, which means that the ant has found a betterpath to join the group from this node. The pheromone inten-sity on this link is also updated to the maximum in order toencourage subsequent FORWARD ANTs to use the same link,as well as to redirect join request to this link instead. If theant comes back with a higher cost, a pheromone amount of1/(1 + cost) is added instead. In case of deterministic ant,the added amount is reduced by half since this link alreadyhas the highest pheromone intensity as it has just been chosenby a deterministic FORWARD ANT. Note that we have men-tioned this procedure before when we explained how a nodeuses it while processing a CORE ANNOUNCE (line 8 of algo-rithm 1). This is because a CORE ANNOUNCE more or lessserves as a deterministic BACKWARD ANT returning from thecore.

Algorithm 5. Node i processing a BACKWARD ANT packet.

1: Input:

2: bant ← incoming BACKWARD ANT

3: lastHop ← the node from which bant

was received

4: Begin:

5: g ← fant.group

6: Invoke UpdatePheromoneAnd

Costg(lastHop, bant.height, bant.cost, bant.d)

7: if i = last entry in fant.visitedNodes then

8: Remove the last entry from bant.visitedNodes

9: if bant.visitedNodes �= φ then

10: bant.cost ← bant.cost + cost(i)

11: broadcast bant

12: end if

13: end if

After updating the pheromone and the best cost tables,k checks if the BACKWARD ANT was intended to itself byexamining the last entry in the visitedNodes field. If its IDmatches, it adds its cost into the cost field, removes the lastentry from visitedNodes, and rebroadcasts as long as there isat least one entry left in visitedNodes. Algorithm 5 presentsthe pseudo code of how a node processes a BACKWARD ANT.

Similar to pheromone evaporation of biological ants, eachnode i updates all the entries τg(i, j, h) in its pheromone table

by reducing their values by DECAYING_FACTOR at everyDECAY_INTERVAL time period:

τg(i, j, h) = (1−DECAYING_FACTOR)×τg(i, j, h), (5)

where 0 < DECAYING_FACTOR < 1.By probabilistically selecting next hops, the majority of

the FORWARD ANTs will choose paths with high pheromoneintensity, while some of them may explore totally differentnew paths. If a BACKWARD ANT comes back with a bettercost on a new branch, the pheromone amount on that branchwill be increased significantly. As a result, a change in multi-cast connectivity (i.e., forwarding set) is triggered due to theperiodic broadcast of JOIN REQUEST packets, as illustratedin figure 4(d).

MANSI also takes advantage of broadcast nature of wire-less communication to speed up the learning process asfollows. When a node i overhears a JOIN REQUEST forgroup g from j but not intended to itself, it invokes Up-datePheromoneAndCostg(j, hj , 0, TRUE), where hj is theheight that j reports in its JOIN REQUEST. This impliesthat i could join the group via j with no cost, given that itsheight is less than hj . However, a drawback of this idea is thatsome members who are not forwarding nodes will broadcastJOIN REQUESTs as well and might be mistaken as forwardingnodes by its neighbors.

3.4. Multicast data forwarding

Since MANSI is a mesh-based protocol which allows for-warding nodes and members to accept data packets arrivingfrom any node, each data packet is assigned a unique se-quence number when it is transmitted from the source. Thesequence numbers are checked by each forwarding node andmember node to make sure that no duplicate data packets arerebroadcast or delivered to the application. When a node i

receives a non-duplicate data packet of group g, it checkswhether it is currently a forwarding node of the group, i.e.,joing(i) �= ∅. If so, it rebroadcasts the packet. Otherwise, thepacket is silently discarded.

3.5. Handling mobility

In MANSI, mobility and other network dynamics are handledinherently rather than as exceptions. With the pheromone lay-ing/following behavior of BACKWARD ANTs and FORWARD

ANTs, each path comprising the forwarding set keeps beingreinforced as long as no link on the path is broken. How-ever, network dynamics can cause optimal connectivity tochange from time to time even though the current connectiv-ity may still be valid. With the probabilistic nature of FOR-WARD ANTs to explore new paths, the multicast forwardingset should be able to evolve into a configuration that is moreefficient for the new topology.

When a link currently used by a member or a forwardingnode to send JOIN REQUESTs breaks, the pheromone tableentries corresponding to that link are also removed. There-fore, all subsequent FORWARD ANTs will be redirect to other


Figure 8. A network of 50 nodes moving at 10 m/s, where members are in black and forwarding nodes are in gray: (a) without mobility-adaptive mechanism,and (b) with mobility-adaptive mechanism where NLFF_THRESHOLD is 0.01.

paths, while the majority of them will take the next hop whosepheromone intensity was the second highest before the linkfailure. If this next hop leads to a forwarding node of a higherheight, BACKWARD ANTs will return and update pheromoneon the new path, hence reestablishing a connection to thegroup. However, in case that FORWARD ANTs fail to find anew path, CORE ANNOUNCEs flooded periodically will even-tually restore the connectivity.

Although MANSI is considered a mesh-based protocol byits way of forwarding data packets, connectivity of the for-warding set may still be fragile if the network is sparse andmembers are far apart from each other, especially with thepresence of mobility. To make data forwarding more effectiveunder mobility, while maintaining good efficiency when thenetwork is static, we incorporate a mobility-adaptive mecha-nism into MANSI. With this mechanism, each node i keepstrack of the normalized link failure frequency, denoted bynlff (i), which reflects the dynamic condition of the area sur-rounding i in terms of the number of link failures per neigh-bor per second. A calculation of nlff (i) is performed everyNLFF_TIME_WINDOW time period as follows:

current_nlff (i)

= f

NLFF_TIME_WINDOW × |ntab(i)| , (6)

nlff (i) = current_nlff (i) + nlff (i)

2, (7)

where f is the number of link failures detected during the lastNLFF_TIME_WINDOW time period. Initially nlff (i) is setto zero.

Each member or forwarding node then uses this nlff to de-termine the stability of its surrounding area. If its nlff is lowerthan a threshold NLFF_THRESHOLD, the node will considerits area stable and join the group by sending JOIN REQUESTs

toward its best next hop as usual. If nlff exceeds the threshold,however, it will add another entry for the second best next hopinto its JOIN REQUESTs. Since all the neighbors are rankedby their goodness in terms of pheromone intensities, the sec-ond best next hop can be easily determined. Formally, if k isthe best next hop for i to join the group g, as defined in (2),then the second best next hop k′ is defined as:

k′ = arg maxn∈ntab(i),n�=k

∑

h>heightg(i)

τg(i, n, h)

bestCostg(i, h) + 1. (8)

Figure 8(a) illustrates the forwarding set created by MANSIwithout the mobility-adaptive mechanism for a multicastgroup of three members in a network of 50 nodes, whereeach node is moving at 10 m/s. The group connectivity isalmost a straight line and is vulnerable to link failures. Withthe mobility-adaptive mechanism enabled, most members andforwarding nodes request two of their neighbors to be in theforwarding set, as shown in figure 8(b), so that the group con-nectivity becomes more robust.

4. Experimental results and discussion

To study the characteristics and evaluate the performance ofMANSI, we have conducted simulation experiments using theQualNet simulator [10]. Ten random networks were gener-ated with 50 nodes uniformly distributed over a terrain ofsize 1000 × 1000 m2. Each node was equipped with a ra-dio transceiver which was capable of transmitting signals upto approximately 250 meters over a 2 Mbps wireless chan-nel, using the two-ray path loss model without fading. Weused IEEE 802.11DCF as the MAC layer protocol, and IPas the network layer. Since MANSI does not rely on any

56 SHEN AND JAIKAEO

Table 2Parameter values for MANSI.

HELLO_INTERVAL 1 secANNOUNCE_INTERVAL 10 secANT_INTERVAL 2 secEXPLORE_LIMIT 3DECAY_INTERVAL 1 secDECAYING_FACTOR 0.1NLFF_THRESHOLD 0.01

Figure 9. Average size of the forwarding set as a function of time for COREand MANSI.

unicast routing protocol, no other routing protocols were em-ployed. For each network, a multicast groups of 5 memberswas setup, where each member generated a constant bit rate(CBR) traffic at 2 packets/sec to the group for 20 minutes.The size of data payload was 512 bytes. The MANSI parame-ter values used in our simulation are shown in table 2. Notethat NLFF_THRESHOLD is used only when the mobility-adaptive mechanism is enabled.

Our first set of experiments were setup without mobility inorder to study how MANSI maintains forwarding sets in staticenvironments. For comparison purposes, we used two base-line protocols: FLOOD and CORE, as references. FLOOD isa simple flooding protocol where a data packet is rebroadcastby every node in the network. And CORE is a generic core-based protocol that operates exactly like MANSI, but withno ants deployed, where CORE ANNOUNCEs are periodicallyflooded as usual. The cost of each node was set to one, whichimplies that MANSI would attempt to reduce the size of theforwarding set.

We first look at the average size of forwarding sets main-tained by CORE and MANSI over time for the ten samplenetworks, as shown in figure 9. Due to random delays addedto avoid packet collisions when broadcasting, the dissemina-tion pattern of a CORE ANNOUNCE is unpredictable when itis flooded, which causes a forwarding set to be formed dif-ferently for each announcement. Consequently, the averagesize of forwarding sets keeps changing from time to time inCORE. In contrast, forwarding sets maintained by MANSIstart of at around the same size as that of CORE but keep re-

Table 3Average size of the forwarding set formed inMANSI, CORE, and FLOOD for each network.

Network Average sizeMANSI CORE FLOOD

1 7.89 9.49 50.002 4.00 3.67 50.003 4.00 4.97 50.004 4.46 4.68 50.005 6.51 8.46 50.006 5.52 6.25 50.007 6.90 7.83 50.008 6.04 7.46 50.009 5.16 7.67 50.00

10 5.02 6.95 50.00

Average 5.55 6.74 50.00

ducing in size during the first 200 seconds. Their size thenbecomes stable and stays low most of the time as each mem-ber or forwarding node tends to join the group via a low-costpath (i.e., small hop count in this case), whose existence wasrecently confirmed by BACKWARD ANTs. Although anotherCORE ANNOUNCE may arrive at a member from a differ-ent node, the member will not send a JOIN REQUEST to thisnew node as long as the current joining cost is low and thepheromone intensity on the link it currently uses to join thegroup is high.

Table 3 summarizes the sizes, averaged over the entire sim-ulation time, of the forwarding sets maintained by MANSI,CORE, and FLOOD on each simulated network. (FLOODdoes not really maintain a forwarding set, but the set con-sists of every node in the network.) The results show that inall cases, except one, MANSI yields forwarding sets that areapproximately 15%–20% smaller than those of CORE, andmuch smaller than FLOOD. Since the size of the forward-ing set indicates how many nodes are involved to relay a datapacket from one member to the others, this demonstrates theefficiency of MANSI in terms of data forwarding.

We have performed another set of experiments to comparethe performance of MANSI, in terms of effectiveness and ef-ficiency, with ODMRP. ODMRP [1] is an on-demand, mesh-based multicast protocol that attempts to establish a forward-ing group – similar to a forward set in MANSI – only whena source of the group has data to send. The nodes in the for-warding group form a mesh that connects the group memberstogether. When a multicast source has data to send for thefirst time, it broadcasts to its neighbors a JOIN QUERY packet,which is a data packet with the query flag set. Upon receivinga non-duplicate JOIN QUERY, each node stores the upstreamnode ID in its routing table and rebroadcasts the packet. Whena member of the multicast group receives a JOIN QUERY, itconstructs and broadcasts a JOIN REPLY packet containingthe source ID and the upstream node ID to all of its neighbors.Upon receiving a JOIN REPLY, a node whose ID matches theupstream ID in the packet realizes that it is on the path be-tween the source and a member, so it becomes a forward-ing node for the group by setting its FG_FLAG (ForwardingGroup Flag). It then constructs and broadcasts its own JOIN


REPLY using its corresponding upstream node ID. The broad-casting of JOIN REPLY packets therefore propagates the infor-mation from all the members back to the source on the reversepaths. Once the source has sent out a JOIN QUERY, it sendsall subsequent data packets normally with no query flag set.This will allow only nodes that are currently in the forward-ing group to rebroadcast these data packets, thus reducingdata forwarding overhead. To deal with dynamics of the net-work topology and group membership, each source floods thenetwork with JOIN QUERYs every REFRESH_INTERVAL aslong as it still has data to be sent to the group. The FG_FLAGon each node will be reset if it has not been refreshed by JOIN

REPLY for some period of time, which implies that the sourcehas no data to send, or it is no longer needed as a forwardingnode. If nodes are equipped with GPS, a mobility predictionmethod can also be used to adaptively adjust the value of RE-FRESH_INTERVAL to suit the current mobility condition.

In this comparison, we used QualNet’s implementationof ODMRP, which followed the specification in the Inter-net Draft draft-ietf-manet-odmrp-02.txt [7] butwithout mobility prediction which requires GPS. The valueof REFRESH_INTERVAL was fixed at 3 seconds. Each nodemoved constantly with the predefined speed, which was var-ied from 0 m/s to 20 m/s. The following statistics were col-lected and used in the comparison, where each measurementwill be shown with a 95% confidence interval:

• Packet delivery ratio. The ratio of the number of non-duplicate data packets successfully delivered to the re-ceivers versus the number of packets supposed to be re-ceived. This metric reflects the effectiveness of a protocol.

• Number of total packets transmitted per data packet re-ceived. The ratio of the number of data and control packetstransmitted versus the number of data packets successfullydelivered to the application. HELLO packets are also con-sidered as packets transmitted. This measure shows effi-ciency of a protocol in terms of channel access. The lowerthe number, the more efficient the protocol.

• Number of total bytes transmitted per data byte received.This metric is similar to the second metric except thatnumber of bytes is considered instead. Here, bytes trans-mitted include everything that is sent to the MAC layer(i.e., IP and UDP headers, as well as HELLO packets),where data bytes received involve only the data payloads.This metric presents efficiency of a protocol in terms ofbandwidth utilization. Similar to the second metric, thelower the number, the more efficient the protocol.

Figure 10 presents packet delivery ratio of the protocolsat different mobility speeds. MANSI without the mobility-adaptive mechanism, denoted by MANSI-Basic, shows sig-nificant performance degradation as mobility increases due tothe fact that the forwarding set lacks redundant paths wheneach member and forwarding node always requests only oneof its neighbor to be part of the forwarding set. However,when the mobility-adaptive mechanism is enabled, as denotedby MANSI-Mobile, its results are comparable with ODMRP

Figure 10. Packet delivery ratio as a function of mobility speed.

Figure 11. Total packets transmitted per data packet received at the destina-tions as a function of mobility speed.

and FLOOD. Although the delivery ratio is a bit lower thanthat of the other two protocols, more than 90% of data packetscan be delivered at every mobility speed.

In terms of efficiency, both MANSI-Basic and MANSI-Mobile give significantly better performance than ODMRPand FLOOD at low mobility in both channel access andbandwidth utilization aspects, as shown in figures 11 and12, respectively. The reason is that every multicast senderfloods JOIN QUERY packets periodically in ODMRP, while inMANSI, only the core of the group performs periodic flood-ing. Moreover, ODMRP requires each member to send a JOIN

REPLY toward each sender via the reverse path on which theJOIN QUERY was received, resulting in a fairly large forward-ing group, especially with high number of senders. MANSI,in contrast, has each member establish connectivity towardthe core, which keeps the number of forwarding nodes low.

Without adapting their behaviors to mobility, data forward-ing characteristics of MANSI-Basic, ODMRP, and FLOODremain almost the same regardless of mobility speeds, whereFLOOD employs the highest number of forwarding nodes,and MANSI-Basic uses the least number, but suffers low

58 SHEN AND JAIKAEO

Figure 12. Total bytes transmitted per data byte received at the destinationsas a function of mobility speed.

packet delivery ratio under high mobility. With its mobility-adaptive mechanism, MANSI-Mobile is shown to perform asefficiently as MANSI-Basic at low mobility2 and as ODMRPat high mobility, while yielding consistently high packet de-livery ratio for the entire range of speeds.

5. Related work

There have been numerous multicast routing protocols pro-posed for ad hoc networks. Protocols such as AMRoute [8],AMRIS [12], and MAODV [9] are based on constructinga tree spanning all the group members, where a node canonly accept packets coming from a node with which a treebranch has been established. Since a tree structure providesonly one forwarding path between a pair of sender and re-ceiver, group connectivity may suffer from frequent topologychanges in dynamic networking environments. Other proto-cols such as CAMP [5] and ODMRP [1], including MANSI,employ a mesh-based approach to increase redundancy by al-lowing packets to be forwarded over more than one path, thusgiving a higher chance of successful delivery.

Based on the way they establish and maintain connec-tivity within each multicast group, multicast protocols canalso be broadly classified as either a source-based approachor a group-shared tree/mesh approach. In a source-basedapproach, a multicast tree or mesh is constructed for eachsender. The construction process is usually initiated by asender that floods a request message to all other nodes inthe network so that the other members of the group can es-tablish connectivity via the reverse paths. In ODMRP, eachsender also exploits periodic flooding of control packets torefresh group connectivity and handle mobility. This is suit-able with dense multicast groups but yields high overhead asthe network size and the number of senders increase. In con-trast, a group-shared tree/mesh approach aims to construct atree/mesh for each multicast group, which is shared by all

2 In fact, they behave exactly the same.

senders within the group. A common technique for creatinggroup connectivity is to designate a node in the network therole of the rendezvous point, or the core, for each group. Eachmember then establishes connectivity, often via the shortestpath, to the core, which in turn connects all the group mem-bers together. For each group, one of the members, the firstsender, or any node in the network can take the role of thecore. Examples of ad hoc multicast protocols that are basedon this technique are MAODV, AMRIS, and CAMP. In con-trast, in MANSI, group connectivity can be made more ef-ficient by having some members share common paths to thecore with other members in order to further reduce the to-tal cost of forwarding data packets. Moreover, the forwardingcost may adopt different performance metrics for different ob-jectives, in general, in addition to the number of nodes usedto forward data. MANSI extends a core-based technique byadopting the metaphor of swarm intelligence to learn a bettermulticast connection that yields a lower total forwarding cost.

Swarm intelligence appears in biological swarms of cer-tain insect species. It gives rise to complex and often intel-ligent behavior through simple, unsupervised interactions be-tween a sheer number of autonomous swarm members. Theend result is the emergence of very complex forms of so-cial behavior which fulfill a number of optimization objec-tives and other tasks. Its metaphor has been applied to manycombinatorial optimization problems like the traveling sales-man problem (TSP) and the quadratic assignment problem(QAP). In communications networks, a number of routing andload balancing mechanisms based on swarm intelligence havebeen proposed. Ant-Based Control (ABC) [11] has appliedswarm intelligence to achieve load balancing in telecommu-nications networks. Simulated on a model of the British Tele-com (BT) telephone network, ABC has been shown to resultin fewer call failures than other methods such as shortest-path routing. In [3,4], a distributed adaptive routing for data-gram networks, called AntNet, has been described. Severalvariations of AntNet have been developed but all of themrely on the same concept where forward ants are launchedtoward destinations and backward ants travel back and up-date pheromone along the backward paths. The amount ofadded pheromone is proportional to the goodness of the pathmeasured by the forward ant. The same concept has beenextended and applied to Adaptive Swarm-based DistributedRouting (Adaptive-SDR) [6] for routing in wireless and satel-lite networks, which incorporates a mechanism to clusternodes into colonies so as to resolve the scalability issue inlarge networks.

We exploit the concept of forward and backward ant de-ployment in the MANSI protocol to provide multicast supportfor ad hoc networks. Within a multicast group, each memberlaunches a forward ant in order to find an existing forwardingnode where it can use to establish connectivity to the groupwith lower cost. Once such a node is found, the forward antturns into a backward ant and returns to its origin via the re-verse path, while depositing pheromone along the way to at-tract more future forward ants. To our best knowledge, no ad


hoc multicast routing protocol has been proposed to exploitthe concept of swarm intelligence.

6. Conclusion and future work

Inspired by swarm intelligence, we have introduced an alter-native approach to solving the multicast routing problem inmobile ad hoc networks. Our protocol, called MANSI (Mul-ticast for Ad hoc Networks with Swarm Intelligence), is anon-demand multicast routing protocol that creates a multicastmesh shared by all the members within each group. The pro-tocol uses a core-based scheme, where each member initiatesa request to the core node to establish multicast connectivitywith other members. Intermediate nodes who receive such arequest become forwarding nodes that are used to relay datapackets from one member to the others. Unlike other core-based protocols, MANSI does not always rely on the shortestpaths between the core and the members to establish groupconnectivity. Instead, each member who is not the core peri-odically deploys a small packet that behaves like an ant to op-portunistically explore different paths. This exploring mech-anism enables the protocol to discover paths that comprise abetter set of forwarding nodes yielding a lower total cost ofdata forwarding, where the “cost” of forwarding (nodes) canbe defined in terms of different application specific perfor-mance metrics. MANSI also incorporates a mobility-adaptivemechanism that allows the protocol to remain effective asmobility increases. The simulation results have shown thatMANSI performs both effectively and efficiently in static orlow-mobility environments, yet still effectively in highly dy-namic environments.

Research is in progress to apply MANSI with other objec-tives such as load balancing, energy conservation, and secu-rity.

References

[1] S. Bae, S. Lee, W. Su and M. Gerla, The design, implementation, andperformance evaluation of the on-demand multicast routing protocol inmultihop wireless networks, IEEE Network (Special Issue on Multicas-ting Empowering the Next Generation Internet) 14(1) (2000) 70–77.

[2] T. Ballardie, P. Francis and J. Crowcroft, Core-based trees (CBT): Anarchitecture for scalable inter-domain multicast routing, in: Commu-nications, Architectures, Protocols, and Applications, San Francisco,CA, USA (13–17 September 1993).

[3] G. Di Caro and M. Dorigo, AntNet: A mobile agents approach to adap-tive routing, Technical Report IRIDIA/97-12, Université Libre de Brux-elles, Belgium (1997).

[4] G. Di Caro and M. Dorigo, Two ant colony algorithms for best-effortrouting datagram networks, in: Tenth IASTED Internat. Conf. on Par-

allel and Distributed Computing and Systems (PDCS’98), Las Vegas,NV (28–31 October 1998).

[5] J. Garcia-Luna-Aceves and E. Madruga, The core-assisted mesh proto-col, IEEE Journal on Selected Areas in Communications 17(8) (1999).

[6] I.N. Kassabalidis, M.A. El-Sharkawi, R.J. Marks II, P. Arabshahi andA.A. Gray, Adaptive-SDR: Adaptive swarm-based distributed routing,in: IEEE WCCI 2002, IJCNN 2002 Special Session: Intelligent Sig-nal Processing for Wireless Communications, Honolulu, Hawaii (12–17 May 2002).

[7] S.-J. Lee, W. Su and M. Gerla, On-demand multicast routingprotocol (ODMRP) for ad hoc networks, IETF Internet Draft,http://www.ietf.org/proceedings/00jul/I-D/manet-odmrp-02.txt (2000).

[8] M. Liu, R.R. Talpade and A. McAuley, AMRoute: Adhoc multicastrouting protocol, Technical Report 99, The Institute for Systems Re-search, University of Maryland (1999).

[9] R. Royer and C. Perkins, Multicast using ad-hoc on demand distancevector routing, in: MOBICOM’99, Seattle, WA (August 1999) pp. 207–218.

[10] Scalable Network Technologies, QualNet Simulator, http://www.scalable-networks.com.

[11] R. Schoonderwoerd, O. Holland, J. Bruten and L. Rothkrantz, Ant-based load balancing in telecommunications networks, Technical Re-port HPL-96-76, Hewlett-Packart Laboraties Bristol, Bristol, UK(21 May 1996).

[12] C.W. Wu and Y.C. Tay, AMRIS: A multicast protocol for ad hoc wire-less networks, in: IEEE Military Communications Conf. (MILCOM),Atlantic City, NJ (November 1999) pp. 25–29.

Chien-Chung Shen received his B.S. and M.S. de-grees from National Chiao Tung University, Taiwan,and his Ph.D. degree from UCLA, all in computerscience. He was a research scientist at Bellcore Ap-plied Research working on control and managementof broadband networks. He is now an assistant pro-fessor in the Department of Computer and Informa-tion Sciences of the University of Delaware, and arecipient of NSF CAREER Award. His research in-terests include ad hoc and sensor networks, control

and management of broadband networks, distributed object and peer-to-peercomputing, and simulation.E-mail: [email protected]

Chaiporn Jaikaeo received his B.Eng. degree incomputer engineering from Kasetsart University,Bangkok, Thailand, in 1996, and his M.S. and Ph.D.degrees in computer and information sciences fromthe University of Delaware in 1999 and 2004, re-spectively. He is now a faculty member in the De-partment of Computer Engineering at Kasetsart Uni-versity, Bangkok, Thailand. His research interestsinclude unicast and multicast routing, topology con-trol, peer-to-peer computing, and network manage-

ment for ad hoc and sensor networks.E-mail: [email protected]


Regional Gossip Routing for Wireless Ad Hoc Networks

XIANG-YANG LI, KOUSHA MOAVENINEJAD and OPHIR FRIEDERDepartment of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA

Abstract. Many routing protocols have been proposed for wireless ad hoc networks, and most of them are based on some variants offlooding. Thus many routing messages are propagated through the network unnecessarily despite various optimizations. Gossip basedrouting method has been used and re-investigated to reduce the number of messages in both wired networks and wireless ad hoc networks.However, the global gossiping still generates many unnecessary messages in the area that could be far away from the line between sendernode and receiver node. We propose a regional gossip approach, where only the nodes within some region forward a message with someprobability, to reduce the overhead of the route discovery in the network. We show how to set the forwarding probability based on the regionand the network density both by theoretical analysis and by extensive simulations. Our simulations show that the number of messagesgenerated using this approach is much less than the simple global gossiping method, which already saves many messages compared withglobal flooding. We expect that the improvement should be even more significant in larger networks.

Keywords: gossip, fault tolerance, routing, wireless ad hoc networks

1. Introduction

Recent years saw a great amount of research in wireless net-works, especially ad hoc wireless networks due to its po-tential applications in various situations such as battlefield,emergency relief, and so on. There are no wired infrastruc-tures or cellular networks in ad hoc wireless network. Twonodes can communicate directly if they are within the trans-mission range of the other. Otherwise, they communicatethrough multi-hop wireless links by using intermediate nodesto relay the message. Consequently, each node in the wire-less network also acts as a router, forwarding data packetsfor other nodes. In addition, we assume that each node hasa low-power Global Position System (GPS) receiver, whichprovides the position information of the node itself. If GPSis not available, the distance between neighboring nodes canbe estimated on the basis of incoming signal strengths andthe direction of arrival. Relative co-ordinates of neighboringnodes can be obtained by exchanging such information be-tween neighbors [1].

The devices in the wireless ad hoc networks are often pow-ered by batteries only. Thus, the power supply is limited andit is often difficult to recharge the batteries, which motivatesmany researches in designing power efficient protocols forpower assignment [2–7], topology control [8–14] and rout-ing [15–17]. In addition, the bandwidth available is muchless compared with the wired networks counterpart due to itsunique transmission characteristics. Moreover, since nodescan be mobile, routes may constantly change. Thus, the de-signed routing protocols for wireless ad hoc networks shoulduse as less messages as possible, which will reduce powerconsumption (thus enlong network life), and signal interfer-ence (thus increase the throughput).

One of the key challenges in the design of ad hoc networksis the development of dynamic routing protocols that can ef-ficiently find routes between two communication nodes. In

recent years, a variety of routing protocols [16,18–32], tar-geted specifically for ad hoc environment, have been devel-oped. For the review of the state of the art of routing pro-tocols, see surveys by Royer and Toh [33], by Ramanathanand Steenstrup [34], and by Mauve et al. [35]. Some routingprotocols assume that the each node knows its own positions(e.g., equipped with GPS receivers). These category of pro-tocols are called Location-Aided Routing (LAR) protocols inwhich the overhead of route discovery is decreased by uti-lizing location information. Some protocols do not rely onposition information, and make use flooding (or some vari-ants of flooding). Thus many routing messages are propa-gated through the network unnecessarily despite possible var-ious optimizations. Gossip based routing method has beenused and re-investigated to reduce the number of messages inboth wired networks and wireless ad hoc networks. When-ever a node receives a message, it tosses a coin to decidewhether to forward a message or not in order to reduce thetotal number of routing messages sent by all nodes. However,the global gossiping still generates many unnecessary mes-sages in the area that could be far away from the line betweensender node and receiver node. We propose a regional gossipapproach, where only the nodes within some region forwarda message with some probability, to reduce the overhead ofroute discovery in the network.

The key observation for all gossiping based routing meth-ods is that the gossiping exhibits a bimodal behavior, whichis well-known in the percolation theory [36,37]. This canbe rephrased as follows. Let p be the uniform probabilitythat a node will forward the routing message to its neighbors.Then, there is a threshold value p0 such that, in sufficientlylarge random networks, the gossip message quickly dies outif p < p0 (p is slightly less than p0) and the gossip messagespreads to all network nodes if p > p0 (p is slightly greaterthan p0). In other words, in almost all executions, either al-

62 LI ET AL.

most no node receives the message or almost all of them do.So ideally, we would set the gossiping probability to somevalue slightly larger than p0 to reduce the routing messagespropagated. When the network is sufficiently large, we can setp sufficiently close to p0, thus save about (1−p0)n messagesoverhead compared with the flooding, since about p0n nodeswill forward the message in gossiping based method com-pared with n nodes forwarding in flooding. Hass et al. [24]conducted extensive simulations to investigate the extent towhich this gossiping probability can be lowered. They foundthat using gossiping probability between 0.6 and 0.8 sufficesto ensure that almost every node gets the message in almostevery routing. They report of up to 35% fewer messages thanflooding (close to our previous explanation). Notice that theirexperimental setting of the network has some special config-urations [24].

Although gossiping reduces the routing messages com-pared with flooding, it still produces lots of unnecessary mes-sages in regions that are far from the line between sendernode and receiver node. Notice that, the traditional gossipwill propagate the message to the whole network. To furtherreduce the number of forwarding messages, we propose re-gional gossiping, in which essentially only nodes inside someregion (derived from the source and target) will execute thegossiping protocol, and nodes outside the region will not par-ticipate in the gossiping at all. The region we select in oursimulations are some ellipses using the source and target asfoci. Notice that here we assume source node knows eitherthe exact or the approximate location of the destination node,we will discuss this later in section 2 in detail. We also dy-namically adjust the forwarding probability based on the nodedensity estimated by the current node. Our results show that,by using appropriate optimization heuristics, we can save upto 94% messages even compared with the global floodingmethod.

The remaining of this paper is organized as follows. In sec-tion 2, we review some known location services techniquesfor wireless ad hoc networks. We study our regional gossipmethod in detail in section 3. We demonstrate its effective-ness by both theoretical study and extensive simulations insection 4 . We also study the effectiveness of the regional gos-siping on constructing multiple paths for any pair of sourceand destination nodes in section 5. We conclude our paperand discuss possible future research directions in section 6.

2. Preliminaries

We consider a wireless ad hoc network (or sensor network)with all nodes distributed in a two-dimensional plane. As-sume that all wireless nodes have distinctive identities andeach static wireless node knows its position information1 ei-ther through a low-power Global Position System (GPS) re-ceiver or through some other way. For simplicity, we also as-

1 More specifically, it is enough for our protocol when each node knows therelative position of its one-hop neighbors. The relative position of neigh-bors can be estimated by the direction of arrival and the strength of signal.

sume that all wireless nodes have the same maximum trans-mission range and we normalize it to one unit. Throughoutthis paper, a broadcast by a node u means that u sends themessage to all nodes within its transmission range. Noticethat, in wireless ad hoc networks, the radio signal sent outby a node u can be received by all nodes within the trans-mission range of u. The main communication cost in wire-less networks is to send out the signal while the receiving andprocessing costs of a message is neglected here.

2.1. Location service

Several proposed routing algorithms [18,22] assume that thesource node knows the position information (or approximateposition) of the destination node. Our regional gossip methodalso assumes that the source node knows the current posi-tion information of the target approximately. Notice that, forsensor networks collecting data, the destination node is oftenfixed, thus, location service is not needed in those applica-tions. However, the help of a location service is needed inmost application scenarios. Mobile nodes register their loca-tions to the location service. When a source node does notknow the position of the destination node, it queries the loca-tion service to get that information. In cellular networks, thereare dedicated position severs. It will be difficult to implementthe centralized approach of location services in wireless ad-hoc networks. First, for centralized approach, each node hasto know the position of the node that provides the locationservices, which is a chicken-and-egg problem. Second, thedynamic nature of the wireless ad hoc networks makes it veryunlikely that there is at least one location server available foreach node. Thus, we will concentrate on distributed locationservices.

For the wireless ad hoc networks, the location service pro-vided can be classified into four categorizes: some-for-all,some-for-some, all-for-some, all-for-all. Some-for-all servicemeans that some wireless nodes provide location services forall wireless nodes. Other categorizations are defined simi-larly.

An example of all-for-all services is the location servicesprovided in the Distance Routing Effect Algorithm for Mo-bility (DREAM) by Basagni et al. [38]. Each node stores adatabase of the position information for all other nodes in thewireless networks. Each node will regularly flood packetscontaining its position to all other nodes. A frequency of theflooding and the range of the flooding is used as a control ofthe cost of updating and the accuracy of the database.

Using the idea of quorum developed in the databases anddistributed systems, Hass and Liang [39] and Stojmenovic[40] developed quorum based location services for wirelessad-hoc networks. Given a set of wireless nodes V , a quorumsystem is a set of subset (Q1, Q2, . . . ,Qk) of nodes whoseunion is V . These subsets could be mutually disjoint or oftenhave equal number of intersections. When one of the nodesrequires the information of the other, it suffices to query onenode (called the representative node of Qi ) from each quo-rum Qi . A virtual backbone is often constructed between

REGIONAL GOSSIP ROUTING 63

the representative nodes using a non-position-based methodssuch as [41–44]. The updated information of a node v is sentto the representative node (or the nearest if there are many) ofthe quorum containing v. The difficulty of using quorum isthat the mobility of the nodes requires the frequent updatingof the quorums. The quorum based location service is oftensome-for-some type.

The other promising location service is based on thequadtree partition of the two-dimensional space [45]. It di-vides the region containing the wireless network into hierar-chy of squares. The partition of the space in [45] is uniform.However, we notice that the partition could be non-uniformif the density of the wireless nodes is not uniform for someapplications. Each node v will have the position informationof all nodes within the same smallest square containing v.This position information of v is also propagated to up-layersquares by storing it in the node with the nearest identity to v

in each up-layer square containing v. Using the nearest iden-tity over the smallest identity, we can avoid the overload ofsome nodes. The query is conducted accordingly. It is easy toshow that it takes about O(log n) time to update the locationof v and to query another node’s position information.

If the location service is not provided, the nodes can cachethe location information of some other nodes. When thesource node wants to send a message to the target, it directlyuses the region gossip if the target location is known. Oth-erwise, it will use flooding (with selective forwarding [46]to control the number of messages sent) to send the messageto all nodes within k hops, where k is a parameter to be set.Then if a node within k hops knows the destination location,that node then starts the regional gossip to send message tothe destination.

2.2. Random deployment and connectivity

Energy conservation is critical for the life of the wireless net-work. One approach to save energy is to use the minimumpower to transmit the signal without disconnecting the net-work. The universal minimum power used by all wirelessnodes, such that the induced network topology is connected,is called the critical power. Determining the critical powerfor static wireless ad hoc networks is well-studied [5,7,13]. Itremains to study the critical power for connectivity for mo-bile wireless networks. As the wireless nodes move around, itis impossible to have a unanimous critical power to guaranteethe connectivity for all instances of the network configuration.Thus, we need to find a critical power, if possible, at whicheach node has to transmit to guarantee the connectivity of thenetwork almost surely, i.e., with high probability almost one.

The wireless nodes are randomly deployed in majoritywireless ad hoc networks either due to its massive number,due to its emergency requirement, or due to harsh environ-ment. For simplicity, we assume that the n wireless devicesare distributed in a unit area square (or disk) according tosome distribution function, e.g., random uniform distribution,denoted by Xn, or Poisson process, denoted by Pn.

Let G(V, r) be the graph defined on V with edges uv ∈ E

if and only if ‖uv‖ � r where ‖uv‖ is the Euclidean dis-tance between nodes u and v. Let G�(Xn, rn) be the set ofgraphs G(V, rn) for n nodes V that are uniformly and in-dependently distributed in a two-dimensional region �. Theproblem considered by Gupta and Kumar [5] is then to deter-mine the value of rn such that a random graph in G�(Xn, rn)

is asymptotically connected with probability one as n goes toinfinity, when � is a unit disk. Specifically, they showed thatG(V, rn) is connected almost surely if nπr2

n � ln n + c(n)

for any c(n) with c(n) → ∞ as n goes to infinity, andG(Xn, rn) is asymptotically disconnected with positive prob-ability if nπr2

n = ln n + c(n) and lim supn c(n) < +∞. It isunknown whether the same result holds if the geometry do-main in which the wireless nodes are distributed is a unit-areasquare instead of the unit-area disk.

Independently, Penrose [47] showed that the longest edgeMn of the minimum spanning tree of n points randomly anduniformly distributed in a unit area square C satisfies that

limn→∞ Pr

(nπM2

n − ln n � α) = e−e−α

,

for any real number α. This result gives the probability of thenetwork to be connected if the transmission radius is set as apositive real number r when n goes to infinity. For example,if we set α = ln ln n, we have

Pr(nπM2

n � ln n + ln ln n) = e−1/ lnn.

It implies that the network is connected with probability atleast e−1/ ln n if the transmission radius rn satisfies nπr2

n =ln n + ln ln n. Notice that e−1/ ln n > 1 − 1/ ln n from e−x >

1 − x for x > 0. By setting α = ln n, the probability thatthe graph G(V, rn) is connected is at least e−1/n > 1 − 1/n,where nπr2

n = 2 ln n. Notice that the above probability isonly true when n goes to infinity. When n is a finite number,then the probability of the graph being connected is smaller.In [48], Li et al. presented the experimental study of the prob-ability of the graph G(V, rn) being connected for finite num-ber n.

Gupta and Kumar [5] conjectured that if every node hasprobability p of being fault, then the transmission range forresulting a connected graph satisfies pπr2

n = log n/n. Thiswas recently confirmed by Wan et al. [49]. It is not difficultto see that whether the global gossip can deliver the packetis related to whether a set of randomly deployed nodes in aregion form a connected graph when each node has a uniformfaulting probability p. Consequently, given a wireless net-work with n nodes , each with transmission range r , the relayprobability of a gossip routing protocol is p = log n/(πnr2

n),when n goes to infinity. We conjecture that this is true for anynon-flat convex region �.

2.3. Fault tolerance and security

Fault tolerance is one of the central challenges in designingthe wireless ad hoc networks. To make fault tolerance pos-sible, first of all, the underlying network topology must have

64 LI ET AL.

multiple disjoint paths to connect any two given wireless de-vices. Here the path could be vertex disjoint or edge disjoint.Considering the communication nature of the wireless net-works, the vertex disjoint multiple paths are often used in theliterature. A graph is called k-vertex connected (k-connectedfor simplicity) if, for each pair of vertices, there are k mu-tually vertex disjoint paths (except end-vertices) connectingthem. A k-connected wireless network can sustain the failureof k − 1 nodes.

The connectivity of random graphs, especially the geomet-ric graphs and its variations, have been considered in the ran-dom graph theory literature [50], in the stochastic geometryliterature [47,51–54], and the wireless ad hoc network litera-ture [2,5,55–61].

Penrose [53] showed that a graph of G(Xn, r) becomesk-connected almost surely at the moment it has minimum de-gree k. However, this does not mean to guarantee a graph overn points is k-connected almost surely, we only have to con-nect every node to its k nearest neighbors. Let V be a set ofn points randomly and uniformly distributed in a unit square(or disk). Xue and Kumar [61] proved that, to guarantee thata geometry graph over V is connected, the number of nearestneighbors that every node has to connect must be asymptot-ically �(ln n). Dette and Henze [51] studied the maximumlength of the graph by connecting every node to its k nearestneighbors asymptotically. For the unit volume sphere, theirresult implies that, when k > 2,

limn→∞ Pr

(nπr2

n,k � ln n + (2k − 3) ln ln n − 2 ln(k − 1)!− 2(k − 2) ln 2 + ln π + 2α

) = e−e−α

.

Li et al. [48] showed that, given n random points V overa unit-area square, to guarantee that a geometry graphover V is (k + 1)-connected, the number of nearest neigh-bors that every node has to connect is asymptotically �(ln n+(2k−1) ln ln n). Li et al. [48] derived a tighter bound on rn fora set V of n two-dimensional points randomly and uniformlydistributed in C such that the graph G(V, rn) is k-connectedwith high probability.

The theoretical value of the transmission ranges gives usinsight on how to set the transmission radius to achieve thek-connectivity with certain probability. These results also ap-ply to mobile networks when the moving of wireless nodesalways generate randomly (or Poisson process) distributednode positions. Bettstetter [2] conducted the experiments tostudy the relations of the k-connectivity and the minimumnode degree using toroidal model. Li et al. [48] also con-ducted experiments to study the probability that a graph hasminimum degree k and has vertex connectivity k simultane-ously using Euclidean model. Recently, Bahramgiri et al. [8]showed how to decide the minimum transmission range ofeach node such that the resulted directed communicationgraph is k-connected. Here it assumes that the unit disk graphby setting each node with the maximum transmission rangeis k-connected. Lukovszki [62] gave a method to construct aspanner that can sustain k nodes or k links failures.

3. Regional gossip

Although gossiping reduces the routing messages comparedwith flooding, it still produces lots of unnecessary messagesin regions that are far away from the line between the sourceand the target node. Notice that, the traditional gossip willpropagate the message to the whole network. To further re-duce the number of forwarding messages, we propose re-gional gossiping, in which essentially only nodes inside someregion (derived from the source and target) will execute thegossiping protocol, and nodes outside the region will not par-ticipate the gossiping at all. The region we select in our sim-ulations are some ellipses using the source and target as foci.

We now describe our regional gossiping routing methodin detail. Assume that wireless mobile hosts are a set V ofn points distributed in a two-dimensional space. Each nodehas a fixed transmission range r: all nodes within distance r

to a node v can receive the signal sent by v. Thus, all mo-bile hosts define a communication graph G(V, r) in whichthere is an edge uv iff ‖uv‖ � r . From now on, we alsoassume that the source node knows the position of the targetnode, the global ellipse factor �, in addition to its own posi-tion. Every mobile host can get its own position through alow-cost GPS. In many applications such as data-centric sen-sor network, there is only a fixed number of destination nodes(called sink), which is often static, thus every node knows thepositions of these possible target nodes. Otherwise, locationservice is needed to find the location of the destination node.The geometry information of the source node and the destina-tion node and also the current route (i.e., the route from sourceto the sender of the message) is piggybacked along with themessage packet. When a node, say v, receives a message, itretrieves the geometry position of the source node and the tar-get node. Node v then checks if it is inside the ellipse definedby using the source point s and the destination point t as foci.Notice that, a node v is inside this ellipse iff

‖vs‖ + ‖vt‖ � �‖st‖,which can be checked trivially. When a node is not inside theellipse, it will just simply discard this message. Otherwise,with a fixed probability p, the node forwards this message toall nodes within its transmission range. Hereafter, we call p

the relay probability and � the ellipse factor of our regionalgossiping method. Obviously, the probability that the desti-nation node receives the message depends on the relay prob-ability p, the ellipse factor �, the number of nodes n, and thetransmission range r .

Gupta and Kumar [5] showed that a random graph G(V, r)

is connected whenever r is larger than some thresholdvalue rn. It is known that the global gossiping (by simply set-ting � to ∞) exhibits some bimodal behavior: the destinationnode receives the message if and only if the relay probabilityis larger than some threshold value. We expect our regionalgossiping method to have the similar transmission phenom-ena.

We then estimate the relay probability for a network of n

nodes. It was shown in [49] that given n wireless nodes dis-


tributed in a unit square and each node has transmission rangern and being off or fault with probability p, then the networkis connected with high probability if pnπr2

n � 2 ln n. Con-sider the network of n nodes distributed in a square regionwith side length a. Assume that the distance between thesource and the target is d and the ellipse factor is �. Thenumber of nodes inside the ellipse is then about

Nd = n

a2 · π�√

�2 − 1

4d2.

Since each node inside the ellipse forwards the message withprobability p after it receives the message, to let the targetreceive the message almost surely, the subnetwork composedof the nodes inside the ellipse with fault probability p mustbe connected. In other words, the relay probability in ourregional gossiping is at least

p � ln Nd + c(Nd)

Ndπ(r/a)2 .

Here r is the transmission range of each wireless node andc(Nd) is a number going to ∞ when Nd goes to ∞. Theprobability that the network (each node is chosen with prob-ability p) is connected is e−e−c(Nd )

. Substituting in Nd , wehave

p � 4a4 ln(nπ�√

�2 − 1d2/(4a2))

π2d2r2�√

�2 − 1 · n= ln(nπ�2d2/4)

nπ2�2d2r2/4.

Here �2 = �√

�2 − 1, d = d/a, and r = r/a. Since for arandom pair of source and target nodes, d �

√2a, we have

p � ln(nπ�2/4)

nπ2�2r2/4.

For example, consider a network of n = 1000 nodes distrib-uted in a square of side length a = 15, and each node hastransmission range r = 1. For ellipse factor � = 1.2, we cancalculate the relay probability p such that the regional gossip-ing routing can deliver the packets almost surely as

p � ln(nπ�2/4)

nπ2�2r2/4= 0.74.

The actual relay probability should be larger since we omitthe number c(Nd) here, which actually decides the successprobability of the regional gossiping. The percentage of allvertices involved is at most

p · Nd/n = ln(nπ�2d2/4)

πr2 · n � 0.46.

Since the distance d between most pairs of source and target issmall compared with a, the actual number of involved verticesis much smaller. Let Pd be the probability that a pair of sourceand target has distance d . The average percentage of numberof vertices (for all source and target pairs) is actually

∫ a

x=0 p ·NxPx/n dx. It is not difficult to show that the percentage ofvertices involved in regional gossiping is at most pNd/2n =

0.23. When the ellipse factor � = ∞, we can estimate therelay probability of the regional gossiping as

p � ln n

nπr2= 0.495.

The actual relay probability should be larger, so do the per-centage of vertices involved in global gossiping. The exper-iments discussed in the following sections verify the abovestudy.

4. Experimental studies

4.1. Simulation environment

We conducted extensive simulations to study the performanceof our region gossiping method. We model the network byunit disk graph and the mobile hosts are randomly placedin a square region. We tried unit disk graphs with differentnumber of vertices that are randomly placed in a 15 × 15square. Notice that the density of the graph must be abovesome threshold to see the effectiveness of the algorithm oth-erwise the properties would be hidden and cannot be seen. Inother words, the algorithm works better for dense graphs thansparse graphs with the same parameters p and �.

There are different parameters involved in our simulations,which are described as follows:

Number of vertices. We tried graphs with 1000, 1500 and2000 vertices. For convenience, we use n to denote the num-ber of vertices.

Ellipse factor. In each iteration of the simulation, the sourcevertex and the target vertex are the foci of an ellipse with el-lipse factor � chosen from 1.2, 1.4, 1.6, 1.8 and 2. We alsoconsider the case where the ellipse factor � is ∞ which is justthe traditional global gossiping method. The smaller the el-lipse factor is, the narrower the ellipse will be. Notice thatellipse factor must be greater than one.

Transmission range. Remember that to make the graphG(V, r) connected, the transmission range has to be greaterthan some threshold value rn. To study the effect of the graphdensity on the delivery rate, we tried different values of trans-mission range: 1, 1.5, 2, 2.5 and 3. From the result by Guptaand Kumar [5], given 1000 nodes in a 15 × 15 square, thetransmission range should be at least about 0.7 to guarantee aconnected network G(V, r) theoretically.

Relay probability. In our simulation, we use different relayprobabilities p. First, we use the relay probabilities p from0.1 to 1.0 with step 0.1 and we find that, when the networkis dense enough, the transmission phenomenon happens be-tween two intervals of relay probabilities. To study this trans-mission phenomenon in detail, we further refine our relayprobabilities. Specifically, we conduct further simulations us-ing relay probabilities from 0.02 to 0.30 with step 0.02.

66 LI ET AL.

Beside the above parameters there are two more constantmetrics used in our simulations as follows:

Source-target pairs. To compute the exact value of the av-erage delivery rate, we have to try all possible pairs for eachgraph, which is n·(n−1), where n is the number of vertices. Itis not feasible to test all pairs when n is large. Instead we ran-domly select 100 pairs for each graph and conduct regionalgossiping based routing for each pair. Although we are nottesting all possible pairs, choosing 100 random pairs wouldgive the results close enough to exact values.

Number of try’s. The delivery probability (called deliveryrate also) of our regional gossiping method for a pair of nodesis defined as the probability that the destination node receivesthe message. To compute the delivery rate, we tried sendingthe message 1000 times for each pair and then the deliveryrate is approximated by the total number of times that themessage reached the target divided by the total number thatthe message is sent (which is 1000 in out simulations).

There are four different types of nodes in each iteration ofour simulations:

(1) Not in ellipse. Nodes that are out of the ellipse region.

(2) Blocked. Nodes that receive the message and do not relayit.

(3) Relayed. Nodes that receive and relay the message.

(4) Initial hops nodes. The nodes within the initial hops al-ways receive the message and from those, the ones whosedistance from source is less than some fix initial hops pa-rameter, always relay the message. Other nodes inside theellipse relay the message with the given relay probability.

Here we want to involve as little nodes as possible. In otherwords, we want to minimize the number of nodes that relay

the message. It is important because sending message con-sumes energy and energy is a bottleneck for wireless nodes.

In all the figures of this paper the Y-axis is either the mes-sage delivery rate or the percentage of vertices that are in-volved in message delivery, and the X-axis is one of the pa-rameters with respect to another parameter which is shownin the legend and the remaining two parameters are fixed.For example, we can show message delivery rate as a func-tion of relay probability p for different values of ellipse fac-tor �, while the transmission range r and the number of ver-tices n are fixed (see figure 1). Each point in each figure rep-resents the average of the 100, 000 iterations since we willtest 100 different source-target pairs, and each pair is tested1000 times, when all four parameters are fixed.

We believe that the relay probability and the graph densityare two major factors of message delivery rate. On the otherhand, the ellipse factor and the relay probability are the majorfactors determining the number of vertices that are involvedin message delivery. Here a node is said to be involved if itrelays the message. In other words, when the Y-axis is themessage delivery rate and X-axis is either relay probability,number of vertices or transmission range, we expect to see ajump in the figures. It means that when the X-axis exceedssome threshold, then the regional gossiping method almostsurely guarantees that the message arrives at the target. Whenthe X-axis is less than some threshold, the target almost nevergets the message.

4.2. Message delivery rate as a function of relay probability

We first conduct extensive simulations to study the effect ofthe relay probability on the message delivery rate. Intuitively,if we increase the relay probability, the message delivery rateincreases. Besides the relay probability, we vary either the el-lipse factor �, or the number of vertices n, or the transmissionrange r . Now we discuss them one by one as follows.

(a) (b)

Figure 1. Message delivery rate as a function of relay probability for different values of ellipse factor. Here transmission range is 1. (a) Number of verticesis 1000. (b) Number of vertices is 2000.


1. Message delivery rate as a function of relay probability fordifferent values of ellipse factor. As can be seen in figure 1,when the probability exceeds some threshold the delivery ratejumps from near 0% to near 100%. In each figure, this thresh-old decreases as the ellipse factor increases because the biggerthe ellipse factor is, the more vertices contribute in messagedelivery, and consequently, the probability of the message toreach the target, which is nothing but the message deliveryrate, increases. For both figures the transmission range isfixed to 1 unit and the number of vertices is also fixed to 1000and 2000, respectively.

From figure 1, we observe that when the graph becomesdenser (more vertices in this case), the curve jumps earlier,and the reason is each time a vertex relays the message,more nodes get the message (due to more neighbors in densegraphs) so the probability that the message reaches the targetincreases.

One important observation is as follows: as we increasethe ellipse factor, the message delivery rate does not increaseproportionally. Surprisingly, when the ellipse factor is around1.8, the message delivery rate is almost as good as the oneusing global gossiping (i.e., the ellipse factor constraint isrelaxed to ∞). The reason is where a bigger ellipse factoris used we are actually considering the vertices that are lesshelpful than the vertices which are already considered. Intu-itively, the vertices, which are far away from the line connect-ing the source and target, do not help improving the messagedelivery rate.

We also observe that, for a fixed relay probability, when thegraph is dense, even a narrow ellipse could guarantee a goodrate of message delivery. Achieving the same delivery rateusing the same relay probability, for a sparser graph, mightnot be possible, even if the ellipse factor is relaxed to infin-ity. In other words, the ellipse factor does not compensate thedescription of the graph density. For example, in figure 1(b),

when the relay probability is 0.3 with ellipse factor of 1.4, thedelivery rate is about 95% for n = 2000, while we cannotachieve this rate when n = 1000 (see figure 1(a)).

2. Number of nodes involved in message delivery as a func-tion of relay probability for different values of ellipse factor.So far, we have concentrated on the transition phenomena ofthe delivery rate over the relay probability. Not only the deliv-ery rate is important for the network performance, but also thenumber of vertices involved in the message delivery is impor-tant for the network life since the wireless devices are oftenpowered by the batteries only.

The challenge is to find an ellipse factor and a relay prob-ability such that not only the delivery rate is high (close to100%) but also the number of vertices involved in the messagedelivery is as small as possible. Actually the ellipse factorand the number of vertices involved in sending the messagefrom source to target, work against each other. It means thatif we choose a bigger ellipse factor, a higher delivery rate isachieved, on the other hand, lots of vertices will be involvedin route discovery. In reverse, if we choose a small ellipsefactor then fewer vertices will be involved but it may not havea good delivery rate.

As can be seen in figure 2, the relation between the num-ber of vertices involved and the relay probability with respectto ellipse factors is close to linear. The bigger the relay prob-ability, the more number of vertices will be involved in themessage delivery. The exact relation between the number ofvertices and relay probability is not simple. Clearly, the far-ther it is from the source, the less probability it will get thethe message to relay.

In figure 2 when the ellipse factor is infinity, we are ac-tually flooding the network with a uniform relay probability,and when this relay probability is 1, the network is completely

(a) (b)

Figure 2. Number of nodes involved in message delivery as a function of relay probability for different values of ellipse factor. Here transmission range is 1.(a) Number of vertices is 1000. (b) Number sof vertices is 2000.

68 LI ET AL.

flooded, i.e., traditional flooding, so all nodes have the chanceto contribute in message delivery.

Assume that we want to have the delivery rate more than99%, first consider the case in which we have 1000 nodes,illustrated in figures 1(b) and 2(b).

We build the table 1 as follows: for each ellipse factor, wecan find the needed relay probability to guarantee the messagedelivery above 99% from figure 1, and then by knowing thevalues of ellipse factor and the relay probability we can findthe percentage of vertices that are involved from figure 2.

For example, to achieve this rate (above 99%) when ellipsefactor is 1.2, the relay probability must be at least 0.9 (see fig-ure 1). Then having these two values fixed, we can find thenumber of nodes that are involved from figure 2, which wouldbe about 15% of all vertices. Doing the same thing for differ-ent values of ellipse factor, we get table 1.

The first column is the different ellipse factors we simu-lated, and the second column is the corresponding relay prob-ability in our regional gossip method to guarantee this fixeddelivery rate 99%, and the third column is the percentage ofvertices that are involved in our regional gossiping (i.e., re-laying the message).

Table 1 shows that we could involve only 15% of verticesto guarantee the message delivery rate above 99% when theellipse factor is 1.2. If we do the same calculations wherethere are 2000 nodes then only 10% of vertices will be in-

Table 1Percentage of the vertices involved in message delivery.

Ellipse factor Relay probability Vertices involved (%)

1.2 0.9 151.4 0.8 221.6 0.7 251.8 0.7 30

infinity 0.7 70

volved (see figures 1 and 2) by choosing ellipse factor 1.2and relay probability 0.5.

So far the transmission rang was fixed to 1. We were mo-tivated to study the effect of transmission range as well. Wethen tried different values of transmission range. Obviouslythe larger the transmission range is, the denser the graph willbe and as mentioned before that causes the jump to occur ear-lier.

In figure 3 the transmission range is 2. See how similarfigure 1 and figure 3 are, the only difference between thesetwo figures is the probability at which the jump occurs forany fixed ellipse factor. Since in delivery rate happens earlierand quicker when the transmission range increases, we plotthe figures using relay probability range [0, 0.3] for r = 2,instead of [0, 1] for r = 1.

Again assume that we want to have the delivery rate morethan 99%. Consider the case in which we have 1000 nodes,but the transmission range is 2 (figures 3(a) and (b)).

We build table 2 as we built table 1: for each ellipse factor.We can find the relay probability that guarantees the messagedelivery rate above 99% from figure 3(a), and then by know-ing the values of ellipse factor and the relay probability wecan find the percentage of vertices involved in message deliv-ery from figure 3(b).

For example, to achieve this rate (above 99%) when el-lipse factor is 1.2, the relay probability must be 0.3 (see fig-ure 3(a)). Then having these two values fixed, we can find thenumber of nodes involved from figure 3(b), which would beabout 8%. Doing the same thing for different values of ellipsefactor, we get table 2.

Table 2 shows that we could involve only 8% of verticesto guarantee the message delivery rate above 99% for net-works of 1000 nodes and with transmission range equal to 2.If we do the same calculations for networks of 2000 nodes

(a) (b)

Figure 3. (a) Message delivery rate as a function of relay probability for different values of ellipse factor. Here number of vertices is 1000 and transmissionrange is 2. (b) Number of nodes involved in message delivery as a function of relay probability for different values of ellipse factor. Here number of vertices

is 1000 and transmission range is 2.


with transmission range equal to 2, then only 6% of verticeswill be involved (figures are not shown here).

3. Message delivery rate as a function of relay probabilityfor different values of transmission range. So far we plottedthe message delivery rate as a function of relay probability fordifferent values of ellipse factor. Let us replace the ellipse fac-tor parameter with transmission range and see how the graphbehaves.

As you can see in figure 4, transmission range plays a veryimportant role in message delivery (see how far the curves arefrom each other). As the transmission range is increased, thedelivery rate improves significantly as opposed to the situa-tion we had earlier with ellipse factor. The reason is when thetransmission range is bigger then each node will be connectedto more nodes, in other words the graph density increases.Thus, each time a node relays the message, more nodes willget it and the probability that the message dies out becomessmaller. Here in figure 4, the ellipse factor is fixed to 1.6.

We built table 3 as follows: for each transmission range,we can find the relay probability that guarantees the mes-sage delivery rate above 99% from figure 4(a), and then byknowing the values of transmission range and the relay prob-ability we can find the percentage of vertices involved fromfigure 4(b).

For example, to achieve this rate (above 99%) when trans-mission range is 1, the relay probability must be at least 0.8,

(see figure 4(a)). Then having these two values fixed, wecan find the percentage of vertices involved from figure 4(b),which would be about 30%. We get table 3 by doing the samecalculation for different values of transmission range.

Table 3 illustrates the number of vertices involved in theregional gossip routing to guarantee a fixed delivery rate 99%for networks of 1000 nodes with ellipse factor 1.6. Observethat, all these curves intersect in a common point when the re-lay probability is 1. Because the ellipse factor is fixed, chang-ing the transmission range does not change the number ofnodes that are inside ellipse, which is total number of verticesinvolved in message delivery when the relay probability is 1.Actually it is possible to have a node in the ellipse which doesnot contribute in message delivery even when the relay prob-ability is 1, but that is very unlikely. It happens only whena node in the ellipse doesn’t have any neighbor inside the el-lipse. In our simulations this scenario happened 2 times outof 180,000,000 iterations.

Another observation is that we get different curves for dif-ferent transmission ranges. Typically, when the transmissionrange is larger, more nodes inside this ellipse will be involvedin the message delivery.

4. Message delivery rate as a function of relay probabilityfor different number of nodes. In our simulations we stud-ied networks with different densities in two different ways.


Ellipse factor Relay probability Vertices involved (%)

1.2 0.3 81.4 0.24 111.6 0.22 131.8 0.20 14

infinity 0.20 15


Transmission range Probability Vertices involved (%)

1.0 0.8 301.5 0.5 202.0 0.3 142.5 0.14 123.0 0.11 11.71

(a) (b)

Figure 4. (a) Message delivery rate as a function of relay probability for different values of transmission range. Here number of vertices is 1000 and theellipse factor is 1.6. (b) Number of nodes involved in message delivery as a function of relay probability for different values of transmission range. Here

number of vertices is 1000 and ellipse factor is 1.6.

70 LI ET AL.

(a) (b)

Figure 5. (a) Message delivery rate as a function of relay probability for different number of nodes. Here ellipse factor is 1.6 and transmission range is 1.(b) Number of nodes involved as a function of relay probability for different number of nodes. Here ellipse factor is 1.6 and transmission range is 1.

First, as described in the previous section, we studied net-works with fixed number of vertices and different transmis-sion ranges. Now we study networks with fixed transmissionrange and different number of vertices placed in a 15 × 15square. In both cases we expect the similar results if the net-work densities are similar.

As you can see in figure 5, the number of vertices playsan important role in message delivery (see how far the curvesare from each other). Here we have the same reasoning as theprevious section. As the number of vertices is increased, thedelivery rate improves significantly. The reason is when thereare more vertices in the same area, the graph becomes denser.Thus, each time a node relays the message more nodes willget it and the probability that the message dies out becomessmaller.

Now let us look at the percentage of nodes that are involvedin message delivery as a function of relay probability for dif-ferent number of nodes (see figure 5). Remember that in thiscase ellipse factor and transmission range are fixed. Here wehave the same ellipse with different number of vertices in-side them. When there are more vertices in the same areathe message is delivered with higher probability since morenodes will relay the message. Notice that, given a fixed relayprobability, when the node density exceeds some threshold(depending on the relay probability) almost all nodes insidethe ellipse will receive the message, thus, have the chance torelay the massage. In other words, if the relay probability islow, high message delivery rate still can be achieved if thegraph is dense enough and if the graph is sparse, high mes-sage delivery rate still can be achieved by increasing the relayprobability. On the other hand, larger relay probability willinvolve more nodes in message delivery (the number of nodesinvolved is almost linear to the relay probability as shown inright figure of figure 5).

4.3. Message delivery rate as a function of ellipse factor

We can look at the problem from a totally different point ofview. So far we have concentrated on the transition phenom-ena of the delivery rate over the relay probability. In otherwords, in all figures the X-axis was the relay probability. Nowlet us see how the network behaves if we use different ellipsefactors while some other parameters are fixed. We found that,regardless of the network density and relay probability, in-creasing the ellipse factor does not improve the message de-livery rate significantly.

1. Message delivery rate as a function of ellipse factor fordifferent values of transmission range. First let us fix the re-lay probability and the number of vertices. Remember that tochange the message delivery rate dramatically we can eitherincrease the relay probability or increase the network density.As can be seen in figure 6 there is no jump. In other words,increasing the ellipse factor does not improve the message de-livery rate dramatically.

Figure 6 shows when the relay probability is fixed, regard-less of the value of ellipse factor, the graph density must beabove some threshold to guarantee a high message delivery.As you can see in figure 6(a) when the transmission rangeis less than 1.5 then the delivery rate is always below 20%even if the ellipse factor constraint is relaxed (the case whereellipse factor constraint is relaxed and not shown in figure 6).

As it is expected if we set the relay probability to a highervalue then the delivery rate would be higher. This is illus-trated in figure 6: if we increase the value of the relay prob-ability (from figure 6(a) to figure 6(b)) all curves will beshifted up.

2. Message delivery rate as a function of ellipse factor fordifferent number of vertices. As mentioned earlier, the net-work density can be increased either by increasing the trans-mission range or by increasing the number of vertices. Now


(a) (b)

Figure 6. Message delivery rate as a function of relay probability for different values of transmission range. Here number of vertices is 1000. Relay probabilityis (a) 0.1, (b) 0.3.

(a) (b)

Figure 7. (a) Message delivery rate as a function of ellipse factor for different number of vertices. Here transmission range is 1 and relay probability is 0.3.(b) Number of nodes involved in message delivery as a function of ellipse factor for different number of vertices. Here transmission range is 1 and relay

probability is 0.3.

we replace the transmission range of the previous section withnumber of vertices and we expect similar results. In otherwords, let us fix the relay probability and the transmissionrange to see the delivery rate as a function of ellipse factor fordifferent number of vertices.

Again, as can be seen in figure 7 there is no jump. In otherwords, increasing the ellipse factor does not improve the mes-sage delivery rate dramatically.

3. Message delivery rate as a function of ellipse factor fordifferent values of relay probability. In the previous two sec-tions, we studied the effect of ellipse factor in networks withdifferent densities, in this section instead of changing the net-work density, we change the relay probability. Thus, in thissection, the network density is fixed. Specifically, we study

the message delivery rate (as a function of ellipse factor fordifferent values of relay probability) by fixing the number ofnodes and the transmission range.

In figure 8 when the relay probability is below somethreshold, a high delivery rate cannot be achieved even whenthe ellipse factor constraint is relaxed. Figure 8 is similar tofigures 6 and 7 due to the fact that a high relay probabil-ity can compensate the sparseness of the network and viceversa.

Intuitively, all the discussions of the two previous sectionsapply to this section too. For example, when the networkdensity is larger than some threshold, the number of verticesinvolved is almost linear to the ellipse factor, see figures 7and 8.

72 LI ET AL.

(a) (b)

Figure 8. (a) Message delivery rate as a function of ellipse factor for different values of relay probability. Here transmission range is 1 and number of verticesis 1000. (b) Number of nodes involved in message delivery as a function of ellipse factor for different values of relay probability. Here transmission range

is 1 and number of vertices is 1000.

(a) (b)

Figure 9. (a) Message delivery rate as a function of transmission range for different values of relay probability. Here ellipse factor is 1.6 and number ofvertices is 1000. (b) Number of nodes involved in message delivery as a function of transmission range for different values of relay probability. Here ellipse

factor is 1.6 and number of vertices is 1000.

4.4. Message delivery rate as a function of transmissionrange

We can look at the problem from a totally different point ofview. So far the X-axis was the relay probability or the ellipsefactor. Thus, for each curve in figures discussed in previoussections, the network density was fixed. But if we choose thetransmission range or number of vertices as the X-axis thenthe graph density changes for each curve. We first study thecase where the X-axis is the transmission range and in thenext section we study the case where the X-axis is the thenumber of vertices.

1. Message delivery rate as a function of transmission rangefor different values of relay probability. First let us fix the

ellipse factor and the number of vertices. We expect to seejump because in each curve the graph density changes andalso we expect to see curves that are far from each other dueto the fact that for each curve the relay probability is fixed.

As you can see in figure 9 when the relay probability isbigger the jump occurs earlier. This figure is similar to fig-ure 4 due to the fact that the relay probability and transmis-sion range both improve the message delivery rate signifi-cantly.

2. Message delivery rate as a function of transmission rangefor different values of ellipse factor. Let us fix the numberof vertices and the relay probability to see the delivery rate asa function of transmission range for different values of ellipsefactor. As you can see in figure 10, like figure 1, as we in-


(a) (b)

Figure 10. (a) Message delivery rate as a function of transmission range for different values of ellipse factor. Here number of vertices is 1000 and relayprobability is 0.2. (b) Number of nodes involved in message delivery as a function of transmission range for different values of ellipse factor. Here number of

vertices is 1000 and relay probability is 0.2.

crease the ellipse factor, the message delivery rate does not in-crease proportionally. The only difference between figure 10and figure 1 is: in figure 10 the network density changes ineach curve but in figure 1 the relay probability changes ineach curve. Since increasing either the relay probability ortransmission range improves the message delivery, exchang-ing those will lead to similar results. Observe that when theellipse factor is 1.8, the delivery rate is almost the same as theglobal gossiping.

Observe that, in figure 10, the number of vertices involvedin message delivery is almost linear after the transmissionrange is larger than some threshold (almost 2). When thetransmission range is small, the number of nodes involved issmall since the message quickly dies out (the relay probabilityis 0.2 here).

3. Message delivery rate as a function of transmission rangefor different number of vertices. Now let us fix the ellipsefactor and the relay probability to study the message deliveryrate (as a function of transmission range for different numberof vertices). Since the transmission range and the number ofvertices are factors that affect the network density, not onlythe network density changes in each curve, but also the net-work density is different for each curve.

In figure 11, not only the jump occurs (due to the changeof graph density), but also the curves are far from each other(again due to the change of graph density).

Observe that, the number of vertices involved in the mes-sage delivery increases almost proportionally to the transmis-sion range when the relay probability is set to 0.2 (see figure11(a)). However, when the relay probability increases, say0.7, the percentage of the number of vertices involved is al-most constant, see figure 11(b).

4.5. Message delivery rate as a function of number ofvertices

The last parameter is the number of vertices. Since both trans-mission range and number of vertices affect the network den-sity, we expect similar results like the previous section.

1. Message delivery rate as a function of number of verticesfor different values of relay probability. Now let us fix theellipse factor and the transmission range to see delivery rateas a function of number of vertices for different values of re-lay probability. As shown in figure 12, if we use a big enoughrelay probability, a high delivery rate is guaranteed. But whenthe relay probability is small then we need a large number ofvertices to compensate this small relay probability to guaran-tee a high delivery rate.

2. Message delivery rate as a function of number of verticesfor different values of ellipse factor. Now let us fix the relayprobability and the transmission range to see delivery rate asa function of number of vertices for different values of ellipsefactor. Illustrated by figure 13, like figure 10, as we increasethe ellipse factor, the message delivery rate does not increaseproportionally.

3. Message delivery rate as a function of number of verticesfor different values of transmission range. Now let us fix theellipse factor and the relay probability to see delivery rate asa function of number of vertices for different values of trans-mission range. As you can see in figure 14, the bigger thenumber of vertices is, the earlier the jump occurs.

Figures 12–14 study the number of vertices that are in-volved in the message delivery. In these figures, we foundthat there are some strange jumps when the number of

74 LI ET AL.

(a) (b)

Figure 11. (a) Message delivery rate as a function of transmission range for different number of vertices. Here ellipse factor is 1.6 and relay probability is 0.2.(b) Number of nodes involved in message delivery as a function of transmission range for different number of vertices.Here ellipse factor is 1.6 and relay

probability is 0.2.

(a) (b)

Figure 12. (a) Message delivery rate as a function of number of vertices for different values of relay probability. Here ellipse factor is 1.6 and transmissionrange is 1. (b) Number of nodes involved in message delivery as a function of number of vertices for different values of relay probability. Here ellipse factor

is 1.6 and transmission range is 1.

vertices is around 1250. We are studying why this hap-pens.

5. Fault tolerance

To study the fault tolerance of the ad-hoc networks, we simu-lated the cases in which the target receives the message morethan once. The figure 15 shows the number of times that themessage is delivered to the target at least twice as a functionof relay probability for different values of ellipse factor. Iftarget has h neighbors inside the ellipse in the best case (i.e.,all neighbors of the target receive the message) we expect themessage to be delivered p × h times. Note that if the targethas only one neighbor inside the ellipse, then the target has no

chance to receive the message more than once. Observe thatfigure 15 is a little bit misleading. It shows that with a narrowellipse and the replay probability fixed to 1 the probabilitythat the target receives the message more than once is below95%. The reason is in our simulations, the source–target pairsare chosen randomly, so in some cases the target is only onehop away from the source, thus the target gets the message forsure but at the same time, due to the closeness of source andtarget, there might not be another neighbor inside the ellipsefor target. Thus the target has no chance to receive the mes-sage more than once. In other words, in some cases, althoughthe message delivery rate is 100%, the chance that the targetreceives the message more than once is 0%.


(a) (b)

Figure 13. (a) Message delivery rate as a function of number of vertices for different values of ellipse factor. Here relay probability is 0.4 and transmissionrange is 1. (b) Number of nodes involved in message delivery as a function of number of vertices for different values of ellipse factor. Here relay probability

is 0.4 and transmission range is 1.

(a) (b)

Figure 14. (a) Message delivery rate as a function of number of vertices for different values of transmission range. Here ellipse factor is 1.6 and relayprobability is 0.2. (b) Number of nodes involved in message delivery as a function of number of vertices for different values of transmission range. Here

ellipse factor is 1.6 and relay probability is 0.2.


We proposed a regional gossip approach, where only thenodes within some region forward the routing message withsome probability, to reduce the overhead of the routing pro-tocol imposed on the network. We showed how to set the for-warding probability based on the region and the estimatednetwork density both by theoretical analysis and by exten-sive simulations. Our simulations showed that the number ofmessages generated using this approach is less than the sim-ple global flooding (up to 94%), which already saves manymessages compared with global flooding.

Hass et al. [24] expected that the global gossiping com-bined with the cluster-based routing can further improve the

performance. We doubt this due to two reasons: (1) the back-bone formed by clusterheads are already very sparse, and toguarantee that all nodes receive the messages, the gossipingprobability is very high; and (2) the communication cost tomaintain the backbone will also offset the benefit gained byglobal gossiping, if there is any. We will conduct simulationsto study this.

One of the main questions remaining to be studied is to usenon-uniform ellipse factors. In our simulations, the ellipsefactor is uniform regardless of the distance between sourceand target. We believe that using a bigger ellipse factor, whenthe source and target are close, will get better results.

Another question is studying networks with different den-sities, meaning that instead of trying different transmission

76 LI ET AL.

Figure 15. The number of times that the message receives the target morethan once as a function of relay probability for different values of ellipse

factor. Here transmission range is 1 and number of vertices is 1000.

ranges and different number of nodes, networks with differ-ent densities can be studied. To generate a network witha given density with respect to transmission range, we cankeep adding nodes to the network until the desired density isreached.

We had assumed that two nodes can always communicateif their distance is no more than the transmission range. How-ever, this is not totally true practically. Some pair of nodescannot communicate at all even if they are close. We canmodel this by assigning another link probability pl : a link ex-ist with probability pl . Here probability pl could be uniformor dependent on the distance between the pair of nodes.

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Xiang-Yang Li has been an Assistant Professor ofComputer Science at the Illinois Institute of Tech-nology since 2000. He hold M.S. (2000) and Ph.D.(2001) degree in computer science from Universityof Illinois at Urbana-Champaign. He received hisBachelor degree in computer science and Bachelordegree in business management from Tsinghua Uni-versity, P.R. China in 1995. His research interestsspan the computational geometry, wireless ad hocnetworks, optical networks, and cryptography. He

is a member of the ACM and IEEE.E-mail: [email protected]

Kousha Moaveninejad received the Bachelor incomputer soft engineering from Sharif University ofTechnology, Tehran, Iran, in 1997. He joined theDepartment of Computer Science of Illinois Insti-tute of Technology in 2000, as a M.S. student andreceived the Masters degree in 2002. He then contin-ued his study as a Ph.D. student at Illinois Institute ofTechnology and his current research interests includewireless ad-hoc networks, computational geometry,algorithm design, and mobile computing.

E-mail: [email protected]

Ophir Frieder is the IITRI Professor of ComputerScience at the Illinois Institute of Technology. Hisresearch interests span the general area of distributedinformation systems. He is a member of ACM and afellow of the IEEE.E-mail: [email protected]


Comparison and Evaluation of Multiple Objective GeneticAlgorithms for the Antenna Placement Problem

LARRY RAISANEN ∗ and ROGER M. WHITAKER ∗∗Centre for Mobile Communications, Department of Computer Science, Cardiff University, Queens Buildings, The Parade,

P.O. Box 916, Cardiff CF24 3XF, UK

Abstract. The antenna placement problem, or cell planning problem, involves locating and configuring infrastructure for cellular wirelessnetworks. From candidate site locations, a set needs to be selected against objectives relating to issues such as financial cost and serviceprovision. This is an NP-hard optimization problem and consequently heuristic approaches are necessary for large problem instances. In thisstudy, we use a greedy algorithm to select and configure base station locations. The performance of this greedy approach is dependent onthe order in which the candidate sites are considered. We compare the ability of four state-of-the-art multiple objective genetic algorithmsto find an optimal ordering of potential base stations. Results and discussion on the performance of the algorithms are provided.

Keywords: genetic algorithms, antenna placement

1. Introduction

The proliferation of cellular wireless services for mobile com-munication has led to the antenna placement problem (APP).For cellular wireless systems, mobile communication is fa-cilitated by base stations which have an appropriate spatialdistribution. The area of service coverage from a single an-tenna at a base station constitutes a cell, which is a regionwhere the radiated signal power from the serving antenna isof sufficient strength to be received by subscribers. As thepower of transmitted signals must be restricted, multiple cellsare required to provide wide area coverage. The collection ofall cells across the network constitutes a cell plan.

The APP involves selecting base station site locations froma set of candidates, which are normally located irregularly.Selected sites must be configured to provide adequate servicecoverage and capacity while adhering to constraints involv-ing regions which can be served by more than one antenna.Such constraints are imposed to ensure that the potential forinterference is controlled while providing regions in the net-work for call handover, which is necessary for seamless calltransfer between cells. Areas covered by more than one cellmust be carefully controlled both to maintain network opera-tion and minimize the total commitment to infrastructure. Theprimary configuration variable at a site is transmission power.This directly affects the cell size, the required number of cells(and therefore financial cost), and handover regions.

The number of factors involved in solving the APP meansthat automatic software for designing cell plans has becomeincreasingly common [50]. We classify software as auto-matic if the associated computer program has autonomy inthe selection of base station locations and the configurationof antennae. The underlying algorithm in the software is re-quired to tackle an NP-hard [33] optimization problem with

∗ Supported by a Doctoral Scholarship from the EPSRC.∗∗ Corresponding author.

multiple and conflicting objectives. In this regard, the APPis an extension of the well-studied facilities location prob-lem [11], namely, the capacitated facilities location prob-lem with unsplittable demands. Consequently, heuristic andmeta-heuristic techniques have become increasingly popularfor solving the APP.

In this paper, we focus on resolving the two fundamentalaspects of cell planning: providing the required service cov-erage at the lowest possible financial cost. These two con-flicting objectives always exist when setting up cellular net-work services, as adding base stations to improve coverageinherently increases the cost of the network. In this study, weproduce cell plans in which base station locations are selectedand allocated a transmission power, assuming an isotropic ra-diation pattern (i.e., power radiates in all directions with equalstrength). As base stations are expensive to commission andmanage, we optimize the total cost and location of base sta-tions commissioned. Despite the importance of finding anoptimal trade off between these objectives, we are not awareof any studies in the literature addressing this issue.

The cell plans we produce are the first step in establishinga cost effective operational network, and only factors whichhave the largest impact on financial cost and service cover-age are considered. Although not the focus of our work, wenote that once the tension between cost and coverage is re-solved, the resultant cell plan is then ready to undergo de-tailed dimensioning of individual cells [15]. This second stagemay involve adjusting additional variables at the selected basestations such as tilt and direction (i.e., azimuth). Addition-ally, multiple directed co-sited antenna may be invoked atthis stage to increase the capacity for multiplexing, givenknowledge of anticipated traffic patterns. Known as sector-ization, this is common operational practice in mobile tele-phony, whereby using multiple co-sited antenna is generallyfar cheaper than commissioning a new site.

80 RAISANEN AND WHITAKER

To optimally resolve the competition between service cov-erage and financial cost, we introduce a multiple objectiveoptimization framework that does not require a priori knowl-edge of the relative importance of service coverage versuscost. This is achieved by providing a range of alternativesite selections which approximate the best possible trade-offs(i.e., Pareto front) between cost and coverage. This meansthat unlike the current convention for cell planning, whichgenerally seeks to generate a single cell plan given informa-tion on the relative importance of objectives, a radio engineerwill be able to choose from a range of alternative cell plans,given visual and detailed information regarding each. This isparticularly beneficial when there is a nonlinear relationshipor unknown dependency between the objective functions asfor the general APP.

As far as we are aware [50], this method has only been con-sidered for cell planning in [36], where a genetic algorithmwas developed specifically for the APP; however, the totalnetwork cost was not considered. Unlike [36], the frameworkwe propose considers financial cost and is flexible becauseit is possible to “plug-in” any multiple objective optimiza-tion algorithm (MOA) which seeks to approximate a Paretofront. This flexibility is achieved by making the cell plan rep-resentation independent from the task of the MOA, which, inthis case, is to find optimal orderings of candidate site loca-tions which optimize the two objective functions. Exploringa search space in this manner is common practice in manydiscrete optimization problems (e.g., the knapsack problem)and has also been successfully applied to the frequency as-signment problem [27].

Using this approach, we explore the relative trade-offs be-tween financial cost and service coverage using four state-of-the-art genetic MOAs, which were selected as they producemany alternative solutions in parallel. The performance ofSEAMO, SPEA2, NSGA-II, and PESA is considered usinga range of synthesized test problems. We argue that the ap-proach we take is more likely to lead to an efficient opera-tional cellular network, as the two fundamental aspects areconsidered in isolation from other cell planning tasks.

2. Solving the APP

The first published paper on optimizing antenna placementdates back to 1994 [6]. Since then a large number of ap-proaches and scenarios have appeared in the literature. Al-though exact approaches are only feasible for relatively smalltest problems, they have been applied in a range of papers[37–39,49]. However, in some papers, they have been re-laxed or selectively applied. Sequential, or greedy, algo-rithms have been less well used and then predominantly forcomparison purposes (see [5,37]). Deterministic heuristicalgorithms have also been proposed by a range of authors[9,17,18,28,34,37,38,44,45,53]. Frequently, these approachesexploit observations (e.g., density of base station locations)about the APP and incorporate them to enhance the perfor-mance of the cell plans obtained. However, meta-heuristic

algorithms based on simulated annealing [1], tabu search[19,20] and genetic algorithms [23] are far more popular.

Simulated annealing has been adopted for the APP in[2,3,6,26,33], and tabu search for the APP in [4,21,32,47].Both these techniques operate by ranking solutions using acost function. Given a solution, small changes are made tocreate a neighborhood of solutions from the current solution.The meta-heuristic then guides the acceptance of new solu-tions available in the neighborhood. The advantage of thisapproach is that it has the ability to escape from local minimain the search space (regarding the cost function), thereby im-proving performance. Differences in the application of theseapproaches involve how the cell planning problem is mod-elled, the formulation of rules to create neighborhoods, andthe cost function used to rank solutions.

Genetic algorithms have also become increasingly popu-lar for the APP [8,21,25,31,32,36,37,40]. With the excep-tion of [36], these approaches predominantly seek to opti-mize a single function (or a linear combination of multipleobjective functions) to create a population of high quality so-lutions. These algorithms mimic evolution and natural se-lection through fitness assignment, selection, recombination,and mutation on a population of solutions. For a genetic al-gorithm to succeed, a suitable representation of the problemneeds to be used. The most popular representation for theAPP in the literature is a binary string with crossover opera-tions defined using geographic information between individ-ual base stations. However, as we show in this paper, geneticalgorithms need not be restricted to this representation. Ourstudy is unique in that it combines an integer string represen-tation with a multiple objective approach.

2.1. Resolving conflicting multiple objectives

Regardless which optimization technique is adopted to solvethe APP, it is necessary to resolve the conflict between com-peting multiple objectives, such as service coverage and fi-nancial cost. The following definition is useful in this context.

Definition 1 (Pareto optimality). Let o1, o2, . . . , on be ob-jective functions which are to be maximized. Let S be the setof all possible solutions. s ∈ S is dominated by t ∈ S (de-noted t � s) if ∃j , j ∈ {1, . . . , n}, such that oj (t) > oj(s)

and ∀i, 1 � i � n, oi(t) � oi(s). A non-dominated solutionis said to be Pareto optimal.

Pareto optimal cell plans are non-dominated in the sensethat it is not possible to improve the value of any objectivewithout simultaneously degrading the quality of one or moreof the other objectives. The set of all possible Pareto optimalsolutions in the entire search space is called the Pareto front.In figure 1, a hypothetical Pareto front is indicated for the ob-jectives of cost and coverage. The most desirable cell plan inthe Pareto front depends on which objective is most impor-tant. However, in the absence of such a relative ranking ofobjectives, solutions from the Pareto front must be regarded

COMPARISON AND EVALUATION OF MULTIPLE OBJECTIVE GENETIC ALGORITHMS 81

Figure 1. Progress towards Pareto front of cost and coverage.

as equivalent. Our approach is to generate a set of alternativesolutions (i.e., cell plans) which approximate the Pareto front.

Despite the potential strength of using Pareto optimalitywithin the context of cell planning in this way, it has not beenaddressed adequately in the literature. This may be partiallydue to the fact that there are a number of alternative strategiesavailable, which can also handle multiple objective problems.These strategies are:

1. Combine all objectives into a single scalar value, typicallyas a weighted sum, and optimize the scalar value.

2. Solve for the objectives in a hierarchical fashion, optimiz-ing for a first objective then, if there is more than one solu-tion, optimize these solution(s) for a second objective, andrepeat.

3. Obtain a set of alternative, non-dominated solutions, eachof which must be considered equivalent in the absence offurther information regarding the relative importance ofeach of these objectives.

Each approach involves exploring the search space of allpossible cell plans to find one or more suitable solutions.Approach one is by far the most popular approach in theliterature (e.g., [4–6,8,17,21,24,34,35,42,45,47,51,52]). Thebiggest problem with this approach is that setting the rela-tive weights of different components in the cost function maylead to inappropriate favoring or penalizing of different ob-jectives. Approach two may be combined with approach one,as in [26,32,40,53], which may involve changing the objec-tive function at different points in the search in a phased orstaged manner. This approach effectively prioritizes differ-ent single optimization objectives a priori and therefore hassimilar problems to the first approach. Only in [36] has amulti-objective search been implemented using approach 3,where the Pareto front, in this case, was approximated usinga problem specific genetic algorithm which did not considerthe financial cost of the cell plan.

2.2. Cell plan model

Before discussing the genetic algorithms used to generate aset of non-dominated solutions, we first turn our attention to

the cell plan model and representation used. Firstly, we definea working area as the region over which transmission is con-sidered. This is characterized by discretized test points. Thefollowing sets form the input to our formulation of the APP:

• A set of candidate sites for locating base stations, denotedL = {L1, . . . , LnBS }.

• A list of possible transmission powers p0, p1, p2, . . . , pk

in ascending order of magnitude. Zero power is denotedby p0.

• A set of service test points (STP), {s1, . . . , snstp}, where asignal must be received above a minimum specified ser-vice threshold Sq to ensure a required quality of service.

• A maximum handover percentage used to consider the vi-ability of the handover region when commissioning a newcell.

For purposes of candidate sites, we assume that each basestation is operating a single omni-directional antenna with anisotropic radiation pattern. The antenna height is assumed tobe fixed at the maximum permitted at the site to enhance po-tential transmission range. Finally, each base station locationLi has a cost $(Li) associated with commissioning it. Thecost of each base station was set to a fixed uniform randomvalue between 1 and 100 for each test problem. We assumethat service test points are regularly spaced every 300 me-ters and the maximum handover parameter has been set at30% throughout. The service threshold has been taken as−90 dBm, which is a realistic value for GSM services andequipment.

2.2.1. PropagationA service test point r is said to be covered by antenna A ifthe received signal strength from A, denoted PA

r , is greaterthan Sq. We assume

PAr = PA − PL − L + G

where PA is the power at which A is transmitting, PL is thepath loss experienced between A and r , L is the aggregationof other losses experienced, and G is the aggregation of gainsexperienced. For experimental purposes, we assume G = L.For each combination of A and r , PL may be recorded in thefield or estimated using a free space path loss or empiricalmodel. In the absence of data from the field, we adopted theempirical model proposed by Hata [22]:

PL = 69.55 + 26.16 log(f ) − 13.82 log(hb) − a(hm)

− K + (44.9 − 6.55 log(hb)

)log(R),

given particular values for variables such as frequency (f ),base station height (hb), and mobile receiver height (hm). Forthis investigation, these values are set as f = 800 MHz,hb = 31 meters, and hm = 1.5 meters. Additional envi-ronmental correction factors include a(hm),K and the prop-agation distance R. As the mobile receiver height was set to1.5 meters, a(hm) is 0. As the standard urban version of themodel was used, K is 0. R is the distance in kilometers from


each base station to each STP. As well as Hata’s model, manyother propagation models would have been equally suitable.

2.2.2. Handover regions and objectivesThe subset of service test points covered by a particular an-tenna A is the cell served by A, denoted cA. Note that cellsserved by different antennae are not necessarily disjoint sincean STP can potentially be covered by more than one antenna.Such an STP is referred to as a handover STP. A handoverSTP which is contained in more than two distinct cells is re-ferred to as a soft handover STP. For a cell cA, the subset ofhandover STPs is denoted hA. For cA, the handover percent-age is defined as

|hA||cA| · 100.

Controlling the size and distribution of handover regions iscrucial for both operational and financial reasons. Handoverregions are a prerequisite for seamless call transfer betweencells for mobile users. However, if very large handover re-gions are permitted, there is a greater potential for inter-ference due to strong signals being received from multiplesources. In frequency division multiple access systems, largehandover regions increase the need for large channel separa-tion between adjacent cells in the frequency assignment prob-lem. Large handover regions may also adversely affect thecost of the network by increasing the total number of basestations required to cover a given area.

The objectives we are concerned with relate to financialcost and area coverage of service test points. The cost of acell plan L′, denoted costL′ , is defined as:

costL′ =∑

Li∈L′$(Li).

The coverage of a cell plan is expressed as the proportion ofSTPs which are covered. Handover is not considered as anobjective in our problem formulation, but imposed as a con-straint controlled by via the decoder.

2.3. Cell plan representation

The potential base station location Li is referred to as the ithbase station. We use a permutation π of the potential basestation locations to represent cell plans. Each permutation π

orders the potential base station locations. Under the permu-tation π , the ith base station location listed is denoted π(i).We introduce a decoder which translates a permutation π intoa cell plan. This approach mimics the way in which the prob-lem might be attempted manually. The decoder is effectivelya greedy, sequential algorithm for creating a cell plan, whichis dependent on the order of inspection for commissioning po-tential sites occurring in π . The decoder adds cells iterativelyto create a cell plan L′ as follows:

• Initially L′ = ∅.

• Potential sites π(1), π(2), . . . , π(n) are inspected (in theorder induced by π) for possible selection.

• At iteration j (1 � j � n), π(j) is considered for addi-tion to the set L′.

– Handover between cπ(j) and L′ is feasible if the han-dover percentage for cπ(j) is less than the maximumpermitted.

– The largest power setting, denoted pmax, is identifiedfrom the list p0, p1, p2, . . . , pk such that handover isfeasible between cπ(j) and L′.

– If pmax �= p0, then π(j) is added to L′, and the trans-mission power of π(j) is recorded as pmax. Otherwiseπ(j) is not added to L′.

A number of observations can be made regarding this ap-proach. Firstly, the approach is greedy in the sense that oncea base station location is added to L′ at power pmax, the basestation cannot be removed from the cell plan L′ nor can itstransmission power be adjusted. Secondly, for a particular listof potential site locations, characteristics (e.g., cost and cov-erage) of the resultant cell plan L′ is entirely dependent onthe order (i.e., permutation π) in which the base stations areconsidered for selection. It is our aim to find the best permu-tations, which lead to Pareto optimal cell plans, using geneticalgorithms.

3. Genetic algorithms

Only over the last decade have genetic algorithms (GAs) beensuccessfully adapted to solve multiple objective problems. Anexcellent overview of this area is given in [12]. The generalprinciple is to breed a new population of solutions (i.e., cellplans) through a process of selection and recombination. Thisoccurs over a number of generations to try to improve the per-formance of the population, as shown in figure 1. The expec-tation is that desirable characteristics in solutions from onegeneration will combine to produce better solutions for thenext generation. Introduced by Holland [23], GAs are sup-ported by theory which identifies the conditions under whichsolutions converge to a high performing set of solutions. GAswhich approximate the Pareto front seek to find, ideally, a di-verse set of solutions spread evenly over the entire range ofthe Pareto optimal front. In this study, we consider the abil-ity of four state-of-the-art GAs, namely, SPEA2, NSGA-II,PESA, and SEAMO to perform this function. Each algorithmis briefly described below.

3.1. Brief description of each GA

The Strength Pareto Evolutionary Algorithm version II(SPEA2) is an enhancement of that originally proposed in[54], and is described in detail in [55]. SPEA2 has been usedin numerous studies (e.g., [7]) where good performance, incomparison to other MOAs, has been reported. In SPEA2,the most fit individuals from the union of archive and childpopulations are determined by computing a fitness value foreach solution which is the sum of two parts. The first part is araw fitness value based on how many solutions it dominates,


and the second is a density estimate based on the its proximityto other solutions in objective space. At each generation, themost fit n solutions are saved to the archive, and genetic oper-ators are applied to form a new child population. This processis repeated until termination.

NSGA-II is a fast elitist non-dominated sorting genetic al-gorithm (see [13] for a full description), which has been wellstudied (e.g., [14,29]). NSGA-II is similar to SPEA2, but usesslightly different mechanisms. For example, in NSGA-II themost fit individuals from the union of archive and child popu-lations are determined by a ranking mechanism (or crowdedcomparison operator) composed of two parts. The first part“peels” away layers of non-dominated fronts, and ranks solu-tions in earlier fronts as better. The second part computes adispersion measure, the crowding distance, to determine howclose a solution’s nearest neighbors are, with larger distancesbeing better. At each generation, the best n solutions withregard to these two measures are saved to the archive, and ge-netic operators applied to form a new child population. Thisprocess is repeated until termination.

The Pareto Envelope-based Selection Algorithm, PESA, isdescribed in [10]. It uses different mechanisms than SPEA2and NSGA-II. The main differences are that its archive pop-ulation is not of fixed size and only allows non-dominatedsolutions to be members, which is a more limited set thanthe previous two GAs allowed. If the archive ever exceedsn solutions, a squeeze factor is calculated for all members ofthe archive. The squeeze factor is the total number of mem-bers in the same subregion of a hyper-grid (which partitionsthe search space into subregions (see [30])). The higher thesqueeze factor, the more local neighbors a solution has. Ran-dom members from the grid region with the highest squeezefactor are then removed until the size of the archive is reducedto n. Genetic operators are then applied to archive membersto form a new child population. This process is repeated untiltermination.

Finally, the Simple Evolutionary Algorithm for Multi-objective Optimization, known as SEAMO, has performedparticularly well on the benchmark test knapsack optimiza-tion problem [46]. The main difference between SEAMO andthe other algorithms, is that it is steady-state and has only onepopulation (of constant size n) to maintain. The main advan-tage of SEAMO is the simple approach it uses to dispose ofall selection mechanisms based on fitness or rank. Instead,the search progresses based on three simple rules:

(1) Parents can only be replaced by their own offspring.

(2) Duplicates in the population are deleted.

(3) Offspring can only replace parents if superior – elitism.

Genetic operators are applied to each parent in turn to forma new child, which is considered for substitution into the par-ent population based on the three rules. This process is re-peated until termination.

3.2. Recombination and mutation

Each of the algorithms considered has a specific method forselecting parents. SPEA2 bases selection on fitness, NSGA-II on the crowded comparison operator, PESA on non-dominated members of its archive set, and SEAMO uni-formly. However, common recombination and mutation op-erators have been used to maintain a fair comparison betweenthe algorithms. The well-known cycle crossover [43] has beenused as the recombination operator and the mutation operatorinvolves the simple transposition of candidate base station lo-cations in a randomly selected pair of positions. This wasgoverned by a mutation rate (set to 1%) to restrict the fre-quency of mutation.

3.3. Measuring the relative performance of GAs

Comparing the performance of multiple objective algorithmsis problematic because a set of solutions rather than a singlesolution is obtained. Although several alternatives have beenproposed (e.g., [16,48,56,57] ) no single approach seems mostprevalent. We adopt the approach first given in [57] to calcu-late a set coverage metric. This involves the concept of weakdomination. Solution A weakly dominates solution B if A

and B have the same performance across all objectives or A

dominates B. For two sets of solutions SA and SB , the setcoverage metric of set SA with respect to SB is the percent-age of solutions in SB which are weakly dominated by at leastone solution from SA. The higher the set coverage metric ob-tained, the greater the superiority of SA over SB .

4. Results

The performance of the algorithms have been compared usinga wide range of synthesized test problems, each of which hasbeen randomly generated. Each test problem gives the loca-tion and cost of candidate sites. Test problems are classifiedin two ways: the size of area in which they are positioned andthe density of candidate sites, as documented in figure 2.

Combining the size of regions and the density of sites leadsto a total of nine test problem classes, as indicated in figure 2.For each test problem class, we produce five incidences onwhich each algorithm is tested. This means that average algo-rithm performance is estimated and compared using five prob-lem instances from each of nine classes, with four different al-gorithms, leading to a total of 180 experiments. All probleminstances are available at: http://www.cs.cf.ac.uk/user/L.Raisanen/downloads.html.

Region size km2 Density of sites per km2

0.03 0.06 0.1215 × 15 7 14 2830 × 30 27 54 10845 × 45 61 122 244

Figure 2. Number of candidate sites in nine problem classes defined by regionsize and density.


Table 1The average set coverage values obtained in each problem class, for all pairwise comparisons of algorithms.

Set coverage metrics

Algorithm Problem instances (km2 – number of candidate sites)

SA SB 15-7 15-14 15-18 30-27 30-54 30-108 45-61 45-122 45-244 Average

SEAMO SPEA2 100.00 92.00 86.83 91.34 43.23 19.03 39.17 7.85 18.00 55.27NSGA-II 100.00 92.00 84.33 92.80 39.81 13.71 37.15 9.20 11.41 53.38PESA 100.00 96.00 97.50 96.80 60.29 47.74 63.32 38.05 29.90 69.96Average 100.00 93.33 85.56 93.64 47.78 26.83 46.55 18.37 19.77 59.53

SPEA2 SEAMO 100.00 100.00 100.00 100.00 95.38 94.55 93.74 92.11 82.64 95.38NSGA-II 100.00 100.00 97.50 100.00 74.80 69.67 86.56 52.04 49.97 81.17PESA 100.00 100.00 100.00 100.00 86.47 86.75 97.65 83.98 77.21 92.45Average 100.00 100.00 99.17 100.00 85.55 83.65 92.65 76.04 69.94 89.67

NSGA-II SEAMO 100.00 100.00 100.00 100.00 94.83 98.18 93.74 84.33 81.79 94.76SPEA2 100.00 100.00 100.00 96.36 75.34 80.96 92.36 65.91 47.14 84.23PESA 100.00 100.00 100.00 100.00 91.67 88.99 100.00 87.22 82.22 94.46Average 100.00 100.00 100.00 98.79 87.28 89.38 95.38 79.16 70.38 91.15

PESA SEAMO 100.00 90.00 70.89 98.33 45.43 80.93 50.57 53.22 65.21 72.73SPEA2 100.00 86.00 67.33 93.01 31.29 75.53 35.80 31.28 22.28 60.28NSGA-II 100.00 86.00 64.83 94.46 35.35 76.13 35.12 26.91 12.31 59.01Average 100.00 87.33 67.69 95.27 37.36 77.53 40.50 37.14 33.27 64.01

Power setting dBW Wattsp1 30 1000p2 27 501p3 24 251p4 21 125p5 18 63

Figure 3. Power settings used in tests.

To maintain a fair comparison between algorithms, com-mon parameter settings have been adopted for each experi-ment. Five nonzero power settings (displayed in units of dBWand Watts) have been used, as specified in figure 3. Unlessotherwise specified, a population size of 100 is adopted us-ing 500 generations. Additionally, the same random startingpopulations have been used for each problem class.

We consider the performance of each GA in four ways:

(1) the average performance (in terms of the objective valuesof members in the final population) compared to otherGAs across all test problems using the set coverage met-ric, with diagrams to show obtained Pareto fronts and cellplans,

(2) the average measure of population spread,

(3) the speed of convergence to solutions in the final popula-tion, and

(4) the average speed of execution.

4.1. Average performance across test problems

In terms of the average performance of each GA comparedto other GAs across all test problems using the set coveragemetric, it was found that NSGA-II achieves the best perfor-mance by weakly dominating an average of 91.15% of solu-tions obtained by other algorithms in terms of service cov-

erage and cost, closely followed by SPEA2 (89.67%), thenPESA (64.01%) and, finally, SEAMO (59.53%). See table 1for details.

In figure 4 we plot the Pareto fronts (i.e., non-dominatedsolutions from the final population) achieved by each algo-rithm on the large region problem at each density. Despitethe differences in relative algorithm performance, the Paretofronts obtained are closely clustered in real terms. Generally,the plots show that that as candidate site density increases, so-lutions with a higher level of coverage are achievable. Also, inthe most dense problem instances, lower cost solutions withhigher coverage are achievable due to more freedom in siteselection and cost.

In figure 5, we display an example of cell plans with thehighest coverage for the large region problem at each density,with density increasing left to right.

4.2. Measure of solution distribution

To measure the distribution, or spread, of solutions along thePareto front, a metric proposed in [41] has been implemented.The spacing measure is based on the range of values for di ,which is the distance (in terms of solution space) between theith element of the solution set and its nearest neighbour. Theaverage of di values for a solution set of size n is denoted d .Then the measure of spread is defined as:

S =√√√√ 1

n − 1

n∑

i=1

(d − di

)2 (1)

for the n members in the final population. Note that S = 0indicates all members of the Pareto front are spaced equidis-tantly in the solution space.

It was found that PESA performed the best on this mea-sure, with an average spacing value of 19.75, followed by


Figure 4. Pareto fronts (coverage versus cost) for large problem size instances (v4: 45 × 45) with 61, 122, and 244 candidate sites.

Figure 5. Example cell plans with the highest coverage at each density for the large region problem.

Algorithm Problem instances (km2 – number of candidate sites)

15-7 15-14 15-28 30-27 30-54 30-108 45-61 45-122 45-244 AverageSEAMO 3.94 19.63 13.07 14.32 39.92 53.68 25.24 38.07 33.71 26.84SPEA2 3.94 21.31 13.90 14.94 13.83 15.37 29.35 33.36 42.17 20.91NSGA-II 3.94 21.31 15.02 18.05 25.63 14.84 27.34 30.16 36.83 21.46PESA 3.94 19.29 8.50 14.09 21.53 29.78 18.57 23.81 38.26 19.75

Figure 6. Average spacing values by algorithm for each test problem class.

Algorithm Problem instances (km2 – number of candidate sites)15-7 15-14 15-28 30-27 30-54 30-108 45-61 45-122 45-244

SEAMO 7.2 17.4 37.8 104.0 242.8 542.2 526.6 1272.8 2854.8SPEA2 7.8 18.4 40.2 107.0 250.8 553.0 543.8 1305.6 3089.0NSGA-II 7.2 17.6 39.0 106.0 247.2 548.6 539.6 1300.0 3119.6PESA 7.4 17.2 38.8 111.0 246.6 555.2 578.4 1293.8 2963.2Average 7.4 17.7 39.0 107.0 246.9 549.8 547.1 1293.1 3006.7

Figure 7. Average speed of execution in seconds.

SPEA2 at 20.91, NSGA-II at 21.46, and SEAMO at 26.84(see figure 6 for details). It is little surprise that the algorithmwhich is specifically designed to encourage spacing in objec-tive space, PESA, performed the best, and that the algorithmwith no direct measure to control dispersion, SEAMO, per-formed the worst.

4.3. Convergence

The ability of an algorithm to rapidly converge to the finalsolution set is desirable. This has been investigated for eachalgorithm using the largest problem class with the highest sitedensity. Each algorithm has been applied for 1500 genera-

tions, and intermediate populations (produced every 250 gen-erations) have been compared against each other, using theset coverage metric (see section 3.3). The results indicatethat PESA and SEAMO converge quickly, by generation 1000dominating 88.88% and 77.77% of solutions in the final gen-eration (1500), respectively, whereas SPEA2 and NSGA-IIconverge more slowly dominating 22.22% and 33.33% re-spectively (see figure 8 for details). However, consideringthat SPEA2 and NSGA-II outperform PESA and SEAMO interms of final solutions (see section 4.1), this may also indi-cate that these algorithms are better at improving solutionsover time than the other two.


Algorithm Generation Generation forming SB

forming SA 250 500 750 1000 1250 1500SEAMO 250 12.50 0.00 0.00 0.00 0.00

500 100.00 28.57 0.00 0.00 0.00750 100.00 100.00 71.43 16.67 14.29

1000 100.00 100.00 100.00 33.33 28.571250 100.00 100.00 100.00 100.00 71.431500 100.00 100.00 100.00 100.00 100.00

SPEA2 250 0.00 0.00 0.00 0.00 0.00500 100.00 27.27 11.11 9.09 9.09750 100.00 100.00 22.22 18.18 9.09

1000 100.00 100.00 100.00 54.55 36.361250 100.00 100.00 100.00 100.00 45.451500 100.00 100.00 100.00 100.00 100.00

NSGA-II 250 0.00 0.00 0.00 0.00 0.00500 100.00 9.09 8.33 33.33 33.33750 100.00 100.00 33.33 33.33 33.33

1000 100.00 100.00 100.00 75.00 58.331250 100.00 100.00 100.00 100.00 58.331500 100.00 100.00 100.00 100.00 100.00

PESA 250 0.00 0.00 0.00 0.00 0.00500 100.00 55.55 55.55 44.44 44.44750 100.00 100.00 88.88 77.77 66.67

1000 100.00 100.00 100.00 77.77 66.671250 100.00 100.00 100.00 100.00 88.891500 100.00 100.00 100.00 100.00 100.00

Figure 8. Comparison of intermediate populations for each algorithm, usingthe set coverage metric for a total of 1500 generations.

4.4. Speed of execution

The average execution times varied from an average of 7.4seconds to complete the smallest problem, to 3006.7 seconds(roughly 50 minutes) to complete the largest, with SEAMOperforming the fastest marginally (see figure 7 for details).All recorded times were obtained using a Pentium IV 1.8 GHzprocessor, 256 megabytes of RAM, Java JDK 1.3.1-04, andWindows XP Professional. It is suspected that obtaining onlymarginal differences in execution time was due to a bottleneckincurred by the computationally intensive decoder.

5. Conclusions

In this paper we have have introduced a general framework forapplying multiple objective genetic algorithms to the antennaplacement problem. The key aspect of this framework is adecoder which uses an ordering of candidate site locations toconstruct a cell plan. Subsequently, the performance of fourGAs to find an optimal ordering of potential site locationswere compared and evaluated using a range of test problemsclassified by size and density.

We found that all the algorithms considered find closelycomparable solutions, in real terms. However, there are differ-ences. NGSA-II and SPEA2 have very similar performancethroughout, confirming the findings in [55] concerning theperformance of these algorithms. Meanwhile, PESA gener-ally obtains slightly lower quality sets of solutions, but hasthe best performance in terms of distribution of solutions andspeed of convergence. Both the advantage and disadvantageof SEAMO lie in its simplicity. This makes the algorithm

conceptually elegant, easy to implement, and fast to run, butthis simplicity appears to impede the overall quality and dis-tribution of the solutions obtained, with SEAMO obtainingthe lowest performance measures in these areas. On balance,we consider NGSA-II to be the strongest performing algo-rithm for purposes of cell planning when using the generalframework proposed. This is mainly based on the consistentcomparative quality of the solutions obtained.

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Larry Raisanen is a second year Ph.D. studentstudying at Cardiff University at the Centre for Mo-bile Communications based in the School of Com-puter Science. The focus of his work is on the devel-opment, testing, and analysis of algorithms to resolvethe base station placement problem in wireless mo-bile communication systems using a multi-objectiveoptimization framework. He graduated in 1993 witha BA – magna cum laude – from Marquette Univer-sity (WI, USA), and in 2003 with a M.Sc. in com-

puting – distinction – from Cardiff University (Wales, UK).E-mail: [email protected]


Roger M. Whitaker holds a Ph.D. degree in discretemathematics (1999) and a B.Sc. degree in mathe-matics and managment science. He is a lecturer anda co-director of the Centre for Mobile Communica-tions, School of Computer Science, Cardiff Univer-sity, UK. Prior to this position, Roger carried outresearch for the UK Radiocommunications Agencyinto spectrum efficiency. His research addresses theapplication of computer science to the design, co-ordination and optimization of wireless networks and

systems. He is currently leading a number of externally supported researchprojects in this area.E-mail: [email protected]


A Characterisation of Optimal Channel Assignments for Cellularand Square Grid Wireless Networks ∗

M.V.S. SHASHANKABoston University, Boston, MA, USA

AMRITA PATIVirginia Polytechnic Institute and State University, Blacksburg, VA, USA

ANIL M. SHENDE ∗∗221 College Lane, Roanoke College, Salem, VA 24153, USA

Abstract. In this paper we first present a uniformity property that characterises optimal channel assignments for networks arranged ascellular or square grids. Then, we present optimal channel assignments for cellular and square grids; these assignments exhibit a high valuefor δ1 – the separation between channels assigned to adjacent stations. We prove an upper bound on δ1 for such optimal channel assignments.This upper bound is greater than the value of δ1 exhibited by our assignments. Based on empirical evidence, we conjecture that the valueour assignments exhibit is a tight upper bound on δ1.

Keywords: wireless computing, channel assignment, cellular and square grids

1. Introduction

The enormous growth of wireless networks has made the ef-ficient use of the scarce radio spectrum important. A “Fre-quency Assignment Problem” (FAP) models the task of as-signing frequencies (channels) from a radio spectrum toa set of transmitters and receivers, satisfying certain con-straints [9]. The main difficulty in an efficient use of theradio spectrum is the interference caused by unconstrainedsimultaneous transmissions. Interferences can be eliminated(or at least reduced) by means of suitable channel assignmenttechniques, which partition the given radio spectrum into aset of disjoint channels that can be used simultaneously bythe stations while maintaining acceptable radio signals. Sinceradio signals get attenuated over distance, two stations in anetwork can use the same channel without interferences pro-vided the stations are spaced sufficiently apart. The minimumdistance at which channels can be reused with no interfer-ences is called the co-channel reuse distance (or simply reusedistance) and is denoted by σ .

In a dense network – a network where there are a largenumber of transmitters and receivers in a small area – inter-ference is more likely. Thus, reuse distance needs to be highin such networks. Moreover, channels assigned to nearby sta-tions must be separated in value by at least a gap which isinversely proportional to the distance between the two sta-tions. A minimum channel separation δi is required be-tween channels assigned to stations at distance i, with i < σ ,such that δi decreases when i increases [8]. σ is said toplace the co-channel reuse distance constraint, and the vector

∗ This work was partially funded by NSF grant 0200823.∗∗ Corresponding author.

�δ = (δ1, δ2, . . . , δσ−1) is said to place the channel separationconstraint on the channel assignment problem.

The purpose of channel assignment algorithms is to assignchannels to transmitters in such a way that (1) the co-channelreuse distance and the channel separation constraints are sat-isfied, and (2) the span of the assignment, defined to be thedifference between the highest and the lowest channels as-signed, is as small as possible [2].

This paper has two significant contributions:

1. A characterisation of optimal channel assignments for cel-lular and square grids. We essentially show a nice unifor-mity across the grid that every optimal assignment mustsatisfy. (See section 3.)

2. Optimal channel assignments for cellular and square gridswhere the channel separation between adjacent stations islarge. We prove an upper bound on δi for such optimalchannel assignments. This upper bound is greater than thevalue of δi exhibited by our assignments. Based on empiri-cal evidence, we conjecture that the value our assignmentsexhibit is a tight upper bound on δ1. (See section 4.)

In section 2 we formally define the problem of channelassignments and its formulation as a colouring problem, andprovide a brief literature survey. We also outline the generalstrategy we use for our optimal colourings discussed in sec-tion 4. In section 3 we first define the cellular (section 3.1) andsquare (section 3.2) grids, and point out some useful proper-ties of these grids. Then, in section 3.3, we prove a charac-terisation of optimal colourings for cellular and square grids.In section 4 we present our colourings and prove that they areoptimal. Then, in section 5, we present an upper bound on

90 SHASHANKA ET AL.

the value of the channel separation among adjacent stationsas witnessed by optimal colourings.

2. Preliminaries

Formally, the Channel Assignment Problem with Separation(CAPS) can be modelled as an appropriate colouring prob-lem on an undirected graph G = (V ,E) representing the net-work topology, whose vertices in V correspond to stations,and edges in E correspond to pairs of stations that can heareach other’s transmission [2]. The colour assigned to a par-ticular vertex corresponds to the frequency channel assignedto the corresponding station. For a graph G, we will denotethe distance between any two vertices in the graph, i.e., thenumber of edges in a shortest path between the two vertices,by dG(·, ·). (When the context is clear, we will denote thedistance as simply d(·, ·).) CAPS is then defined as:

CAPS (G,σ, �δ). Given an undirected graph G, an integerσ > 1, and a vector of positive integers �δ = (δ1, δ2, . . . ,

δσ−1), find an integer g > 0 so that there is a functionf : V → {0, . . . , g}, such that for all u, v ∈ G, for each i,1 � i � σ − 1, if d(u, v) = i, then |f (u) − f (v)| � δi .

This assignment is referred to as a g-L(δ1, δ2, . . . ,

δσ−1) colouring of the graph G [7], and CAPS (G, σ, �δ) issometimes referred to as the L(�δ) colouring problem for G.Note that a g-L(δ1, δ2, . . . , δσ−1) uses only the (g + 1)

colours in the set {0, . . . , g}, but does not necessarily use allthe (g + 1) colours. A g-L(δ1, δ2, . . . , δσ−1) colouring of G

is optimal iff g is the smallest number witnessing a solutionfor CAPS (G, σ, �δ).

Finding the optimal colouring for general graphs hasbeen shown to be NP-complete. The problem remainsNP-complete even if the input graphs are restricted to planargraphs, bipartite graphs, chordal graphs, and split graphs [4].Most of the work on this problem has dealt with specificgraphs such as grids and rings, for small reuse distance (σ )

values, and for small channel separation (δi) values, e.g., op-timal L(1, 1) colourings for rings and bidimensional grids [1],optimal L(2, 1) and L(2, 1, 1) colourings for hexagonal, bidi-mensional, and cellular grids [2], etc. Recently, Bertossiet al. [3] exhibited optimal L(δ1, 1, . . . , 1) colourings, forδ1 � �σ/2�, for bidimensional grids and rings. (See [3] for asuccinct literature survey of this problem.) Below, we refer toL(·, 1, . . . , 1) colourings by L(·, 1k) colourings.

As pointed out in [2], a lower bound for the L(1, �1k)

colouring problem is also a lower bound for the L(δ, �1k),δ > 1. Given an instance of CAPS, consider the augmentedgraph obtained from G by adding edges between all thosepairs of vertices that are at a distance of at most σ −1. Clearly,then, the size (number of vertices) of any clique in this aug-mented graph places a lower bound on an L(1, �1σ−1) colour-ing for G; the best such lower bound is given by the size of amaximum clique in the augmented graph.

In each graph G, for each σ , we identify a canonical sub-graph, T (G, σ), of the graph so that the vertices of T (G, σ)

induce a clique in the augmented graph of the graph. We willrefer to T (G, σ) as a tile. When the context is clear, we willrefer to the size of T (G, σ) simply as c(σ ).

Most (but not all) of the assignment schemes described inthis paper follow the pattern: for a given graph G, and for agiven σ ,

(1) identify T (G, σ),

(2) find the number of vertices in T (G, σ), and hence a lowerbound for the given assignment problem,

(3) describe a colouring scheme to colour all the vertices ofT (G, σ),

(4) demonstrate a tiling of the entire graph made up ofT (G, σ) to show that the colouring scheme describedcolours the entire graph, and

(5) show that the colouring scheme satisfies the given reusedistance and channel separation constraints.

3. A characterisation of optimal colourings

We first introduce the conventions we follow to representsquare grids and cellular grids. We explain tilings in bothgrids, and define some notation. Then we present our charac-terisation of optimal colourings in cellular and square grids.

For any d-dimensional lattice L, the minimal distance inthe lattice is denoted by µ(L). The infinite graph, denotedG(L), corresponding to the lattice L consists of the set of lat-tice points as vertices; each pair of lattice points that are at adistance µ(L) constitute the edges of G(L).

The lattice Zd is the set of ordered d-tuples of integers, andAd is the hyperplane that is a subset of Zd+1, and is charac-terised as the set of points in Zd+1 such that the coordinatesof each point add up to zero. µ(Zd) = 1, and the minimallength vectors in Zd are the unit vectors in each dimension.For each d > 0, for each i, j, 0 � i, j � d, i �= j , defineλd

ij = (x0, . . . , xd), where xi = 1, xj = −1, and for each

k, 0 � k � d , k �= i, j, xk = 0. Then, µ(Ad) = √2, and

the set of minimal length vectors in Ad is {λdij | i, j, 0 � i,

j � d, i �= j }. (See [5,10] for more on these lattices.)The infinite 2-dimensional square grid is, then, G(Z2), and

the infinite 2-dimensional cellular grid is G(A2).

3.1. Cellular grids

For a given value of σ , two kinds of tiles can be identified:triangular and hexagonal. The tiles are shown in figure 1(a).

It can be easily shown that:

Lemma 1. 1. The number of vertices in a triangular tile cor-responding to reuse distance σ , denoted by cT (σ ), is given bycT (σ ) = σ(σ + 1)/2.

2. The number of vertices in a hexagonal tile corre-sponding to reuse distance σ , denoted by cH (σ), is given bycH (σ) = (3σ 2 +(σ mod 2))/4. Thus, when σ = 2k+1(odd),c(σ ) = 3k2 + 3k + 1.

A CHARACTERISATION OF OPTIMAL CHANNEL ASSIGNMENTS 91

Figure 1. Cliques in cellular and square grids.

Figure 2. Basis vectors in A2.

σ = 0 mod 4 σ = 2 mod 4 σ = 1, 3 mod 4(0, 0) (0, 0) (0, 0)

(σ

2− 1, 0

) (σ

2, 0

) (⌊σ

2

⌋, 0

)

(σ − 1,

σ

2

) (σ − 1,

σ

2− 1

) (σ − 1,

⌊σ

2

⌋)

(σ − 1, σ − 1) (σ − 1, σ − 1) (a − 1, σ − 1)(σ

2− 1, σ − 1

) (σ

2, σ − 1

) (⌊σ

2

⌋, σ − 1

)

(0,

σ

2

) (0,

σ

2− 1

) (0,

⌊σ

2

⌋)

Figure 3. Coordinates of the corners of a hexagonal tile.

From the above lemma, we observe that for a given valueof σ , the size of a hexagonal tile is greater than the size of atriangular tile. As mentioned in the previous section, the sizeof the maximum clique, i.e. cH (σ) places a lower bound onthe colouring of G(A2). Henceforth, for G(A2), we consideronly hexagonal tiles and the word tile refers to hexagonal tileunless otherwise mentioned. Also, we refer to cH (σ) simplyas c(σ ).

For a particular σ , hexagons are regular with sides ofσ/2 vertices if σ is odd. In case a is even, alternate sidesof the hexagon are equal and consecutive sides have σ/2 and(σ + 1)/2 vertices, respectively.

Figure 2 shows the coordinate system we use for represent-ing vertices in A2 where (0, 1,−1) and (1,−1, 0) indicate thebasis vectors i and j . In the table in figure 3, we list the co-ordinates of the corners of a hexagon in clockwise order, forvarious values of σ . We start with the left-most vertex, whichwe refer to as the origin and assign (0, 0) as its coordinates.Consider the arrangement of tiles as shown in figure 4. It isclear that such an arrangement will tile all of A2. Note thatany translation of this tiling will also tile A2.

In such a tiling, we will refer to two tiles as neighbours ifthere is an edge e1 of one and an edge e2 of the other suchthat at least two points on e1 have neighbours on e2 and viceversa. In such a tiling of A2, every tile is surrounded by

Figure 4. Tiling of A2 and Z2.

six neighbouring tiles. For any tile H , we will refer to theneighbouring tiles as H0,H1, . . . , H5 as shown in figure 4.If the coordinates of the origin of H are (0, 0), then the ori-gins of H0,H1, . . . , H5 will have coordinates (−�σ/2�,−σ),(σ/2,−�σ/2�), (σ, σ/2), (�σ/2�, σ ), (−σ/2, �σ/2�)and (−σ,−σ/2), respectively. These points are marked infigure 4. The edge of tile H which is adjacent to the tile Hi

will be denoted by ti .

Definition 1. In a cellular grid tile, we define a diagonal to bea line formed by all vertices having the same ith coordinate.In a tile with origin (io, jo), diagonal corresponding to thecoordinate ic is represented as Lic−io and (ic − io) is calledthe diagonal number.

In a tile corresponding to reuse distance σ , there are σ

diagonals. Figure 5 shows the diagonals in a cellular grid.

3.2. Square grids

As mentioned in section 2, the size of the maximum cliquefor a particular σ places a lower bound on the colouring ofG(Z2). The following lemma gives a formula for the size ofsuch a clique which is also referred to as a tile.

Lemma 2. The number of vertices in a tile corresponding toreuse distance σ , denoted by c(σ ) is given by c(σ ) = σ 2/2.Thus, when σ = 2k + 1 (odd), c(σ ) = 2k2 + 2k + 1.

For a particular σ , tiles are diamonds with their diagonalsalong the X and Y axes (as shown in figure 1(b)) and everyside contains σ/2 vertices. They tile the entire grid Z2. Inthe case of odd σ , every corner of the tile corresponds toa vertex on the grid. We use the vectors (1, 0) and (0, 1)

as the basis vectors i and j for representing points in Z2.Then, the coordinates of the vertices of the corners of a tile

92 SHASHANKA ET AL.

Figure 5. Verticals and diagonals.

Figure 6. Possible tiling of Z2 and A2 for odd σ .

(in clockwise order, starting with the left-most vertex) are(0, 0), (�σ/2�, �σ/2�), (σ − 1, 0) and (�σ/2�,−�σ/2�). Inthe case of even σ , only opposite corners of the tile along theX direction correspond to vertices on the grid, their coordi-nates being (0, 0) and (σ − 1, 0).

Consider the arrangement of tiles as shown in figure 4. Itis clear that such an arrangement will tile all of Z2. Note thatany translation of this tiling will also tile Z2.

In such a tiling, we will refer to two tiles as neighbours ifthere is an edge e1 of one and an edge e2 of the other suchthat at least two points on e1 have neighbours on e2 and viceversa. In such a tiling of Z2, every tile is surrounded by fourneighbouring tiles. For any tile H , we will refer to the neigh-bouring tiles as H0,H1,H2 and H3. If the coordinates of theleft-most vertex of H (which we refer to as the origin) are(0, 0), then the origins of H0,H1,H2 and H3 will have coor-dinates (−�σ/2�, σ/2), (σ/2, �σ/2�), (�σ/2�,−σ/2)and (−σ/2,−�σ/2�), respectively. These points are shownin figure 4. The edge of tile H which is adjacent to the tile Hi

will be denoted by ti .There is another kind of tiling possible in both cellular and

square grids for odd reuse distances, as shown in figure 6 forσ = 5. We shall refer to the tiling shown in figure 4 as tiling A

and the one in figure 6 as tiling B.

Definition 2. In a square grid tile, we define a vertical to bea line formed by all vertices having the same X-coordinate.In a tile with origin (io, jo), a vertical corresponding to thecoordinate ic is represented as Vic−io and (ic − io) is calledthe vertical number.

Figure 7. Bounding box B(p) with edges marked.

Definition 3. In a square grid tile, we define a diagonal tobe a line of the form i − j = c, where c is a constant. Itis represented as Di , where i, called the diagonal number isgiven by (i − j) mod σ .

In a tile corresponding to reuse distance σ , there are σ di-agonals/verticals as the case may be. Figure 5 shows verticalsand diagonals of a square grid tile.

Definition 4. 1. Consider a point p in a square/cellular gridand consider all points which are at a distance σ from p,where σ is the reuse distance. In the case of square grids,all these points form a diamond centered at p and in the caseof cellular grids, they form a hexagon centered at p. This di-amond/hexagon will be called the bounding box surroundingpoint p and will be denoted by B(p). The edges, consideredin a clockwise direction, are denoted by d0, d1, . . . , d3 in caseof square grids and d0, d1, . . . , d5 in case of cellular grids, asshown in figure 7.

2. Consider the bounding box for point p. Every edgecontains σ − 1 vertices apart from the two corners. Each cor-ner, which belongs to two edges di and di+1, is taken to bea part of the second edge di+1, where i refers to (i mod s), s

being 4 in case of square grids and 6 in case of cellular grids.For each edge, we number the vertices consecutively, clock-wise, starting with 0 being assigned to the left-corner vertex.These numbers are called position numbers. This is shown infigures 8–11.


Figure 8. Cellular grid bounding box for σ = 4 with position numbers.

Figure 9. Cellular grid bounding box for σ = 5 with position numbers.

Figure 10. Square grid bounding box for σ = 4 with position numbers.

3.3. Optimal colouring schemes

A colouring scheme is optimal if it uses the smallest possiblenumber of colours. In other words, a colouring which usescolours from the set {0, 1, . . . , g} will be optimal if it uses thesmallest possible value for g. From lemmas 1 and 2, we knowthat c(σ ) is a lower bound on the number of colours used. Weare concerned only with such colouring schemes which useexactly c(σ ) different colours.

We already know that σ is the minimum distance at whichchannels can be reused. In other words, the same colour canbe used for vertices which are at distance σ or greater. Thefollowing lemma establishes that in an optimal colouring thenearest vertex where a colour is reused is no more than dis-tance σ away.

Figure 11. Square grid bounding box for σ = 5 with position numbers.

Lemma 3. Consider an optimal colouring scheme for a wire-less network modelled as an infinite square or cellular gridwith reuse distance σ . For a given point p, there exists at leastone point at distance σ from p which has the same colouras p.

Proof. Let us assume that, on the contrary, there is no pointat distance σ from p which has the same colour as p. Thus,no point inside, or on the boundary of, B(p) is assigned thesame colour as that of p.

Now, consider one of the edges of B(p), say d0 and a tileinside B(p) such that one of its edges t0 is completely con-tained in this edge of B(p). Clearly, p is not in this tile. Sincewe have an optimal colouring, one of the points in the tilemust be assigned the same colour as the colour assigned to p.This is a contradiction, and hence the result. �

We now present a theorem using which we will be able toestablish an important property of optimal colouring schemes.

Theorem 1. Consider an optimal colouring scheme for awireless network modelled as an infinite square or cellulargrid with reuse distance σ . For every point p, there is a po-sition number n, such that each point corresponding to thisposition number on each edge of the bounding box surround-ing p has the same colour as p. Moreover, n = σ/2 − 1 orn = σ/2.

Proof. Consider the edge d0 of the bounding box aroundp,B(p). Consider the k different tiles, each of whose edget0 is a part of the edge d0 of B(p), where σ = 2k + 1 for oddσ and σ = 2k for even σ . Refer to figures 8–11.

Let P(i) denote the sequence of position numbers on d0 ofB(p) that are on the edge t0 of the ith of these k tiles. In thecase of odd σ , P(i) are given by:

P(1) = 〈1, . . . , k + 1〉,P (2) = 〈2, 3, . . . , k + 2〉,

...

P (k) = ⟨2 + (k − 2), 2 + (k − 1), . . . , 2k

⟩.

(1)

94 SHASHANKA ET AL.

In case σ is even, P(i) are given by:

P(1) = 〈1, 2, . . . , k〉,P (2) = 〈2, 3, . . . , k + 1〉,

...

P (k) = ⟨2 + (k − 2), 2 + (k − 1), . . . , 2k − 1

⟩.

(2)

Since the colouring is optimal, the colour c, that thepoint p is coloured in, must appear somewhere on each ofthese tiles. Except for the edge t0, each of these tiles is com-pletely contained within B(p). Thus, the colour c must appearon the edge t0 of each of these tiles, otherwise, the reuse con-straint is violated. Since no pair of vertices with position num-bers 1, 2, . . . , 2k on the edge d0 of B(p) are at a distance σ ,it must be the case that the colour c is assigned to some vertexthat is common to all the above tiles. In case of odd σ , as seenfrom equation (1), the only two common vertices are the oneswith position numbers k and k + 1, and hence, one of thesetwo vertices must be assigned the colour c. In case of even σ ,we see from equation (2) that the only common vertex is theone corresponding to position number k, and hence, it has tobe assigned the colour c. A similar argument establishes thaton each edge of B(p), the vertices corresponding to positionnumbers k and k + 1 in case of odd σ and k in case of even σ

are the only possible candidates for being assigned colour c.Now, in case of odd σ , let q be the vertex, corresponding

to position number k on the edge d0 of B(p), that is assignedcolour c (see figures 9 and 11). Suppose, by way of contra-diction, the vertex with position number k + 1 on the edge d1

of B(p) is assigned colour c. Let us name this vertex x. Wewill now consider cellular and square grids separately in twodifferent cases.

Case 1 (Cellular grids). Consider the bounding box B(q).The edge d2 of B(q) passes through the vertex with positionnumber k on the edge d1 of B(p) as shown in figure 9. Bythe above argument, one of the two vertices with positionnumbers k or k + 1 on this edge d2 of B(q) must be as-signed colour c. But both these vertices are at a distance lessthan σ from the vertex x. Therefore, x cannot be assignedcolour c, implying that the vertex with position number k onthe edge d1 must be assigned colour c.

Case 2 (Square grids). Consider the bounding boxes B(q)

and B(x). Let r be the point of inter-section of the edges d1

of B(q) and d0 of B(x) (see figure 11). If both q and x arecoloured c, it follows that r should be assigned the colour c.This is not possible because r lies within the bounding boxB(p) of point p which is also coloured c. This implies thatthe vertex with position number k on the edge d1 must beassigned colour c.

Similar arguments in both cases above establish that, if thevertex with position number k on any one edge of B(p) iscoloured the same as the colour of p, then on each edge ofB(p), the vertex with position number k is also coloured thesame as p. �

The following characterisation of optimal colourings ofcellular and square grids is an immediate consequence of the-orem 1.

Theorem 2. Given σ , and given a tiling of a cellular orsquare grid by tiles (for σ ), a colouring with reuse distance σ

is optimal iff all the tiles in the tiling are identical in theircolour assignment.

Recall the definition of tilings A and B from section 3.2(see figures 4 and 6). From the proof of theorem 1, we makethe following observation.

Corollary 1. Suppose σ = 2k + 1, and we have an optimalcolouring of the cellular (square) grid. If for any point p inthe grid, the vertex corresponding to position number k on anedge of the bounding box of p has the same colour as that as-signed to p, then the tiling of the grid, by identically colouredtiles, corresponds to tiling B; if the position number is k + 1,then the tiling of the grid corresponds to tiling A.

4. Optimal L(δ1, �1σ−2) colourings for G(A2) and G(Z2)

In this section, we deal with optimal frequency assignmentschemes for wireless networks modelled as cellular gridsand square grids. We first present an L(δ1, �1σ−2) colouringscheme of G(A2) for the case where reuse distance is oddi.e., σ = 2k + 1, k ∈ {1, 2, . . .}. This is followed by anL(δ1, �1σ−2) colouring scheme of G(Z2) for all values of σ .The colouring schemes presented here correspond to tiling A.

4.1. Cellular grids

We present a colouring scheme where δ1 varies as the squareof σ , for σ � 5, σ odd. We note that the colouring of theentire cellular grid is achieved by colouring one tile and re-producing the same colouring in all the tiles present in thegrid. Recall that the number of vertices c(σ ), in a tile cor-responding to an odd reuse distance σ = 2k + 1 is equal to3k2 + 3k + 1 (see lemma 1). From the above fact and fromtheorem 2, we make the following observations (refer to fig-ure 12).

Lemma 4.

1. Colouring c(σ ) points starting from the vertex of a tilealong the direction j is equivalent to colouring all the di-agonals of a tile in the following order:

L0, Lk+1, L1, Lk+2, . . . , Lk−1, L2k, Lk.

2. Along a line i = m, where m is a constant, any pair ofpoints which are at a distance c(σ ) apart will have the samecolour assigned to them.

3. Consider a point (p, q) on the line i = p. The point(p + 1, q − 3k − 1) on the line i = p + 1 will have thesame colour as (p, q).


Figure 12. L(δ1, �1σ−2) colouring for σ = 7.

From lemma 4, we see that a colouring for c(σ ) pointsalong a line i = m for some arbitrary m describes the colour-ing for the entire grid.

The following colouring scheme is shown in figure 12. Tocolour along the line i = 0, we proceed as follows: Startingwith the point (0, 0) which is assigned the colour 0, we assignconsecutive colours to every third vertex, and wrap aroundafter the c(σ )th vertex. This will colour all the c(σ ) points inthree passes uniquely. This can be easily seen because c(σ ) =c′(k) = 3k2 + 3k + 1 = 1 (mod 3). Consecutive sets of c(σ )

vertices along this line follow the same colouring pattern.Formally, this colouring scheme can be expressed as fol-

lows. Let χ(i, j) represent the colour assigned to the vertex(i, j) and χ ′(j) represent the colour assigned to the vertex(0, j), i.e. χ ′(j) = χ(0, j). We first give a formula for χ ′(j)

and then derive an expression for χ(i, j).

χ ′(j) =

ρ, j = 0 (mod 3),

ρ + 2k2 + 2k + 1, j = 1 (mod 3),

ρ + k2 + k + 1, j = 2 (mod 3),

where j = j mod c(σ ) and ρ = �j /3�. Now, from lem-ma 4.3, we can easily derive that

χ(i, j) = χ ′(j + i(3k + 1)).

Theorem 3. For all σ = 2k+1, k = {1, 2, . . .}, the colouringscheme described above is an optimal L(δ1, �1σ−2) colouringfor G(A2), with δi = k2. Moreover, this is a constant timecolouring scheme.

Proof. From lemma 1, c(σ ) is a lower bound. We can easilysee that each vertex in the tile is assigned a unique colour fromthe set {0, 1, . . . , c(σ ) − 1}. This implies that the optimalitycondition is satisfied.

Again, the above scheme ensures that correspondingpoints in neighbouring tiles have the same colour and are ex-actly σ distance apart. Thus, the re-use constraint is satisfied.

To derive the value of δ1, we proceed as follows. Considera point (i, j) in the grid. Its six neighbours are (i, j − 1), (i +1, j), (i + 1, j + 1), (i, j + 1), (i − 1, j) and (i − 1, j − 1).

(i, j − 1) (χ(i, j) + k2 + k)

(i + 1, j) (χ(i, j) + 2k2 + 3k + 1)

(i + 1, j + 1) (χ(i, j) + k2 + 2k + 1)

(i, j + 1) (χ(i, j) + 2k2 + 2k + 1)

(i − 1, j) (χ(i, j) + k2)

(i − 1, j − 1) (χ(i, j) + 2k2 + k)

Figure 13. Colours (mod c(σ )) assigned to neighbours of (i, j) by thescheme χ .

The colour assigned to (i, j) according to the abovescheme will be χ(i, j) = χ ′(j + i(3k + 1)).

The table in figure 13 shows the colours assigned to theneighbours of (i, j). (All the colour expressions are moduloc(σ ).)

From the table in figure 13, we see that the least differencebetween the colours assigned to neighbouring points is k2.Hence, δ1 = k2.

From the formula for χ(i, j), it can be easily seen thatgiven any arbitrary point (i, j) in the grid, the colour assignedto (i, j) can be computed in constant time. �

Lemma 5 notes the values of δ2 and δ3 for the abovecolouring.

Lemma 5. For all σ = 2k + 1, k = {1, 2, . . .}, the colouringscheme described above has the properties that δ2 = k andδ3 = 1.

Proof. Similar to the proof of theorem 3 above, using thetable in figure 13 twice proves the value of δ2. The value ofδ3 is 1 by construction. �

4.2. Square grids

We present colouring schemes where δ1 varies as the squareof σ , for σ � 4. There are two different schemes, one forthe case where σ is odd and one for even σ . We note that thecolouring of the entire square grid is achieved by colouringone tile and reproducing the same colouring in all the tilespresent in the grid.

Odd σ . Recall that the number of vertices c(σ ), in a tilecorresponding to an odd reuse distance σ = 2k + 1 is equalto 2k2 + 2k + 1 (see lemma 2). From the above fact andfrom theorem 2, we make the following observations (refer tofigure 14):

Lemma 6.

1. Colouring c(σ ) points starting from the vertex of a tilealong the direction j is equivalent to colouring all the di-agonals of a tile in the following order:

V0, Vk, V2k, Vk−1, . . . , Vk+2, V1, Vk+1.

2. Along a line i = m, where m is a constant, any pair ofpoints which are at a distance c(σ ) apart will have the samecolour assigned to them.

96 SHASHANKA ET AL.


3. Consider a point (p, q) on the line i = p. The point(p + 1, q + 2k + 1) on the line i = p + 1 will have thesame colour as (p, q).

From lemma 6, we see that a colouring for c(σ ) pointsalong a line i = m for some arbitrary m describes the colour-ing for the entire grid.

The following colouring scheme is shown in figure 14. Tocolour along the line i = 0, we proceed as follows: Startingwith the point (0, 0) which is assigned the colour 0, we assignconsecutive colours to every second vertex, and wrap aroundafter the c(σ )th vertex. This will colour all the c(σ ) points intwo passes uniquely. This can be easily seen because c(σ ) =c′(k) = 2k2 + 2k + 1 is odd and hence, points coloured in thefirst pass will not be repeated again. Consecutive sets of c(σ )

vertices along this line follow the same colouring pattern.Mathematically, this colouring scheme can be expressed

as follows. Let �(i, j) represent the colour assigned to thevertex (i, j) and �′(j) represent the colour assigned to thevertex (0, j), i.e. �′(j) = �(0, j). We first give a formulafor �′(j) and then derive an expression for �(i, j).

�′(j) ={

ρ, j even,

ρ + k2 + k + 1, j odd,

where j = j mod c(σ ) and ρ = �j /2�. Now, from lem-ma 6.3, we can easily derive that

�(i, j) = �′(j − i(2k + 1)).

Theorem 4. For all σ = 2k+1, k = {1, 2, . . .}, the colouringscheme described above is an optimal L(δ1, �1σ−2) colouringfor G(Z2), with δ1 = k2. Moreover, this is a constant timecolouring scheme.


(i − 1, j) (�(i, j) + k2 + 2k + 1)

(i, j + 1) (�(i, j) + k2 + k + 1)

(i + 1, j) (�(i, j) + k2)

(i, j − 1) (�(i, j) − k2 − k − 1)

Figure 15. Colours (mod c(σ )) assigned to the neighbours of (i, j) by thescheme � when σ is odd.


Again, the above scheme ensures that correspondingpoints in neighbouring tiles have the same colour and are ex-actly σ distance apart. Thus, the re-use constraint is satisfied.

To derive the value of δ1, we proceed as follows. Con-sider a point (i, j) in the grid. Its four neighbours are(i − 1, j), (i, j + 1), (i + 1, j), and (i, j − 1).

The colour assigned to (i, j) according to the abovescheme will be �(i, j) = �′(j − i(2k + 1)).

The table in figure 15 shows the colours (modulo c(σ ))assigned to the neighbours of (i, j).

From the table in figure 15, we see that the least differencebetween the colours assigned to neighbouring points is k2.Hence, δ1 = k2.

From the formula for �(i, j), it can be easily seen thatgiven any arbitrary point (i, j) in the grid, the colour assignedto (i, j) can be computed in constant time. �

Even σ . We now present a colouring scheme for even σ ,σ � 4, i.e. σ = 2k, k ∈ {2, 3, . . .}. We first note that the totalnumber of points in a tile in terms of k will be equal to 2k2.Since colouring of the entire grid is achieved by colouring onetile and reproducing the same colouring in all tiles of the grid,description of the colouring for a single tile is sufficient.

The colouring scheme is shown in figure 16. Alternate di-agonals are coloured consecutively starting with D0, i.e. thefollowing diagonals D0,D2, . . . ,Dσ−2,D1,D3, . . . ,Dσ−1are coloured in order. Starting with the origin of the tile whichis assigned colour 0, points are coloured consecutively withineach diagonal.

Let �(i, j) be the colour assigned to the point (i, j) in thegrid. It can be mathematically expressed as follows:

�(i, j) =⌊

(i − j) mod 2k

2

⌋k +

⌊(i + j) mod 2k

2

⌋

+ ((i + j) mod 2

)k2.


(i − 1, j) �(i, j) − k2

(i, j + 1) �(i, j) − k2 − k + 1,if (i + j) mod 2k = 2k − 1

�(i, j) − k2 + 1, otherwise(i + 1, j) �(i, j) − k2 + 1,

if (i + j) mod 2k = 2k − 1�(i, j) − k2 + k + 1, otherwise

(i, j − 1) �(i, j) − 2k2 + k,if (i − j) mod 2k = 2k − 1

�(i, j) − k2 + k, otherwise

Figure 17. Colours (mod c(σ )) assigned to the neighbours of (i, j) by thescheme � when σ is even.

Theorem 5. For all σ = 2k, k = {2, 3, . . .}, the colouringscheme described above is an optimal L(δ1, �1σ−2) colouringfor G(Z2), with δ1 = k2 − k − 1. Moreover, this is a constanttime colouring scheme.


Again, from the formula, we see that corresponding pointsin neighbouring tiles have the same colour and are exactly σ

distance apart. Thus, the re-use constraint is satisfied.To derive the value of δ1, we proceed as follows. Consider

the neighbours of an arbitrary point (i, j) in the grid. They are(i −1, j), (i, j +1), (i +1, j) and (i, j −1). We will find thedifferences between the colours assigned to (i, j) and each ofits neighbours. The least difference will be equal to δ1.

There are two cases to consider: (1) (i + j) is even, and(2) (i + j) is odd. Note that (i + j) value for alternate pointsin both X and Y directions will be of the same parity. If weconsider a point for which (i + j) is odd, (i + j) for all itsneighbours will be even and vice versa. It follows that weneed to consider only one case, as considering the other casewill yield the same expressions for the differences.

Consider a point (i, j) and suppose (i + j) is odd. Let thecolour assigned to (i, j) be �(i, j). The table in figure 17shows the colours assigned to the neighbours of (i, j).

Clearly, from the table in figure 17, the least differencebetween the colours assigned to neighbouring points is k2 −k − 1. Hence, δ1 = k2 − k − 1.

From the formula for �(i, j), it can be easily seen thatgiven any arbitrary point (i, j) in the grid, the colour assignedto (i, j) can be computed in constant time. �

5. Upper bound on δ1

The previous subsections presented colouring schemes forodd reuse distances where δ1, the channel separation con-straint, has a value of k2, where σ = 2k + 1. Lemma 7provides an upper bound on δ1.

Lemma 7. For all σ = 2k + 1, k = {1, 2, . . .}, for any op-timal L(δ1, �1σ−2 colouring for G(A2) (G(Z2), respectively)δ1 < k2 + k.

Proof. Suppose C is an optimal L(δ1, �1σ−2) colouring forG(A2). Since there are 3k2 + 3k + 1 vertices in a tile ofG(A2), C assigns each number in {0, . . . , 32 + 3k} to somevertex in G(A2). Let δ1 = δ be the separation between thecolours assigned by C to any two adjacent vertices.

Consider the vertex v assigned the colour δ−1. Each of itssix neighbours must be assigned a colour that is at least 2δ−1.Since (1) the neighbours of v form a cycle of length 6, and (2)each adjacent pair of vertices in the cycle must be assignedcolours differing by at least δ, it follows that at least three ofthese vertices must each be assigned a colour that is at least3δ − 1. Thus, at least one neighbour of v must be assigned acolour that is at least 3δ + 1. Then,

3δ + 1 � 3k2 + 3k ⇒ δ < k2 + k.

A similar argument can be used to show that for an optimalL(δ1, �1σ−2) colouring of G(Z2), δ < k2 + k. �

If σ = 2k is even, the size of the tile in G(A2) and in G(Z2)

is 3k2 and 2k2, respectively. Then, an argument similar to theone in the proof of lemma 7 can be used to show that

Lemma 8. For all σ = 2k, k = {1, 2, . . .}, for any opti-mal L(δ1, �1σ−2) colouring for G(A2) (G(Z2), respectively)δ1 < k2.

Based on experimental verification by means of an exhaus-tive search for all values of k � 4, we conjecture that:

Conjecture 1. For all σ , for any optimal L(δ1, �1σ−2)-col-ouring for G(A2) (G(Z2), respectively) δ1 � (�σ/2�)2.

The conjecture implies a tighter upper bound, for odd σ ,than the one presented in lemma 7 above. Note that, forodd σ , our assignments presented in sections 4.1 and 4.2 dorealise this value for δ1.

6. Conclusions and open problems

We characterised optimal channel assignment schemes forcellular and square grids, and hence showed that any suchscheme must be uniform across the entire grid. More specif-ically, in an optimal colouring, the colouring of a tile (for agiven σ ) will be identically repeated in all the tiles through-out the grid. We also presented optimal L(δ1, �1σ−2) colour-ing schemes, with a high value for δ1, for square grids for allσ � 4 and for cellular grids for the case where reuse distanceis odd, i.e., σ = 2k + 1, k ∈ {1, 2, . . .}. The previous bestknown results have been restricted to δ1 � 3k2/8 [6], in caseof cellular grids and δ1 � �(σ − 1)/2� [3] in case of squaregrids. We prove an upper bound for δ1 for optimal colouringsof cellular and square grids. In the case of σ being odd, weconjecture that our value of δ1 is a tight upper bound on δ1 foroptimal colouring schemes for these grids.

Several interesting open questions arise from the work pre-sented here. We list a few of them here:

98 SHASHANKA ET AL.

(1) Find optimal colouring schemes for cellular grids withhigh δ1 values for the case when σ is even.

(2) Find and prove the existence of tight upper bounds forδ1, δ12, . . . for a general σ .

References

[1] R. Battiti, A.A. Bertossi and M.A. Bonuccelli, Assigning codes in wire-less networks: Bounds and scaling properties, Wireless Networks 5(1999) 195–209.

[2] A.A. Bertossi, C.M. Pinotti and R.B. Tan, Efficient use of radio spec-trum in wireless networks with channel separation between close sta-tions, in: Proc. of the DIAL M Workshop (2000) pp. 18–27.

[3] A.A. Bertossi, C.M. Pinotti and R.B. Tan, Channel assignment withseparation for special classes of wireless networks: Grids and rings, in:Proc. of IPDPS (2002).

[4] H.L. Boedlander, T. Kloks, R.B. Tan and J. van Leeuwen, λ-coloringof graphs, in: Proc. of STAGS (2000) pp. 395–406.

[5] J. Conway and N. Sloane, Sphere Packings, Lattices and Groups, 2nded. (Springer, Berlin, 1993).

[6] A. Dubhashi, A. Pati, M.V.S. Shashanka, R. Shashank andA.M. Shende, Channel assignment in wireless networks modelled ascellular and d-dimensional square grids, http://citeseer.nj.nec.com/ (2002).

[7] J.R. Griggs and R.K. Yeh, Labeling graphs with a condition at dis-tance 2, SIAM J. Discrete Math. (1992) 586–595.

[8] W.K. Hale, Frequency assignment: Theory and application, Proc. IEEE68 (1980) 1497–1514.

[9] R.A. Murphey, P.M. Pardalos and M.G.C. Resende, Frequency assign-ment problems, in: Handbook of Combinatorial Optimization (KluwerAcademic, Dordrecht, 1999).

[10] D.S. Rajan and A.M. Shende, A characterization of root lattices, Dis-crete Math. 161 (1996) 309–314.

M.V.S. Shashanka obtained a B.E. (Hons.) from theBirla Institute of Technology & Science, Pilani, In-dia. He is currently a graduate student in the De-partment of Cognitive and Neural Systems at BostonUniversity, Boston, MA, USA.E-mail: [email protected]

Amrita Pati obtained a B.E. (Hons.) from the BirlaInstitute of Technology & Science, Pilani, India. Sheis currently a graduate student in the Department ofComputer Science at the Virgina Polytechnic Insti-tute and State University, Blacksburg, VA, USA.E-mail: [email protected]

Anil M. Shende is an Associate Professor of Com-puter Science at Roanoke College, Salem, VA, USA.Part of this work was done while he was at the BirlaInstitute of Technology & Science, Pilani, India onsabbatical leave.E-mail: [email protected]


CARD: A Contact-based Architecture for Resource Discovery inWireless Ad Hoc Networks

AHMED HELMY ∗, SAURABH GARG and NITIN NAHATA3740 McClintock Avenue, EEB 232, Electrical Engineering Department, University of Southern California, Los Angeles, CA 90089-2562, USA

PRIYATHAM PAMUComputer Science Department, University of Southern California, Los Angeles, CA 90089-2562, USA

Abstract. Traditional protocols for routing in ad hoc networks attempt to obtain optimal or shortest paths, and in doing so may incursignificant route discovery overhead. Such approaches may be appropriate for routing long-lived transfers where the initial cost of routediscovery may be amortized over the life of the connection. For short-lived connections, however, such as resource discovery and smalltransfers, traditional shortest path approaches may be quite inefficient. In this paper we propose a novel architecture, CARD, for resourcediscovery in large-scale wireless ad hoc networks. Our mechanism is suitable for resource discovery as well as routing very small datatransfers or transactions in which the cost of data transfer is much smaller than the cost of route discovery. Our architecture avoids expensivemechanisms such as global flooding and complex hierarchy formation and does not require any location information. In CARD resourceswithin the vicinity of a node, up to a limited number of hops, are discovered using a proactive scheme. For resources beyond the vicinity,each node maintains a few distant nodes called contacts. Contacts help in creating a small world in the network and provide an efficient wayto query for distant resources. Using contacts, the network view (or reachability) of the nodes increases, reducing the discovery overhead andincreasing the success rate. On the other hand, increasing the number of contacts also increases control overhead. We study such trade-offin depth and present mechanisms for contact selection and maintenance that attempt to increase reachability with reduced overhead. Ourschemes adapt gracefully to network dynamics and mobility using soft-state periodic mechanisms to validate and recover paths to contacts.Our simulation results show that CARD is scalable and can be configured to provide desirable performance for various network sizes.Comparisons with other schemes show overhead savings reaching over 93% (vs. flooding) and 80% (vs. bordercasting or zone routing) forhigh query rates in large-scale networks.

Keywords: energy efficient, sensor networks, routing

1. Introduction

Ad hoc networks are wireless networks composed of mobiledevices with limited power and transmission range. Thesenetworks are rapidly deployable as they neither require awired infrastructure nor centralized control. Because of thelack of fixed infrastructure, each node also acts as a relayto provide communication throughout the network. Appli-cations of ad hoc networks include coordination betweenvarious units (e.g., in a battlefield), search and rescue mis-sions, rapidly deployable networks, and vehicular networks,among others. Although research on mobile ad hoc networks(MANets) has attracted a lot of attention lately, little attentionhas been given to resource discovery in large-scale MANets.In addition, a very important mode of communication thathas been largely ignored in the ad hoc networks literature isthat of short flows and small transactions, where the com-munication cost of discovering shortest routes is usually thedominant factor (not the data transfer as in long flows). Forsuch short flows reducing overhead (not route optimization)is the main design goal. Current routing protocols in gen-

∗ A. Helmy was supported by NSF CAREER Award 0134650, and researchgrants from Intel Corp. and Pratt & Whitney Institute for CollaborativeEngineering.

eral attempt to discover optimal (shortest path) routes. Inour study, instead of obtaining shortest paths, we focus onreducing the overhead of resource (or route) discovery forshort flows. Examples of resource discovery and small trans-fers in ad hoc networks include discovering servers, objectsand capabilities (e.g., GPS capable nodes), instant and textmessaging, short transactions, DNS-like queries, paging, anddissemination of sensory data in sensor and vehicular net-works.

In ad hoc networks, lack of infrastructure renders resourcediscovery a challenging problem. In addition, mobility in-duces frequent route changes. Traditional protocols proposedfor resource discovery employ either global flooding or com-plex hierarchy formation schemes. While flooding is ineffi-cient and does not scale well, hierarchy formation involvescomplex coordination between nodes and therefore may suf-fer significant performance degradation due to frequent, mo-bility induced, changes in network connectivity.

To overcome these limitations we propose a new architec-ture for efficient resource discovery in large-scale ad hoc net-works, called CARD. Our study targets resource discoveryand routing for short flows. CARD is not a general routingprotocol, as we make a design decision to trade-off shortestpaths for drastic reduction in discovery overhead. CARD,

100 HELMY ET AL.

however, may be integrated easily with zone routing proto-cols to compose a general routing solution.

Nodes in ad hoc networks are usually portable devices withlimited battery power. Therefore to save power the resourcediscovery mechanism should be efficient in terms of commu-nication overhead. Our architecture is designed to meet re-quirements for power-efficient resource discovery and smalltransfers in large-scale ad hoc networks with (potentially)thousands of wireless devices. Scalability is one of our maindesign goals.

Our architecture is based on the concept of small worlds[8,26,27] where the addition of a small number of short cutsin highly clustered networks results in significant reductionin the average path length (or degrees of separation) to ap-proach that of random networks. In our architecture we adopta hybrid approach in which a node uses periodic updates toreach its vicinity within a limited number of hops, R, and re-active querying beyond the vicinity via contacts. Contactsact as short cuts that attempt to transform the network into asmall world by reducing the degrees of separation between thesource and destination of the transfer. They help in providinga view of the network beyond the vicinity during resource dis-covery. Each node maintains state for a few contacts beyondits vicinity. Contacts are polled periodically to validate theirpresence and routes. For discovering resources efficiently,queries are sent to the contacts that leverage the knowledgeof their vicinity. As the number of contacts increases, the net-work view (or reachability) increases. However, at the sametime the overhead involved in contact selection and mainte-nance also increases. Our results show this trade-off. Weintroduce and study alternative mechanisms for contact selec-tion and identify a novel scheme (called the edge method forcontact selection) that is able to achieve a balanced trade-offand good performance in terms of increased reachability andreduced overhead.

Once the contacts are selected by a node they are used inthe resolution of resource discovery queries. Only the contactnodes are queried without the need for flooding, resulting indrastic reduction in per-query communication overhead. Thetotal overhead, however, is the resultant of

(i) the query overhead, which is a function of the per-queryoverhead and the query rate,

(ii) the vicinity establishment and maintenance overhead,which is a function of the node mobility, and

(iii) the contact selection and maintenance overhead.

Our study elaborates on the interplay between these var-ious overhead components, the query rate, and the mobilityrate using the call-to-mobility-ratio (CMR) metric.

Extensive simulation-based comparisons with floodingand bordercasting [5,20] show our architecture to be more ef-ficient, especially for high query rates. Simulation results alsoshow that our protocol is scalable and can be configured toprovide good performance for various network sizes. Over-head savings are function of the query rate, reaching 93%(vs. flooding) and 80% (vs. bordercasting) in communication

savings for high query rates in large-scale networks; a drasticimprovement in performance.

The rest of this document is organized as follows. Sec-tion 2 discusses related work. Section 3 describes our designgoals and provides an overview of our architecture, CARD,and introduces the contact selection, maintenance and queryalgorithms. Section 4 presents analysis of CARD, and com-pares it to flooding, smart flooding and bordercasting. Weconclude in section 5.

2. Related work

Related research lies in the areas of routing and resource dis-covery in ad hoc networks. Due to lack of infrastructure in adhoc networks, resource (and route) discovery is a challeng-ing problem. Most of the routing protocols can be broadlyclassified as: proactive (table-driven), reactive (on-demand),hybrid, or hierarchical.

Proactive schemes such as DSDV [21], WRP [18] andGSR [2] flood periodic updates throughout the network. Thisis resource consuming, especially for large-scale networks.Reactive schemes such as AODV [22] and DSR [13] attemptto reduce the overhead due to periodic updates by maintainingstate only for the active resources and using route caching. Inthese schemes a search is initiated for new discovery requests.However, the search procedure generally involves flooding (orexpanding ring search), which also incurs significant over-head. Furthermore, the performance of on-demand routingwith caching has been shown to degrade significantly withsmall transfers in large-scale mobile networks.

Hybrid schemes such as the zone routing protocol (ZRP)[5,20] try to combine the benefits of both the proactive and re-active schemes. ZRP limits the overhead of periodic updatesto a limited number of hops (called the zone radius). Re-sources beyond the zone are discovered in a reactive mannerby sending queries through nodes at the edges of the zones(bordercasting). The zone concept is similar to the vicin-ity concept in our study. However, instead of bordercastingwe use contact queries. The design principles upon whichour CARD architecture was designed – employing contactsas short cuts to create a small world, and trading off optimalpaths for energy efficiency – are fundamentally different fromthose used for ZRP bordercast. In our study, through detailedcomparison we show that the contact-based approach is muchmore efficient than bordercasting for our purposes. Further-more, CARD maybe easily integrated with ZRP to provide acomplete routing protocol in which ZRP is used to discoverroutes for long-lived flows and CARD is used for resourcediscovery and small transfers.

Hierarchical schemes, such as CGSR [3,15], tend to havegood scalability, but involve election of cluster-heads, thathave greater responsibilities than other nodes. A cluster-headis responsible for routing traffic in and out of the cluster.Cluster-based hierarchies rely on complex coordination andthus are susceptible to major re-configuration due to mobilityand node failure, leading to serious performance degradation

CARD: A CONTACT-BASED ARCHITECTURE FOR RESOURCE DISCOVERY 101

in highly dynamic networks. Also, a cluster head may be asingle point of failure and a potential bottleneck. In our ar-chitecture each node has its own view of the network, andhence there is very little coordination between various nodes.This enables our architecture to adapt gracefully to networkdynamics. GLS [14] provides a location-discovery service forgeographic routing. GLS requires nodes to know of a networkgrid map and assumes knowledge of node locations (via GPSor other). CARD does not require location information.

Related work on smart or efficient flooding has been pro-posed in [4,6,16,19]. These techniques attempt to reduce theredundancy inherent in flooding, and may be integrated in ourwork to provide more efficient vicinity establishment insteadof regular link state protocol. One major difference betweensmart flooding and CARD is that smart flooding reduces theredundant messages in querying every node in the network,whereas CARD attempts to create a small world and onlyqueries a small number of nodes on the order of the degrees ofseparation from source to target. In relatively sparse networks(some of which we include in our study) smart flooding willnot be very effective since there is no significant redundancyin flooding anyway. Section 4.3 discusses this issue further.

In [8] we have shown the relationship between smallworlds and wireless networks. In this paper, we buildupon that relationship by introducing the contacts to act asshort cuts in the highly clustered multi-hop wireless network,proposing and evaluating – in details – two proactive contact-selection mechanisms. We first introduced the high level ideaof using contacts in [7]. The initial work on the CARD archi-tecture was presented in [11]. This work extends the analysisof the CARD architecture and explores the important inter-play between the query rate and mobility rate. The MARQarchitecture [9] provides a mobility-assisted contact selec-tion mechanism, the efficiency of which increases with mo-bility. In cases of static networks (e.g., sensor networks), orwhen mobility is low, CARD may be used in conjunction withMARQ for efficient query resolution. TRANSFER [10] pro-vides a reactive (on-the-fly) contact selection mechanism toreduce node-contact vicinity overlaps, but does not explicitlyreduce the contact-contact vicinity overlap because contactsare selected in parallel. CARD, by virtue of selecting contactsproactively and using the edge method for contact selectionin serial is able to guarantee non-overlapping node-contactvicinities and reduce the contact-contact vicinity overlap, butmay incur more overhead for periodic contact maintenance.ACQUIRE [23,24] is an architecture for multi-variable queryresolution in sensor networks, that uses the look-ahead tech-nique to optimize overhead. The query is forwarded from onequerying node to another d hops away, randomly. The workprovides an analytical framework to get optimal d for givenlevel of network and event dynamics. A variant of CARD’scontact selection may be used to reduce the overlap betweenthe look ahead zones for successive querying nodes to im-prove the performance of ACQUIRE.

3. CARD architectural overview

In this section we provide an overview of the CARD archi-tecture. In particular, we describe the design requirements forour architecture, present definitions and terminology used inthis document, and introduce and investigate alternative con-tact selection, maintenance and query mechanisms.

3.1. Design requirements

The design requirements of our CARD resource discovery ar-chitecture for large-scale Ad hoc networks include (a) scala-bility, (b) power-efficiency, (c) robustness, (d) decentralizedself-organization, and (e) independence of location informa-tion.

(a) Scalability. Applications of large-scale ad hoc networksinclude military and sensor network environments that mayinclude thousands of nodes. Therefore the resource discoverymechanism should be scalable in terms of control overheadwith increase in network size. We shall show that CARD maybe configured to perform very well over a wide array of net-work sizes and conditions.

(b) Power and communication efficiency. Ad hoc networksinclude portable devices with limited battery power. There-fore, resource discovery mechanisms should be power-effi-cient. CARD achieves dramatic reduction in communica-tion overhead (in terms of transmitted and received messages)over the several existing schemes considered in our study.

(c) Robustness. The mechanism should be robust in the faceof network dynamics. A periodic soft-state mechanism is pro-vided to handle node failures and frequent link failures due tomobility.

(d) Decentralized operation. For the network to be rapidlydeployable, it should not require any centralized control.CARD does not require or assume any centralized entity orspecial infrastructure.

(e) Independence of location information. GPS (or other lo-cation information) may not be available in many context(e.g., indoors, or in simple devices and sensors). Hence, as-suming availability of location information limits the applica-bility of the proposed scheme. We avoid such limitation in ourdesign and do not assume or require any location information.

3.2. Definitions

An overview of the CARD architecture is shown in figure 1.Following are some terminology definitions we use through-out this document.

• Vicinity (of a node). All nodes within a particular numberof hops (R) from the node. R is the radius of the vicinity.

• Edge nodes (of a node’s vicinity). All nodes at a distanceof exactly R hops away from the node.

• Maximum contact distance (r). The maximum distance (inhops) from the source within which a contact is selected.

102 HELMY ET AL.

Figure 1. Architectural overview for CARD: Node S (potentially any source) keeps track of nodes and resources in its vicinity, up to R hops away. S alsoelects and maintains routes to a small number of contacts (NoC) (in this case NoC = 3 contacts: C1, C2, and C3). Contacts are selected within r hops away

from S. Nodes exactly R hops away from S are called the edge nodes (Ei).

• Overlap. Overlap between nodes represents number ofcommon nodes between their vicinities.

• Number of Contacts (NoC). NoC specifies the value ofthe maximum number of contacts to be selected by eachsource node. The actual number of contacts chosen is usu-ally less than this value. This is due to the fact that fora particular value of R and r , there is only a limited re-gion available for choosing contacts. Once this region hasbeen covered by vicinities of the chosen contacts, choos-ing more contacts in the same region is not possible, astheir vicinities would overlap with the vicinities of the al-ready chosen contacts. This is according to our contactselection policy to minimize overlap.

• Depth of search (D). D specifies the levels of contacts(i.e., contacts of contacts) queried by a source.

• Reachability. The reachability of a source node refers tothe number of nodes that can be reached by the sourcenode. This includes the nodes within the vicinity that canbe reached directly and the nodes that lie in the contacts’vicinities, and their contacts’ vicinities, and so on, up to D

levels of contacts. This is also considered a measure of thediscovery success rate.

3.3. Establishing and maintaining vicinity information

Our architecture employs a hybrid of proactive and reactiveapproaches for resource discovery. As shown in figure 1, allnodes within R hops from a node form the node’s vicinity.

Each node proactively (e.g., using a link state protocol) main-tains state for resources within its vicinity. Alternatively, asmart flooding scheme (e.g., based on dominating sets) maybe used to reduce the vicinity establishment and maintenanceoverhead. For comparison reasons, however, in this studywe shall use a link state protocol similar to that used in ZRPto maintain vicinity information. The overhead of such linkstate protocol increases with node mobility and the number ofnodes within the vicinity. Such overhead under mobility sce-narios will be thoroughly studied later in section 4.3, and willbe factored into the overall overhead of CARD. As we shallsee, when the vicinity overhead is amortized over a reasonablenumber of queries the overall gain is still quite significant.

Each node also maintains state for (a few) nodes that lieoutside the vicinity. These nodes serve as contacts for ac-cessing resources beyond the vicinity. Contacts are selected,maintained and queried using the mechanisms described be-low.

3.4. Contact selection, maintenance and query mechanisms

Contacts are key to the efficient resolution of resource discov-ery queries. In CARD, the contacts are selected proactively inanticipation of queries, and paths to these contacts are main-tained using a periodic soft state mechanism to capture net-work dynamics and mobility effects. Since the contacts areselected proactively, the contact selection delays become lessof a concern (than they would otherwise in a reactive, on-the-fly, scheme, for example). Hence, contacts are selected in


a serial fashion, one after the other, with information aboutpreviously selected contacts being utilized to effectively se-lect new contacts. When a resource discovery query is issued,and the resource is not found in the vicinity, the source nodequeries its contacts first, and the contacts may query their con-tacts, and so on, until the query is resolved. Below, we in-troduce the details of the contact selection, maintenance andquery mechanisms.

3.4.1. Contact selection mechanismAny potential source of query or small transfer may chooseto select contacts. The procedure starts when a node s sendsa Contact Selection (CS) message through each of its edgenodes (Ei), one at a time, until NoC number of contacts areselected or until all edge nodes have been attempted. An edgenode receiving a CS forwards it to a randomly chosen neigh-bor (X).

A node receiving a CS decides whether or not to be a con-tact for s based on a contact selection method. This deci-sion is made using either a probabilistic method (PM) or edgemethod (EM). These methods are described later in this sec-tion. After using either procedure PM or EM for decidingwhether (or not) to be a contact, if the node receiving a CSdoes not choose to be the contact, it forwards the CS to one ofits randomly chosen neighbor (excluding the one from whichthe CS was received).

The CS traverses in a depth-first manner until a contactis chosen or the distance traversed by the CS from s reachesr hops. If a contact is still not chosen (due to overlap), CSbacktracks to the previous node, which forwards it to anotherrandomly chosen neighbor. When a contact is selected, thepath to the contact is returned and stored at s.

3.4.2. Contact selection methodsWe introduce and compare two different methods for contactselection: (a) the probabilistic method (PM), and (b) the edgemethod (EM).

(a) Probabilistic Method (PM). Contacts increase a node’sview (reachability) of the network beyond its own vicinity.To increase the reachability of a node, the vicinities of thatnode, call it s, and its contacts should be disjoint, i.e., thereshould be reduced (or no) overlap between the vicinity of s

and the vicinity of any of its contacts. The vicinities of differ-ent contacts of the same node should also be non-overlapping,to achieve good increase in reachability. To achieve this, theCS contains the following information: (i) ID of node s, (ii) alist of already-selected-contacts of s (Contact_List; typicallysmall of ∼5 IDs), and (iii) the hop count d .

This information is used as follows. When a node X

receives a CS, it first checks if s lies within its vicinity.This check is easily performed since each node has completeknowledge of its vicinity. So a node knows the IDs of all theother nodes in its vicinity. X also checks if its vicinity con-tains any of the node IDs contained in the Contact_List.

If neither s nor any of its already-selected-contacts lie inthe vicinity of X, then X probabilistically chooses itself as

(a) (b)

Figure 2. Overlap in (a) due to the use of P . (a) Heavy overlap; (b) nooverlap.

the contact. This probability (P ) of choosing to be a contactis defined as follows:

P = d − R

r − R, (1)

where d is the number of hops traversed from s to X. Thevalue of d is included in the CS as hop count. From the aboveequation, when d = R, P = 0, and when d = r , P = 1. Thisaims to select contacts between R and r hops away from s,and is formulated to provide an increase in reachablility withthe addition of new contacts outside the vicinity of s, i.e., withdistance >R hops from s. The probability P increases withthe number of hops traversed, d . However, there are caseswhere equation (1) does not provide the maximum benefit ofadding a contact. An example case is shown in figure 2(a)where c is the contact for node s and the contact route (route 1in the figure) is R + 2 hops.

In this figure although the distance between s and its con-tact c is greater than R hops, there is still heavy overlap be-tween the two vicinities. Such situations will arise whenever anode within R hops from the edge node becomes the contact.To alleviate this effect, equation (1) is modified to:

P = d − 2R

r − 2R. (2)

In this equation P = 0 when d = 2R and P = 1 whend = r . Hence, contacts are chosen after traversing between2R and r hops from the source s.

Figure 3 explains the contact selection procedure with anexample. In the figure R = 3 and r = 6. Nodes a, b, cand d are the edge nodes for node s. Node s sends a Con-tact Selection (CS) message through its edge node a. Nodea randomly chooses one of its neighbors, e, and forwards theCS to that node. Node e calculates the probability P , say ac-cording to equation (1). If the probability of being the contactfails at e, it forwards the CS to one of its neighbors f (chosenrandomly). Node f again forwards the CS to g. As g is atr hops from s, the probability P at g is 1. However, g still

104 HELMY ET AL.

Figure 3. Selecting contacts.

cannot become a contact for s as there already exists anothercontact h (which was selected through a previous selectionvia another edge node d) in the vicinity of g. So g returns theCS to f (backtracking). Node f then forwards CS to anotherneighbor, and so on.

(b) Edge Method (EM). Even with equation (2) the proba-bilistic method can result in a situation where there is someoverlap between the vicinity of the contact and the vicinityof s. This is possible due to the fact that the nodes do nothave a sense of direction once the CS message is forwardedout of the vicinity (i.e., d > R). Therefore, it is possiblethat a contact may be selected at a location where the CS hastraversed more than 2R hops, but the contact may in fact becloser than 2R hops from the source, as shown in figure 2(a)route 2, leading to heavy overlap.

More seriously, the probabilistic method for contact selec-tion can be expensive in terms of the amount of traffic gener-ated by the CS. This is due to the extra traffic generated dueto backtracking, and lost opportunities when the probabilityfails, even when there is no overlap. To reduce the possibil-ity of such a situation, the probability equations (1) and (2)are not used. The probability equations were formulated tohave a higher possibility of choosing the contact that lies ei-ther between R and r hops (equation (1)) or between 2R andr hops (equation (2)). To maintain this non-overlapping prop-erty without the probability equations, the contact selectionprocedure is modified as follows.

The list of all edge nodes (Edge_List) of s is added tothe CS. Also, the query and source IDs are included to pre-vent looping. Note that the Edge_List is readily availablethrough the vicinity information and obtaining it does not re-quire any extra overhead. Upon receiving a CS, in additionto checking for overlap with s’s vicinity and the vicinities ofall the already-selected-contacts (Contact_List), the receivingnode also checks for overlap with the vicinities of any of thenodes on the Edge_List as well. It can be easily proven that

Figure 4. Reachability for (1) PM and (2) EM. 1

Figure 5. Overhead for (1) PM and (2) EM.1

this scheme guarantees non-overlap between the node and thecontacts vicinities, as follows. Any node that lies at a dis-tance of R hops or less from the edge will have an overlap-ping vicinity with the s’s vicinity, and hence will have at leastone of s’s edge nodes in its vicinity. Thus, checking for non-overlap with the edge nodes ensures that a contact is chosen atleast 2R+1 hops away from s. This eliminates the possibilityof an overlap due to the lack of direction. The Edge_List maybe added to the CS in a communication-efficient manner byusing bloom filters [17] to represent membership in the edge-list. Figures 4 and 5 show a comparison of the probabilisticand edge methods. As can be seen from figure 4 the reach-ability saturates in both PM and EM. However the saturationoccurs much earlier in the case of probabilistic method. Alsoas compared to EM, the reachability achieved is less for PM,for the same values of NoC. Figure 5 shows the backtrack-ing overhead for PM and EM. Due to the reasons explainedearlier, overhead is significantly reduced for EM.

3.4.3. Contact maintenance mechanismNode mobility may cause the path to a contact to change.Therefore a node needs to keep track of its contacts and theirpaths. This is done using soft-state periodic polling of thecontacts as follows.

1 Shown: 500 nodes, 710 m × 710 m, Tx range = 50 m, R = 3, r = 20,D = 1. Similar trends were obtained for other simulation scenarios.


(1) Each node periodically sends a validation message to-wards each of its contacts. These validation messagescontain the path from a node s to the contact.

(2) Each node on the path that receives the validation mes-sage checks if the next hop in the path is a directly con-nected neighbor. If so, it forwards the validation messageto the next hop node. If the next hop is missing, the nodetries to salvage the path using local recovery, discussedlater in this subsection.

(3) If a path cannot be salvaged using local recovery, the con-tact is considered to be lost.

(4) If the path to a contact is validated but the number of hopsto the contact does not lie between 2R and r , the contactis considered to be lost.

(5) After validating all the contacts, if the number of contactsleft is less than the specified NoC, then a new contact se-lection procedure is initiated.

The local recovery mechanism is illustrated using an ex-ample of a contact path (a → b → c → d → e). Assumingreasonable values of node velocities and validation frequency(section 1 in our study), there is a high probability that if anode (say c) has moved out of a contact path (i.e., moved outof transmission range of b), that it is still within the vicinityof the previous hop (b) in the path. Even in the case whenthe moving node (c) is completely lost (because it has movedout of the vicinity of the previous hop, b), some other nodefurther down the path (say dor e) might have moved into thevicinity of the previous node (b). Local recovery takes advan-tage of these cases to recover from changes in the path when

possible, without having to initiate new searches from s. Thuslocal recovery provides an efficient mechanism for validatingcontacts and recovering from changes in the contact paths. Ifthe next hop on the path (node c) is missing, the node thatreceived the validation message (node b) looks for the nexthops (c, d and e) in its vicinity routing table. If any of thenext hops (c, d or e) is found the vicinity, the path is updatedand the validation message is forwarded to that next hop. Ifthe lookups for all next hops fail, an error message is returnedto the source s, and another contact selection is initiated. Fig-ure 6 further illustrates an example of local recovery whentwo nodes along the path to the contact (nodes c and d in thiscase) move.

3.4.4. Query mechanismWhen a source node s (potentially any node), needs to reacha destination or target resource T it first checks its vicinitytable to see if T exists in its own vicinity. If T is not foundin the vicinity, s sends a Destination Search Query (DSQ) toits contacts. The DSQ contains the following information:(1) depth of search (D), and (2) target resource ID (T ). Uponreceiving a DSQ, each contact checks the value of D. If D

is equal to 1, the contact performs a lookup for T in its ownvicinity. If T exists, then the path to T is returned to s, andthe query is considered successful. Otherwise, if D > 1, thecontact receiving the DSQ decrements D by 1 and forwardsthe DSQ to its contacts. In this way the DSQ travels throughmultiple levels of contacts until D reduces to 1.

The source node s first sends a DSQ with D = 1 to itscontacts. So only the first level contacts are queried with thisDSQ. After querying all its contacts if the source does not

Figure 6. Contact maintenance using local recovery: (A) Path to the contact node e goes through a → b → c → d → e. Node c is moving away from b’stransmission range, and node d is moving away from e. (B) During validation, node b loses contact with node c but finds node d in its range. Also, node d

loses direct contact with e but finds a path in its vicinity to node e through node f . The updated part of the contact path is thus a → b → d → f → e.

106 HELMY ET AL.

Table 1Description of the various scenarios used for simulating CARD.

Scenario Nodes Area Transmission range No. of links Aver. node degree Network diameter Aver. path length (hops)

1 250 500 × 500 50 837 6.75 23 9.3782 250 710 × 710 50 632 5.223 25 9.6143 250 1000 × 1000 50 284 2.57 13 3.764 500 710 × 710 30 702 4.32 20 5.87445 500 710 × 710 50 1854 7.416 29 11.6416 500 710 × 710 70 3564 14.184 17 7.067 1000 710 × 710 50 8019 16.038 24 8.758 1000 1000 × 1000 50 4062 8.156 37 14.33

receive a path to the target within a specified time, it createsa new DSQ with D = 2 and sends it again to its contacts.Each contact observes that D = 2 and recognizes that thisquery is not meant for itself. So it reduces the value of D inthe DSQ by 1 and forwards it to its contacts. These contactsserve as second level contacts for the source. Upon receivingthe DSQ, a second level contact observes that D = 1 and itdoes a lookup for the target T in its own vicinity and returnsthe path to T , if found. In this way the value of D is usedto query multiple levels of contacts in a manner similar to theexpanding ring search. However, querying in CARD is muchmore efficient than the expanding ring search as the queriesare not flooded with different TTLs but are directed to indi-viual nodes (the contacts). Contacts leverage knowledge oftheir vicinity (gained through the proactive scheme operatingwithin the vicinity) to provide an efficient querying mecha-nism.

4. Evaluation and analysis

In this section we present detailed simulation based evalua-tion and analysis of our architecture. NS-2 [1] along with ourCARD extensions and other utilities were used to generatevarious scenarios of ad hoc networks. The mobility modelused for these simulations was the random way-point model.Our simulations so far did not consider MAC-layer issues. Inrandom way point model a node is assigned a random velocityfrom [0, Vmax] and assigned a destination location randomly.Once the node reaches its destination it is assigned a randomvelocity and random destination again, so on. In the reacha-bility analysis experiments the mobility was set to ‘0’ to un-derstand the basic effects of the various architectural parame-ters on reachability characteristics. For the maintenance over-head, total overhead and comparison experiments, continuousmobility was used (with no pauses) with Vmax = 20 m/s.

First we try to understand the effect of various parameterssuch as vicinity radius (R), maximum contact distance (r),the number of contacts (NoC), the depth of search (D) andnetwork size (N) on reachability and overhead. Reachabilityhere is defined as the percentage of nodes that are reachablefrom a source node. For overhead we consider the numberof control messages; the contact selection (CS) messages andthe periodic contact maintenance validation messages. Hav-ing developed an understanding of the various parameters inour architecture, we then compare it to other schemes such

as flooding and bordercasting in terms of query overhead andquery success rate.

Table 1 shows the scenarios used in our simulations. Thesescenarios vary in number of nodes, network size, node den-sity and transmission range. The variation is considered tocapture the effect of these factors on CARD. As was shownin figures 4 and 5, the edge method outperforms the proba-bilistic method. Therefore, we use the edge method (EM) forcontact selection in the rest of our study.

4.1. Analysis of reachability

Analysis of the reachability, or query success rate, was con-ducted to understand how contacts help in increasing the viewof the network. Here we present results for a topology of 500nodes spread over area of 710 m × 710 m. The details canbe seen from table 1, scenario number 5. Similar trends wereobserved for other scenarios.

4.1.1. Varying vicinity radius (R)Figure 7 shows the effect of increasing the vicinity radius (R)on reachability. As R increases, the reachability increasesand the reachability distribution in figure 7(a) shifts to theright; i.e., more nodes achieve higher percentage of reacha-bility. This increase in reachability with the increase in R isdue to increase in the number of nodes within the vicinity.As the value 2R approaches the maximum contact distance r

(r = 16 in this experiment), the region available for contactselection (between 2R and r) is reduced. This results in lessnumber of contacts being chosen. In figure 5, when R = 7,contacts can only be selected between 2R = 14 and r = 16hops from the source. This small region for contact selec-tion significantly reduces the number of contact and hencethe reachability reduces as seen in figure 7(b). At this pointmost reachability is due to the vicinity of the source.

4.1.2. Varying maximum contact distance (r)Figure 8 shows the effect of increasing r on reachability.Since contacts are selected between 2R and r hops from thesource, higher values of r provide a wider region for contactselection. The mechanisms for the edge method for contactselection described earlier provide selection of contacts thathave vicinities with reduced overlaps. This implies that as r

increases a larger number of contacts can be selected with-out having vicinity overlaps. Therefore reachability increases


(a) (b)

Figure 7. Effect of vicinity radius (R) on reachability. N = 500, area = 710 m×710 m, propagation range = 50 m, r = 16, NoC = 10, D = 1. (a) Histogramof reachability for different values of R; (b) average reachability with R.

(a) (b)

Figure 8. Effect of maximum contact distance (r) on reachability. N = 500, area = 710 m × 710 m, propagation range = 50 m, R = 3, NoC = 10, D = 1.(a) Histogram of reachability for different values of r; (b) average reachability with r .

(a) (b)

Figure 9. Effect of number of contacts (NoC) on reachability. N = 500, area = 710 m × 710 m, propagation range = 50 m, R = 3, r = 10, D = 1.(a) Histogram of reachability for different values of NoC; (b) average reachability with NoC.

with increase in r . Larger values of r also mean that the av-erage contact path length would increase (as more contactsare chosen at larger distances from the source). However,once the vicinities of the contacts and the source become non-overlapping, for r > (2R + 8), we see no significant increasein reachability with further increase in r .

4.1.3. Varying number of contacts (NoC)NoC specifies the maximum number of contacts to be selectedfor each node. The actual number of contacts chosen maybe less than this value. This is because of the limited re-gion available for choosing contacts for given R and r accord-ing to the contact selection mechanism. Once this region has

108 HELMY ET AL.

been covered by vicinities of chosen contacts, choosing morecontacts in the same region is not possible as their vicinitieswould overlap with the vicinities of the already chosen con-tacts. Therefore the contact selection mechanism prevents theselection of more contacts. This can be seen in figure 9, inwhich the reachability initially increases sharply as more andmore contacts are chosen. However, the increase in reacha-bility saturates beyond NoC = 6 as the actual number of con-tacts chosen saturates due to the effect of overlapping vicini-ties.

4.1.4. Varying depth of search (D)D specifies the levels of contacts that are queried in a breadthfirst manner. When D = 1, a source node looking for a re-source beyond its vicinity, queries its first level contacts only.When D = 2, if none of the first level contacts contain the re-source in its vicinity, second level contacts (contacts of thefirst level contacts) are queried through the first level con-tacts. As can be seen from the figure 10, reachability increasessharply as the depth of search, D is increased. The depth ofsearch, D, results in a tree-like structure of contacts, improv-ing the reachability and success rate of CARD.

4.1.5. Varying network sizeFigure 11 shows the reachability distribution for three differ-ent network sizes, N = 250, 500 and 1000 nodes. The area ofthe three networks has been chosen so that the node densityis almost same across the three networks. Figure 11 showsthat for a given network (specified by the values of N and thearea), the values of R and r can be configured to provide adesirable reachability distribution in which most of the nodeshave a high value of reachability.

4.2. Contact selection and maintenance overhead analysis

The overhead analysis measures the number of control mes-sages required for contact selection and maintenance. Thequery overhead is considered in the next section. The over-head considered in this section includes:

1. Contact selection overhead: This is the amount of CS traf-fic generated for selecting new contacts. This includesoverhead due to backtracking as described earlier.

2. Contact maintenance overhead: This is the traffic gener-ated by the contact path validation messages. Local recov-

Figure 10. Effect of depth of search (D) on reachability. N = 500, area =710 m × 710 m, Tx range = 50 m, R = 3, NoC = 10, r = 10.

Figure 11. Reachability for different network sizes (D = 1).

(a) (b)

Figure 12. Effect of number of contacts (NoC) on contact selection overhead. N = 500, area = 710 m × 710 m, Tx range = 50 m, R = 3, r = 10, D = 1.(a) Overhead over time for different values of NoC; (b) average (per sec) overhead for different values of NoC.


ery, as described earlier, helps in reducing this part of thetotal overhead.

Results are shown for scenario number 5 in table 1 (N =500, area = 710 m × 710 m). Similar trends were observedfor other scenarios.

4.2.1. Varying number of contacts (NoC)As shown in figure 12, as the number of contacts increasesthe maintenance overhead increases sharply as more nodesare periodically maintained through the validation scheme.

4.2.2. Varying maximum contact distance (r)As r increases the number of selected contacts increases. Theincrease in the number of contacts is due to the availabil-ity of a wider area for choosing contacts. Moreover, withhigher values of r , contacts may lie at greater distances fromthe source. That is, the contact path length is expected to behigher for larger values of r . This suggests that the mainte-nance overhead should increase with increase in r . However,as shown in figure 13, the overhead actually decreases with in-crease in r . Figure 14 explains this decrease in maintenanceoverhead. Figure 14 shows that as the value of r increasesthe backtracking overhead decreases significantly. Recall that

Figure 13. Effect of maximum contact distance (r) on contact selection over-head. N = 500, area = 710 m × 710 m, Tx range = 50 m, NoC = 5, R = 3,

D = 1.

backtracking occurs when a node receiving a CS cannot be-come a contact due to overlap with already existing contacts.As r increases, the possibility of this overlap decreases dueto availability of a wider area for contact selection. This de-crease in back-tracking overhead is significantly more thanthe increase in overhead due to increased number of contactsand contact path length. Therefore, the total contact selectionand maintenance overhead decreases.

4.3. Maintenance overhead over time

Figure 15 shows the maintenance overhead per node over a20 s period for Vmax = 20 m/s. The maintenance overheaddecreases steadily with time. However, the number of con-tacts increases slightly. This suggests that the source nodesfind more stable contacts over time. Stable contacts may bedefined as those nodes that have low velocity relative to thesource node. For example, a node moving in the same direc-tion as source node with similar velocity could prove to bea stable contact. Hence, over time, CARD leads to sourcenodes finding more stable contacts.

4.4. Comparison with related schemes (query overhead andtotal overhead)

We compare the performance of CARD to that of flooding,smart flooding [19] and bordercasting [20], in terms of aver-age query overhead and overall overhead. Simulations wererepeated several times with various random seeds to filter outthe noise.

Figure 16 shows the average traffic generated per queryfor the three protocols. We select random source-destinationpairs in the network (the same pairs were used for all the threeprotocols). The graph shows the average overhead for randomqueries with different network sizes, for each protocol. Theoverhead includes number of transmissions as well as numberof receptions. Therefore the overhead for flooding is abouttwice the number of links (as expected). Bordercasting isimplemented as described in [20]. We implemented querydetection (QD1 and QD2) and early termination (ET) as de-scribed in [20] to improve the performance. For smart flood-

(a) (b)

Figure 14. Effect of maximum contact distance (r) on backtracking overhead. N = 500, area = 710 m × 710 m, propagation range = 50 m, NoC = 5,R = 3, D = 1. (a) Backtracking over time for different values of r; (b) contact selection and backtracking overheads (per sec).

110 HELMY ET AL.

Figure 15. Variation of overhead with time. N = 250, area = 710 m×710 m,Tx range = 50 m, NoC = 6, R = 4, r = 16, D = 1.

ing we investigated several techniques (probabilistic flood-ing, minimum dominating set, counter based methods) andwe show the results for those settings that achieved successrate of 90%. This was equivalent to probabilistic flooding asin [19] with p = 0.65. For CARD the values of R and r usedwere chosen as the values that gave maximum reachability forthat particular network size. This information was obtainedfrom previous results shown under the analysis of CARD withrespect to various parameters (see figure 11. Reachability fordifferent network sizes). Flooding and bordercasting result in100% success in queries, smart flooding achieved 90% suc-cess rate, and CARD showed a 95% success rate with D = 3.CARD’s success rate can be increased by increasing D, orwith resource replication. No replication is assumed in ourstudy. As can be seen from figure 16, CARD leads to sig-nificant savings in communication overhead over the othertwo approaches. CARD incurs, on average, around 5% ofthe query overhead for flooding, and around 10% or more ofthe query overhead of bordercasting or smart flooding. Wenote that smart flooding achieves the least success rate. Toincrease the success rate for smart flooding the overhead ap-proaches that of flooding.

What is not shown in figure 16, however, is the effectof contact and vicinity maintenance. For that we show thefollowing ‘total overhead’ comparison results. Maintenanceoverhead (for contacts and vicinity) is a function of mobil-ity and simulation time. Its cost is amortized over the num-ber of queries performed during that period. Hence, wepresent our results as function of the query rate per mobil-ity per node (i.e., query/sec/(m/s) or query/m); this is re-ferred to as call-to-mobility ratio (CMR) or q query/m pernode. We show results for simulations with Vmax = 1 m/sand 20 m/s, for various query rates q for 20 seconds of sim-ulated time. These results take into consideration the con-tact selection and maintenance overhead, the vicinity estab-lishment and maintenance overhead and the query overhead.As can be seen from figures 17 and 18, the advantage of us-ing contacts becomes clearer for higher query rates, wherethe cost of maintenance is amortized over a large number ofqueries. For low mobility, in figures 17(a) and (b), the main-tenance overhead is low and the advantages of using con-tacts are the clearest (46–85% savings for low query rates

Figure 16. Query overhead for CARD, flooding and bordercasting.

q = 0.005 query/m, and 86–94% savings for high query ratesq = 0.05 to 0.5 query/m).

For high mobility, in figures 18(a), (b) the savings are lessthan low mobility scenarios, nonetheless they are still signif-icant for moderate to high query rates (22–75% savings forq = 0.05 query/m, 79–93% savings for q = 0.5 query/mover flooding or bordercast). For low query rates and highmobility however, e.g., for 20 m/s and q = 0.005 query/m,CARD and bordercasting perform worse than flooding, wheremaintenance overhead dominates and only very few queriesare triggered (an unlikely scenario in mobile ad hoc net-works). For high mobility, large-scale, high query rates (1000nodes, 20 m/s, 0.5 query/m), we get savings between 79%(vs. bordercasting) and 87% (vs. flooding).

To further understand the effect of query rate and mobilityon the total overhead we investigate the overhead ratio (OR)metric for CARD over the total overhead of bordercast andflooding. This metric enables us to have a more comprehen-sive view of the operating conditions under which CARD isfavorable. Let OR(C/B) be the overhead ratio for CARD overbordercast, and OR(C/F) and OR(C/S) be the overhead ratioof CARD over flooding and smart flooding. Let CSM be thecontact selection and maintenance overhead, and let ZO bethe zone (or vicinity) maintenance overhead, both in packetsper node per m/s. Also, let CQO be the CARD query over-head in packets per query, hence q · CQO is the overhead inpackets per node per m/s. Define BQO as the query overheadfor bordercast. Hence, we get

OR(C/B) = CSM + ZO + q · CQO

ZO + q · BQO.

Similarly, we have

OR(C/F) = CSM + ZO + q · CQO

q · FQOand

OR(C/S) = CSM + ZO + q · CQO

q · SQO,

where FQO and SQO is the flooding and smart flooding over-head in packets per query, respectively.

OR(C/B) and OR(C/F) were evaluated for q = 0.01 to100 query/sec/(m/s) per node. Figure 19 shows results forOR(C/B) and figure 20 shows results for OR(C/F). From thefigures we note that, in general, when q is quite small (e.g.,


(a) (b)

Figure 17. Total overhead for low mobility and different query rates. (a) Vmax = 1 m/s, CMR q = 0.005 query/m. (b) Vmax = 1 m/s, CMR q = 0.05 to0.5 query/m.

(a) (b)

Figure 18. Total overhead for high mobility and different query rates. (a) Vmax = 20 m/s, CMR q = 0.05 query/m. (b) Vmax = 20 m/s, CMRq = 0.5 query/m.

Figure 19. OR(C/B): the overhead ratio for CARD over bordercast for vari-ous values of q.

q < 0.01) then CARD incurs more overhead than floodingand bordercasting. This is due to the fact that CARD ex-pends communication overhead to select and maintain con-tacts, as well as vicinities. If the nodes are relatively idle,resulting in very small q , then there is not enough query toamortize the cost of the maintenance overhead. This sce-nario is unlikely though, as we expect idle nodes to transitinto sleep mode (to conserve energy) and not participate inperiodic activities (such as vicinity and contact maintenance)while idle. From that perspective, one may consider q to be

the call-to-mobility ratio during active periods. Hence, it isunlikely that q will become too small for most practical pur-poses. As q becomes moderate (around q = 0.01 query/m)we start noticing the advantage of CARD in overhead sav-ings. In figure 18 we see that OR(C/B) becomes less than 1(the cross over point) for q ∼ 0.01–0.025 query/m. Also,OR(C/B) becomes less than 0.2 (i.e., 80% overhead savings)for q ∼ 0.295–0.315 query/m for 500 and 1000 nodes andq = 0.810 query/m for 250 nodes. For q ∼ 10 query/mOR(C/B) approaches 0.11 for 500 and 1000 nodes and 0.18for 250 nodes; i.e., over 80% saving in overhead.

In figure 20 we observe that OR(C/F) becomes less than 1for q ∼ 0.015–0.02 query/m. Furthermore, OR(C/F) < 0.2for q ∼ 0.14–0.155 query/m, and OR(C/F) < 0.1 forq ∼ 0.43–0.51 query/m. For q ∼ 10 query/m, the over-head ratio OR(C/F) approaches 0.066; i.e., over 93% savingin overhead. In figure 21 the overhead ratio with respect tosmart flooding is shown. In addition to achieving better suc-cess rate than smart flooding, CARD also achieves less to-tal overhead for all values of q > 0.035 query/m. The ratioOR(C/S) goes below 0.2 for q ∼ 0.22–0.3 query/m, and goesbelow 0.1 for q ∼ 0.92–9.8 query/m, approaching 6.6–9.9%(i.e. more than 90% in overhead savings) when q approaches10 query/m.

112 HELMY ET AL.

Figure 20. OR(C/F): the overhead ratio for CARD over flooding for variousvalues of q.

Figure 21. OR(C/S): the overhead ratio for CARD over smart flooding forvarious values of q.

5. Conclusions

In this paper we presented the CARD architecture for re-source discovery and small transfers in large-scale ad hocnetworks. The main contributions of this paper include theintroduction of a contact-based architecture that explicitlytrades-off route optimality (as in shortest path routes) forcommunication and energy efficiency, along with a proactivecontact selection scheme to reduce vicinity overlap. Unlikeexisting routing protocols, instead of expending significantoverhead to discover shortest path routes, CARD explicitlyfocuses on route discovery or query delivery with the leastoverhead, even if the routes used are suboptimal. We be-lieve such trade-off is appropriate for our target applications,mainly resource discovery and small transfers. Salient fea-tures of our architecture include its ability to operate withoutrequiring any location information or any complex coordina-tion. In our architecture, each node proactively discovers re-sources within its vicinity. Based on small world concepts,we have utilized the notion of contacts to serve as short cutsthat increase reachability beyond the vicinity. Two protocolsfor contact selection were introduced and evaluated: (a) prob-abilistic method and (b) edge method. The edge method wasfound to result in more reachability and less overhead during

selection due to reduced backtracking, and was thoroughlyanalyzed over the various dimensions of the parameter space(including R, r , D, NoC, and network size). We further com-pared our approach to flooding and bordercasting. The overalloverhead experienced by CARD was found to be significantlylower than the other approaches. Overhead savings are func-tion of the query rate, reaching over 93% (vs. flooding andsmart flooding) and over 80% (vs. bordercasting) in commu-nication saving for high query rates; a drastic improvement inperformance.

These results show a lot of promise for the contact-basedapproach to support short transfers in many applications of adhoc networks. One possible future research direction to inves-tigate is to integrate CARD with other routing protocols (e.g.,ZRP), where CARD may be used as the resource discovery(and transaction routing) protocol. Similarly, we plan to in-vestigate the integration of CARD in other data disseminationprotocols for sensor networks, such as directed diffusion [12].Instead of using flooding, CARD maybe use for efficient re-source discovery.

References

[1] L. Breslau, D. Estrin, K. Fall, S. Floyd, J. Heidemann, A. Helmy,P. Huang, S. McCanne, K. Varadhan, Y. Xu and H. Yu, Advances innetwork simulation, IEEE Computer (May 2000).

[2] T.-W. Chen and M. Gerla, Global state routing: A new routing schemefor ad-hoc wireless networks, in: Proc. of the IEEE Internat. Conf. onCommunications (ICC) (1998).

[3] C.-C. Chiang, Routing in clustered multihop, mobile wireless networkswith fading channel, in: Proc. of IEEE SICON’97 (April 1997).

[4] T. Clausen, P. Jacquet, A. Laouiti, P. Muhlethaler, A. Qayyum andL. Viennot, Optimized link state routing protocol, in: Proc. of IEEEINMIC (2001).

[5] Z. Haas and M. Pearlman, The zone routing protocol (ZRP) for ad hocnetworks, IETF Internet draft for the Manet group (June 1999).

[6] W. Heinzelman, J. Kulik and H. Balakrishnan, Adaptive protocols forinformation dissemination in wireless sensor networks, in: The ACMMOBICOM Conf., Seattle, WA (August 1999).

[7] A. Helmy, Architectural framework for large-scale multicast in mo-bile ad hoc networks, IEEE Internat. Conf. on Communications (ICC),Vol. 4, New York (April 2002) pp. 2036–2042.

[8] A. Helmy, Small worlds in wireless networks, IEEE CommunicationsLetters 7(10) (2003) 490–492.

[9] A. Helmy, Mobility-assisted resolution of queries in large-scale mo-bile sensor networks (MARQ), Computer Networks (Special Issue onWireless Sensor Networks) 43(4) (2003) 437–458.

[10] A. Helmy, TRANSFER: Transactions routing for ad-hoc networks withefficient energy, in: IEEE Global Communications Conf. (GLOBE-COM) (December 2003).

[11] A. Helmy, S. Garg, P. Pamu and N. Nahata, Contact-based architecturefor resource discovery (CARD) in large scale MANets, in: IEEE/ACMIPDPS Internat. Workshop on Wireless, Mobile and Ad Hoc Networks(WMAN) (April 2003) pp. 219–227.

[12] C. Intanagonwiwat, R. Govindan and D. Estrin, Directed diffusion:A scalable and robust communication paradigm for sensor networks,in: ACM MobiCOM Conf. (August 2000).

[13] D.B. Johnson and D.A. Maltz, The dynamic source routing protocol formobile ad hoc networks, IETF Internet draft (October 1999).

[14] J. Li, J. Jannotti, D. Couto, D. Karger and R. Morris, A scalable locationservice for geographic ad hoc routing, in: The ACM MOBICOM Conf.(2000).


[15] J. Liu, Q. Zhang, W. Zhu, J. Zhang and B. Li, A novel framework forQoS-aware resource discovery in MANets, in: IEEE Internat. Conf. onCommunications (ICC) (May 2002).

[16] W. Lou and J. Wu, On reducing broadcast redundancy in ad hoc wire-less networks, IEEE Transactions on Mobile Computing 1(2) (2002).

[17] M. Mitzenmacher, Compressed bloom filters, in: The Twentieth ACMSymposium on Principles of Distributed Computing (PODC) (August2001).

[18] S. Murthy and J.J. Garcia-Luna-Aceves, An efficient routing protocolfor wireless networks, Mobile Networks and Applications (Special Is-sue on Routing in Mobile Communication Networks) (October 1996).

[19] S. Ni, Y. Tseng, Y. Chen and J. Sheu, The broadcast Storm problemin a mobile ad hoc network, in: Proc. of the ACM MOBICOM Conf.(August 1999) pp. 151–162.

[20] M. Pearlman and Z. Haas, Determining the optimal configuration forthe zone routing protocol, IEEE Journal on Selected Areas in Commu-nications 8 (1999) 1395–1414.

[21] C.E. Perkins and P. Bhagwat, Highly dynamic destination-sequenceddistance-vector routing (DSDV) for mobile computers, ACM ComputerCommunications Review (October 1994) 234–244.

[22] C.E. Perkins, E.M. Royer and S.R. Das, Ad hoc on-demand distancevector routing, IETF Internet draft (October 1999).

[23] N. Sadagopan, B. Krishnamachari and A. Helmy, Active query for-warding in sensor networks (ACQUIRE), Ad Hoc Networks Journal(2004) to appear.

[24] N. Sadagopan, B. Krishnamachari and A. Helmy, The ACQUIREmechanism for efficient querying in sensor networks, in: First IEEEInternat. Workshop on Sensor Network Protocols and Applications(SNPA), in conjunction with IEEE ICC, Anchorage (May 2003)pp. 149–155.

[25] S. Wang and A. Helmy, Effects of small transfers and traffic patternson performance and cache efficacy of ad hoc routing, (poster), in: TheACM MOBICOM Conf. (The Ninth Annual Internat. Conf. on MobileComputing and Networking), San Diego, CA (September 2003).

[26] D.J. Watts, The dynamics of networks between order and randomness,in: Small Worlds (Princeton Univ. Press, Princeton, 1999).

[27] D. Watts and S. Strogatz, Collective dynamics of ‘small-world’ net-works, Nature 393 (4 June 1998).

Ahmed Helmy received his Ph.D. in computer sci-ence (1999), M.S. in electrical engineering (1995)from the University of Southern California, M.S.Eng. Math. (1994) and B.S. in electronics and com-munications engineering (1992) from Cairo Univer-sity, Egypt. Since 1999, he has been an AssistantProfessor of Electrical Engineering at the Universityof Southern California. In 2002, he received the Na-tional Science Foundation (NSF) CAREER Award.In 2000 he received the USC Zumberge Research

Award, and in 2002 he received the best paper award from the IEEE/IFIPInternational Conference on Management of Multimedia Networks and Ser-vices (MMNS). In 2000, he founded – and is currently directing – thewireless networking laboratory at USC. His current research interests liein the areas of protocol design and analysis for mobile ad hoc and sensornetworks, mobility modeling, design and testing of multicast protocols, IPmicro-mobility, and network simulation.E-mail: [email protected]

Saurabh Garg received his M.S. in computer science from the University ofSouthern California in May 2003 and B.S. in engineering from University ofDelhi, India. He is currently working as a Programmer for National Centerfor Ecological Analysis and Synthesis (NCEAS) affiliated with Universityof California, Santa Barbara. He has worked with researchers in fields ofwireless networking, linguistics dialogue management and ecology. Beforejoining NCEAS he was working for Institute of Creative Technologies at theUniversity of Southern California in the field of dialogue management. Inwireless networking, he has worked on network simulation, protocol design

and analysis.E-mail: [email protected]

Nitin Nahata received his M.S. in computer science from the University ofSouthern California in December 2002. He is currently working as a softwareengineer at Dynamix Technologies Inc.E-mail: [email protected]

Priyatham Pamu received his M.S. in computer science from the Universityof Southern California in December 2002.E-mail: [email protected]


Energy-Balanced Task Allocation for Collaborative Processingin Wireless Sensor Networks ∗

YANG YU and VIKTOR K. PRASANNADepartment of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2562, USA

Abstract. We propose an energy-balanced allocation of a real-time application onto a single-hop cluster of homogeneous sensor nodesconnected with multiple wireless channels. An epoch-based application consisting of a set of communicating tasks is considered. Eachsensor node is equipped with discrete dynamic voltage scaling (DVS). The time and energy costs of both computation and communicationactivities are considered. We propose both an Integer Linear Programming (ILP) formulation and a polynomial time 3-phase heuristic. Oursimulation results show that for small scale problems (with �10 tasks), up to 5x lifetime improvement is achieved by the ILP-based approach,compared with the baseline where no DVS is used. Also, the 3-phase heuristic achieves up to 63% of the system lifetime obtained by theILP-based approach. For large scale problems (with 60–100 tasks), up to 3.5x lifetime improvement can be achieved by the 3-phase heuristic.We also incorporate techniques for exploring the energy-latency tradeoffs of communication activities (such as modulation scaling), whichleads to 10x lifetime improvement in our simulations. Simulations were further conducted for two real world problems – LU factorizationand Fast Fourier Transformation (FFT). Compared with the baseline where neither DVS nor modulation scaling is used, we observed up to8x lifetime improvement for the LU factorization algorithm and up to 9x improvement for FFT.

Keywords: sensor networks, single-hop wireless networks, ILP, energy saving

1. Introduction

Wireless sensor networks (WSNs) are being developed for awide range of civil and military applications, such as targettracking, infrastructure monitoring, habitat sensing, and bat-tlefield surveillance [6,10]. WSNs usually contain a numberof networked sensor nodes with each sensor node consistingof computation, communication, and sensing devices. Thesesensor nodes collaborate with each other to realize certain ap-plications.

For instance, in a target tracking application, up to thou-sands of sensor nodes are dispersed over a specific area ofinterest. The sensor nodes are usually organized into clus-ters [13,31] with each cluster consisting of tens of sensornodes. Distributed signal detection and collaborative dataprocessing are performed within each cluster for detecting,identifying, and tracking vehicles. Some of the operationsinvolved in such data processing include the LU factoriza-tion [5] and the Fast Fourier Transformation (FFT) [7].

Energy efficiency is a key concern in WSNs. The largenumber of sensor nodes involved in the system and the needto operate over a long period of time require energy-awaredesign and operation at all levels of abstraction, from thephysical layer to the application layer. However, while manyhardware techniques [1,14], network protocols [13,16], anddata processing algorithms [18,19] have been proposed forenergy-aware design, systematic mechanisms for designingenergy-aware collaborative processing between sensor nodesstill need to be addressed.∗ This work is supported by the DARPA Power Aware Computing and Com-

munication Program under contract no. F33615-C-00-1633. A preliminaryversion of this paper appears in ACM LCTES 2003.

The state of the art in WSN design is largely ad-hoc – sys-tem planning and resource management are done without asystematic methodology. This can lead to inefficient utiliza-tion of the system. The main motivation of our efforts is todevelop techniques for systematic and rapid design and de-ployment of WSN applications [3,25,31].

We focus on the development of energy-efficient collabo-rative algorithms for WSNs based on high-level computationmodels of WSNs. Such high-level models allow designers tomake informed decisions regarding energy and time tradeoffsat the node and network level – creating a modular, layeredparadigm for application development. Toward such a goal,we study the following problem in this paper.

Energy-balanced task allocation problem. We consider asingle-hop cluster of homogeneous sensor nodes connectedthrough multiple wireless channels. Each sensor node isequipped with dynamic voltage scaling (DVS) [30]. The tar-get application consists of a set of communicating tasks.Throughout the paper, the term activity refers to either a com-putation task or a communication request. We consider anepoch-based scenario [18], where an instance of the appli-cation is executed during the beginning of each epoch andmust be completed before the end of the epoch. Such a re-quirement is usually called the latency constraint. We usethe term period to indicate the length of each epoch. Also,we assume that time-synchronization schemes (e.g., [9]) areavailable within the cluster.

We consider the exclusive access constraint. Specifically, anon-preemptive scheduling policy is employed by each sensornode and each wireless channel. Also, at any time, a sensornode can receive or send data by using at most one channel.

116 YU AND PRASANNA

The underlying network protocol is assumed to be capable ofscheduling a communication activity over a specified chan-nel according to the start and finish time of the activity. Sucha scheduling policy requires coarse-level bandwidth reserva-tion mechanisms, which can be provided by, for example, atime-division multiple-access (TDMA) protocol. Moreover,we consider the task placement constraint, which is typicallyrequired when certain tasks for sensing the raw data must beallocated onto different sensor nodes.

A task allocation is defined as (1) the assignment of tasksonto sensor nodes, (2) the voltage settings of tasks, (3) theassignment of communication activities onto channels, and(4) the scheduling of computation and communication activ-ities. Our general goal is to find an allocation in order tomaximize the lifetime of the cluster. Toward such a goal, wepropose an energy-balanced task allocation such that the max-imal energy dissipation among all sensor nodes during eachperiod is minimized, subject to the latency, exclusive access,and task placement constraints.

Our contributions. The idea of energy-balanced task alloca-tion to a single-hop cluster in WSNs is proposed. As we shallsee in section 2, most of the previous efforts in energy-awaretask allocation or resource management try to minimize theoverall energy dissipation of the system. This strategy maynot be suitable in the context of WSNs, since each sensor nodeis equipped with its own energy source. Moreover, for event-driven systems, applications often need to be executed afterthe system has been working for sometime. In such a case, anenergy-balanced task allocation should also consider the factthat the remaining energy can vary among sensor nodes.

To the best of the authors’ knowledge, this is the first workfor task allocation in WSNs that considers the time and en-ergy costs of both the computation and communication ac-tivities. We first present an integer linear programming (ILP)formulation of our problem. The optimal solution of the prob-lem can be obtained by using a commercial software pack-age such as [26], though the running time of such a softwarecan be large. Next, we propose a polynomial time 3-phaseheuristic. Finally, we incorporate techniques that explore thelatency-energy tradeoffs of communication activities, such asmodulation scaling [23].

Our simulation results show that for small scale problems,up to 5x lifetime improvement is achieved by the ILP-basedapproach, compared with the case where no DVS is used.Also, the 3-phase heuristic achieves up to 63% of the sys-tem lifetime obtained by the ILP-based approach. For largescale problems, the 3-phase heuristic achieves up to 3.5x life-time improvement when only DVS is used. By incorporat-ing modulation scaling, up to 10x lifetime improvement wasobserved. Simulations were also conducted for applicationgraphs from two real world problems – LU factorization andFFT. We observed a lifetime improvement of up to 8x for theLU factorization algorithm and up to 9x for FFT.

Paper organization. We discuss the related work in sec-tion 2. The energy-balanced task allocation problem is de-fined in section 3. The ILP formulation of the problem is

given in section 4. The 3-phase heuristic is described in sec-tion 5. Techniques, such as modulation scaling, are incor-porated into our approaches in section 6. Simulation resultsare demonstrated in section 7. Finally, we give concludingremarks in section 8.

2. Related work

Extensive research efforts have studied the problem of energy-efficient task allocation and scheduling with DVS in uni-processor real-time systems, including [2,15,24,30]. Re-cently, research interests have been shifted to multi-processorsystems. A list-scheduling based heuristic is proposed in [12],to dynamically recalculate the priority of communicatingtasks. In [17], static and dynamic variable voltage schedul-ing heuristics for real-time heterogeneous embedded systemsare proposed. An approach based on critical-path is used forselecting the voltage settings of tasks. However, both [12]and [17] assume that the task assignment is given. A sim-ilar problem to the one studied in this paper is investigatedin [33]. A two-phase framework is presented to first deter-mine the allocation of tasks onto processors and then the volt-age settings of tasks using convex programming. In [34], adynamic processor voltage adjustment mechanism for a ho-mogeneous multi-processor environment is discussed. How-ever, the time and energy costs for communication activitiesare not addressed in any of [12,33,34].

The goal of all the above works is to minimize the overallenergy dissipation of the system. While such a goal is reason-able for tightly coupled systems, it does not capture the natureof WSNs. The reason is that to minimize the overall energydissipation can lead to heavy use of energy-effective sensornodes, regardless of their remaining energy. The consequentshort lifetime of such sensor nodes will very likely hinder thesystem from delivering required performance. This weaknessis a major motivation of the proposed energy-balanced taskallocation.

Our work considers the energy and time costs of both com-putation and communication activities. As indicated by sev-eral researches, wireless communication is a major source ofenergy dissipation in WSNs. By incorporating techniquessuch as modulation scaling, we can greatly improve theenergy-efficiency of the system.

Energy-balanced task allocation bears some resemblanceto load-balance in distributed computing. However, the com-munication activities over the same wireless channel need tobe serialized such that run-time contentions can be avoided.The serialization imposes new challenges that distinguish ourproblem from most of the existing works for load-balance orreal-time scheduling in distributed systems.

3. Problem definition

3.1. System model

We consider a set of m homogeneous sensor nodes, PE ={PEi : i = 1, . . . ,m}, connected by a single-hop wireless

ENERGY-BALANCED TASK ALLOCATION 117

network with K communication channels. The homogene-ity refers to the same processor and radio capabilities. Eachsensor node is equipped with D discrete voltage levels, listedas V = {Vi: i = 1, . . . ,D} in decreasing order. Each voltagelevel in V corresponds to a specific computation speed (givenin cycles per second) of the processor. Let SPj denote thespeed of Vj . Let Ri denote the remaining energy of PEi . Forease of analysis, we assume that the processors consume zeropower during idle state.

Regarding the exclusive access constraint, we assume thata non-preemptive scheduling policy is employed by each sen-sor node and each wireless channel. In other words, thetime duration scheduled for different computation (commu-nication) activities over the same sensor node (wireless chan-nel) cannot overlap with each other. Moreover, the under-lying communication protocols are assumed to be capableof scheduling communication activities according to the starttime of each activity in order to avoid run-time contentions.We assume all channels have the same bandwidth. Let τ de-note the time for transmitting one data unit between two sen-sor nodes over any channel. For ease of analysis, we assumethat such a transmission costs the same amount of energy atboth the sender and the receiver, denoted by ε. Let τs and εs

denote the startup time and energy costs for communication.The data transmission between two tasks on the same sensornode is performed through the local memory with zero timeand energy costs.

For ease of analysis, we assume that the radios are com-pletely shutdown in idle state. The energy cost for shuttingdown and restarting the radio is assumed to be included in εs .Low power paging or signaling channel mechanisms can beused for synchronization between sensor nodes when the ra-dios are shutdown. However, the modeling of the power con-sumption for such mechanisms is beyond the scope of thispaper. We also assume that computation and communicationactivities can be parallelly executed on any sensor node.

3.2. Application model

An epoch-based application [18] consisting of a set of com-municating tasks is considered. Let P denote the period of theapplication, which is the length of each epoch. An instance ofthe application is activated at time kP , and must be completedby the relative deadline, (k + 1)P , where k = 0, 1, 2, . . . .

The structure of the application is represented by a directedacyclic graph (DAG), G = (T ,E), where node set T denotesthe set of n tasks, {Ti : i = 1, . . . , n}, and edge set E de-notes the set of e directed communication activities betweentasks, {Ei : i = 1, . . . , e}. Every edge in E pointing fromnode Ti to Tj , denoted as (i, j), means that the output oftask Ti needs to be transmitted to Tj before Tj can start com-putation. There is a precedence constraint on two tasks Ti

and Tj , if there is a path of alternate nodes and edges from Ti

to Tj in the DAG. Similarly, there is a precedence constrainton two communication activities, (i, j) and (i ′, j ′), if there isa path from Tj to Ti′ . A task with no incoming edges is called

a source task. A task with no outgoing edges is called a sinktask.

For most applications in WSNs, the source tasks are usedfor sensing or gathering raw data. For ease of analysis, thetask placement constraint is defined as that no two sourcetasks can be assigned to the same sensor node. Nevertheless,our models and approachs can be extended to handle the gen-eral case that any pair of tasks must be or must not be assignedto the same sensor node.

For any task Ti ∈ T , let Ci denote its workload in terms ofthe worst-case number of required computation cycles. Theexecution time of Ti on any voltage level Vj ∈ V , tij , canbe calculated as tij = Ci/(SPj ). The voltage level of a sen-sor node is assumed to be dynamically switched, if necessary,upon the arrival of a task instance. Because at most one switchis needed for executing a task instance, the associated timeoverhead is assumed to be included in the workload of thetask. From [4], the power consumption for executing a taskfollows a monotonically increasing and strictly convex func-tion of the computation speed, gi(·), which can be representedas a polynomial function of at least second degree. Hence, theenergy dissipation for executing Ti on Vj , eij , can be calcu-lated as eij = gi(SPj )tij . The exact forms of gi(·) can varyfor different tasks based on their instruction components.

The communication load of any edge Ei ∈ E is rep-resented by its weight wi , as the number of data units tobe transmitted. We assume that all the data of an edge istransmitted in one data packet with variable size. For anedge Ei = (j, k), let t ′i and e′

i denote the time and energycosts of the corresponding communication activity, if tasksTj and Tk are not assigned to the same sensor node. We havet ′i = τs + τwi and e′

i = εs + εwi .

3.3. Task allocation

Based on the above system and application models, a task al-location is defined as (1) the assignment of tasks onto sensornodes, (2) the voltage settings of tasks, (3) the assignment ofcommunication activities onto channels, and (4) the schedul-ing of computation and communication activities. Each taskcan be assigned to exactly one sensor node with a fixed volt-age setting. Also, each communication activity can be as-signed to exactly one channel. An allocation is feasible ifit satisfies the latency, exclusive access, and task placementconstraints.

The system lifetime is defined as the time duration from thetime when the application starts execution to the time whenany sensor node in the cluster fails due to depleted energy.A general solution to maximize the system lifetime is to allowvariable task allocations in different periods. Consequently,the energy cost for each sensor node may vary in different pe-riods. However, due to the high complexity raised by such asolution, we assume that the task allocation remains the samefor all application periods. That is, the behavior of the systemrepeats for each period and every sensor node spends the sameenergy duration each period. Let Ei denote the energy dissi-pation of PEi ∈ PE during each application period. Given an

118 YU AND PRASANNA

allocation, the system lifetime (in number of periods) can becalculated as mini{�Ri/Ei�}. A feasible allocation is optimalif the corresponding system lifetime is maximized among allthe feasible allocations.

Note that a more complex definition of the system lifetimewould be the time period from the beginning of the applica-tion execution to the time when not enough sensor nodes arealive to deliver required performance. However, such a defin-ition is quite application-specific. Thus, a simple but generaldefinition of the system lifetime is adopted in this paper. Now,our task allocation problem can be informally stated as:

Find an allocation of a set of communicating tasks onto asingle-hop cluster that minimizes the maximal energy dis-sipation among all sensor nodes during each applicationperiod, normalized by their remaining energy.

4. Integer linear programming formulation

In this section, we present an ILP formulation of our task allo-cation problem that captures the behavior of the system duringone application period. We first list the notations used in theformulation as follows:

P : period of the applicationtij , eij : time and energy costs of executing task Ti using

voltage level Vj

t ′i , e′i : time and energy costs of edge Ei = (j, k), if Tj

and Tk are not assigned to the same sensor nodea||b: no precedence constraint exists for computation

(or communication) activities a and b

{xij }: a set of 0–1 variables such that xij equals oneiff Ti is assigned to PEj

{yij }: a set of 0–1 variables such that yij equals oneiff the voltage level of Ti is set to Vj

{zij }: a set of 0–1 variables such that zij equals oneiff Ei is assigned to the j th channel

{rij }: a set of 0–1 variables such that rij equals oneiff Ti and Tj are assigned to the same sensor node

{sij }: a set of 0–1 variables such that sij equals oneiff Ei and Ej are assigned to the same channel

{α(i)}: a set of real variables indicating the time whenTi starts execution

{β(i)}: a set of real variables indicating the time whenTi completes execution

{γ (i)}: a set of real variables indicating the time whenEi starts transmission

{δ(i)}: a set of real variables indicating the time whenEi completes transmission

{pij }: a set of 0–1 variables such that zij equals oneiff the execution of Ti finishes before Tj starts

{qij }: a set of 0–1 variables such that qij equals oneiff the transmission of Ei finishes before Ej starts.

To capture the relative order imposed by the precedenceconstraints among activities, we define the constraint set 1shown in figure 1. It is easy to verify that the exclusive accessconstraint for activities with precedence constraints is also

enforced by constraint set 1. However, for activities that donot have precedence constraints between them, an extra set ofconstraints are needed (constraint set 2 in figure 2) to enforcethe exclusive access constraint. In addition, the task place-ment constraint is captured by the constraint set 3 in figure 2.

The complete ILP formulation is given in figure 3, whereE is an auxiliary variable. In the figure, the factor |xik − xjk|means that the energy cost for (i, j) is counted if exactly oneof Ti or Tj is assigned to PEk , but not both. Clearly, the pre-sented formulation is nonlinear. It can be transformed intoan ILP formulation by standard linearization techniques [29].Due to the space limitation, we omit the details of lineariza-tion in this paper.

5. Heuristic approach

In this section, we describe an efficient 3-phase heuristic forsolving the task allocation problem. Initially, we assumethat the voltage levels for all tasks are set to the highest op-tion (V1). In the first phase, the tasks are grouped into clusterswith the goal to minimize the overall execution time of theapplication. In the second phase, task clusters are assigned tosensor nodes such that the highest energy dissipation amongall sensor nodes, normalized by their remaining energy, isminimized. In the last phase, the system lifetime is maxi-mized by lowering the voltage levels of tasks. The details ofthe heuristic are as follows.

Phase 1. A task cluster is defined as a set of tasks assigned tothe same sensor node with a specific execution order. Com-munication between tasks within a cluster costs zero time andenergy. In this phase, we assume an unlimited number of sen-sor nodes, implying that the number of clusters is also unlim-ited. The main purpose of this phase is to eliminate communi-cation activities in order to reduce the overall execution timeof the application.

The idea of phase 1 is similar to the algorithm pro-posed in [22, pp. 123–131]. However, traditional approachesfor task clustering usually assume a full connection amongprocessors such that all communication can be parallelized,whereas in our problem, communication activities over thesame channel must be serialized. Thus, a new challenge isto select a policy for the serialization that facilitates the re-duction of the execution time of the application. We use asimple first-come-first-serve policy to order the communica-tion activities ready at different times. Activities ready at thesame time (such as those initiated by the same task) are exe-cuted in a nondecreasing order of their communication loads.Nevertheless, more sophisticated policies are also applicable.

The pseudo-code for phase 1 is shown in figure 4. In thecode, L denotes the overall execution time of the applicationand C(i) denotes the cluster that contains task Ti . Initially,every task is assumed to constitute a cluster by itself. Wethen examine all the edges in a non-increasing order of theirweights. For each edge (i, j) if the execution time of the ap-


Constraint set 1:∀Ti ∈ T∑

j xij = 1, // every task can be assigned to exactly one sensor node∑j yij = 1, // every task can be executed using exactly one voltage level

α(i) � maxEl=(j,i)∈E{δ(l)}, // Ti starts execution after receiving all input dataβ(i) = α(i) + ∑

j (yij tij ), // execution time of Ti depends on its voltage level∀Ti, Tj ∈ T

rij = 1 iff ∀k = 1, . . . ,m, xik = xjk // rij equals one if Ti and Tj are assigned to the same sensor node∀Ei = (a, b) ∈ E∑

j zij = 1, // Ei can be assigned to exactly one channelγ (i) � β(a), // Ei starts transmission after Ta completes execution

(*) δ(i) = γ (i) + t ′i (1 − rab), // the transmission time of Ei depends on the locations of Ta and Tb

for any source tasks Ti

α(i) � 0, // all source tasks can start execution at time 0for any sink task Ti

β(i) � P . // all sink tasks must complete before the relative deadline

Figure 1. Constraint sets 1 for the ILP formulation.

Constraint set 2:∀Ti, Tj ∈ T , such that i �= j and Ti ||Tj

pij = 1 − pji , // pij is the inverse of pji

α(j) � pij rij β(i), // if Ti and Tj are assigned to the same sensor node, Ti

// completes before Tj starts execution iff pij = 1α(i) � pjirij β(j), // if Ti and Tj are assigned to the same sensor node, Tj

// completes before Ti starts execution iff pji = 1∀Ei,Ej ∈ E, such that Ei = (a, b), // Communication activities from the same sensor node

Ej = (a, c), b �= c

qij = 1 − qji , // qij is the inverse of qji

γ (j) � qij (1 − rab)(1 − rcd )δ(i), // Ei completes before Ej starts transmission iff qij = 1γ (i) � qji(1 − rab)(1 − rcd )δ(j), // Ej completes before Ei starts transmission iff qji = 1

∀Ei,Ej ∈ E, such that Ei = (a, b), // Communication activities to the same sensor nodeEj = (c, b), a �= c


γ (j) � qij (1 − rab)(1 − rcd )δ(i), // Ei completes before Ej starts transmission iff qij = 1γ (i) � qji(1 − rab)(1 − rcd )δ(j), // Ej completes before Ei starts transmission iff qji = 1

∀Ei,Ej ∈ E, such that Ei = (a, b),Ej = (c, d), a �= c, b �= d , and Ei ||Ej


sij = 1 iff ∀k = 1, . . . ,K , zik = zjk , //sij equals one if Ei and Ej are assigned to the same channelγ (j) � qij (1 − rab)(1 − rcd )sij δ(i), // if Ei and Ej are assigned to the same channel, Ei completes

// before Ej starts transmission iff qij = 1γ (i) � qji(1 − rab)(1 − rcd )sij δ(j). // if Ei and Ej are assigned to the same channel, Ej completes

// before Ei starts transmission iff qji = 1Constraint set 3:∀Ti, Tj ∈ T , such that Ti and Tj are source tasks and i �= j

rij = 0 // any two source tasks cannot be assigned to the same sensor node

Figure 2. Constraint sets 2 and 3 for the ILP formulation.

plication can be reduced by merging C(i) with C(j) with-out violating the task placement constraint, we perform themerge. Otherwise, Ti and Tj remain in two different clusters.In lines 3 and 6, the function Traverse() is called to traversethe DAG in order to determine the schedule of the tasks andhence L.

The pseudo code for Traverse() is shown in figure 5. Inthe code, we maintain a queue of activities, Qact, that storesall the ready computation or communication activities in theirexpected execution order. We also maintain a timestamp foreach task cluster that indicates the finish time for all scheduledtasks within the cluster. Similarly, we maintain a timestamp

120 YU AND PRASANNA

Minimize ESubject to∀PEk∑

Ti∈T {xik

∑j (yij eij )} + ∑

Ei=(a,b)∈E{e′i |xak − xbk|}

Rk

� Eand constraint sets 1, 2 and 3

Figure 3. ILP formulation for the energy-balanced task allocation problem.

1. Each task is assumed to constitute a cluster by itself

2. Set E as the list of edges in a nondecreasing orderof the edge weights

3. L ← Travese()

4. While E is not empty Do

5. Remove the first edge from E, denoted as (i, j)

6. L′ ← Traverse() as if C(i) and C(j) are merged

7. If L′ < L and to merge C(i) and C(j) does notviolate the task placement constraint

8. Merge C(i) and C(j)

9. L ← L′

10. If L > P , Return failure

Figure 4. Pseudo-code for phase 1.

for each channel that indicates its nearest available time. Thetimestamps are used to schedule the computation and com-munication activities in lines 7, 13 and 14. In lines 9 and 14,the timestamps are updated based on the execution time ofthe scheduled activities. The actions in lines 17 and 18 areimportant to ensure that the radio can be tuned to at most onechannel at any time.

Phase 2. In this phase, we assign the task clusters fromphase 1 onto the actual sensor nodes in PE. Note that multipleclusters can be assigned to the same sensor node. Based onthe contained tasks and the corresponding communication ac-tivities, we first calculate the energy dissipation of each clus-ter. Let � = [π1, π2, . . . , πc] denote the list of all tasks clus-ters and ξi denote the energy dissipation of πi . The normal-ized energy dissipation (norm-energy for short) of a sensornode is given as the sum of the energy dissipation of the clus-ters assigned to the sensor node, normalized by the remainingenergy of the sensor node.

The pseudo-code of phase 2 is shown in figure 6. Ini-tially, � is sorted into a non-increasing order of energy dis-sipation of clusters. Then, for each cluster in �, we calcu-lated the norm-energy of every sensor node as if the clusteris assigned to the sensor node (called expected norm-energy).We then assign the cluster to the sensor node that gives theminimal expected norm-energy. In the code, function Tra-verseAssigned() is used to find the execution time of the ap-

1. Initialize Qact

2. Set the timestamps for all task clusters and channels to zero

3. Append all source tasks to Qact with ready time set to zero

4. While Qact is not empty Do

5. Remove the first activity from Qact

6. If the removed activity is a computation activity, denotedas Ti

7. Set α(i) ← max{ready time of Ti , timestamp of C(i)}8. Set β(i) to the expected completion time of Ti , i.e.,

β(i) ← α(i) + ti1

9. Set the timestamp of C(i) to β(i)

10. Insert all communication activities initiated by Ti into Qactwith ready time set to β(i) in a nondecreasing order oftheir communication loads

11. Else

12. Let Ei = (a, b) denote the removed communicationactivity

13. Find the channel with the smallest timestamp, say the j thchannel

14. Set γ (i) ← max{ready time of Ei , timestamp of the j thchannel}

15. Set δ(i) to the expected completion time of Ei , i.e.,δ(i) ← γ (i) + t ′

i

16. Set the timestamp of the j th channel to δ(i)

17. Set the ready time of any unscheduled communicationactivities from Ta to δ(i)

18. Set the ready time of any unscheduled communicationactivities to Tb to δ(i)

19. If all the communication activities to Tb have beenscheduled

20. Insert Tb into Qact with ready time set to δ(i)

21. Return the largest timestamp among all clusters

Figure 5. Pseudo-code for function Traverse().

plication based on the resulting assignment. Compared withTraverse(), the modification in TraverseAssigned() is that inline 7 of figure 5, each computation activity is scheduled onthe sensor node that it is assigned to. Thus, timestamps aremaintained for all sensor nodes, instead of clusters.

Phase 3. The voltage levels of tasks are adjusted in this phasewith the goal to maximize the system lifetime. An iterativegreedy heuristic is used (shown in figure 7). Let E denotethe maximum of the norm-energy among all sensor nodes.The sensor node that determines E is called the critical node.In each iteration, we find the task such that by lowering itscurrent voltage level to the next level, E can be decreased themost. The increased latency caused by lowering the voltage


1. Sort � in a non-increasing order of the energydissipation of clusters

2. While � is not empty Do

3. Select the first element π in �

4. Calculate the expected norm-energy for eachsensor node (set to infinity if two source tasksare assigned to the same sensor node)

5. Assign π to the sensor node that gives the minimalexpected norm-energy

6. Update the norm-energy of the sensor node

7. Remove π from �

8. L ← TraverseAssigned()

9. If L > P , Return failure


1. For each PEi , sort EDi in a non-increasing order

2. Do

3. i ← 1

4. Let PEr denote the critical sensor node andE denote the norm-energy of PEr

5. While i � |EDr | Do

6. Select the ith item in EDr ;let Tj denote the corresponding task

7. If L + tdj � P

8. L ← L + tdj

9. Lower the voltage of Tj to the next level

10. Update edj in EDr ; resort EDr if necessary

11. Find the new critical sensor node, PEr ′ ; update E

12. If r �= r ′

13. r ← r ′; i ← 1

14. Else i ← i + 1


16. Until E can not be reduced any more


is added to L. Since the schedule of activities can be changedby the latency increment, L is re-computed by traversing theDAG every time it reaches P (in line 15).

In figure 7, edj denotes the energy gain by lowering thecurrent voltage of Tj to the next level, while tdj denotes theincurred increment in latency. The array composed by edj ’sfor all tasks assigned to PEi is denoted as EDi .

Time complexity analysis. In phase 1 (figure 4), the Whileiteration is executed e times. Function Traverse() in line 6

(a)

Task time cost energy cost

Vh Vl Vh Vl

T1 10 33 20 6T2 60 199 120 36T3 10 33 20 6T4 10 33 20 6T5 20 66 40 12T6 10 33 20 6T7 10 33 20 6

(b)

Figure 8. An application example: (a) application graph; (b) time and energycosts for executing tasks at voltage levels Vh and Vl .

takes O(n + e) time. Thus, phase 1 needs O(e(n + e)) time.In phase 2 (figure 6), the ordering in line 1 takes O(c log c)

time. The outer iteration is executed c times. The resultsof m possible assignments are compared in line 5. The tra-verse in line 8 takes O(n + e) time. Hence, phase 2 takesO(c log c+mc+n+e) time. In phase 3 (figure 7), the sortingin line 1 takes O(n log n) time. The number of voltage switch-ing in line 9 is bounded by dn. To update EDr in line 10needs O(log n) time. Let p denote the number of times forcalling TraverseAssigned() in line 12. The time complexityof phase 3 is O(dn log n + p(n + e)). Although p equals dn

in the worst case, it was observed in our simulations that p

usually equals 1 or 2. Thus, the overall time complexity ofthe heuristic is O((e + p)(n + e) + mc + dn log n + c log c),which is O(dn(n + e + log n) + e2 + mn) in the worstcase.

An illustrative example. We illustrate the execution of theabove heuristic through a simple example. We assume a clus-ter of 3 sensor nodes connected by 2 channels. Each sensornode have two voltage levels, Vh and Vl , with SPh = 1 andSPl = 0.3. We assume that it costs one time and energy unitfor transmitting one data unit over any channel. The applica-tion graph is shown in figure 8(a), with each circle represent-ing a task. The number close to each circle is the required

122 YU AND PRASANNA

workload, while the number on each edge is the weight of theedge. The time and energy costs for executing tasks at thetwo voltage levels are given in figure 8(b). We assume thatP = 250 time units.

The clustering steps in phase 1 is shown in figure 9. In thisphase, the voltage levels of all tasks are set to Vh. The sortededge list with respect to edge weights is {(T4, T6), (T1, T2),(T3, T6), (T6, T7), (T2, T7), (T5, T6), (T1, T3)}. The table infigure 9 traces the execution of the algorithm, where Li is theexecution time of the application at the completion of step i.The subfigures (a) through (e) correspond to the applicationgraph at the completion of steps 0, 1, 2, 3 and 5, respectively.The clusters are marked with polygons in dash line. Note thatin steps 6 and 7, the clustering is not performed due to thetask placement constraint.

During phase 2, we first calculate the energy dissipa-tion for each cluster – 190 energy units for cluster π1 ={T1, T2, T7}, 100 for the cluster π2 = {T3, T4, T6}, and 50 forcluster π3 = {T5}. Since the remaining energy for the threesensor nodes are the same, we simply assign π1 to PE1, π2 toPE2, and π3 to PE3.

Finally, we adjust the voltage levels of tasks. Since PE1 isthe critical node, we first set the voltage level of T2 to Vl ,which reduces E1 to 106 and increases L from 80 to 219.Next, we set the voltage level of T1 to Vl , which further de-creases E1 to 92 and increases L to 242. After this step, thecritical node becomes PE2 with E2 = 0.1. Since the latencyconstraint is 250, our heuristic terminates.

In the above example, we decreases the norm-energy ofthe critical sensor node from 0.19 to 0.1, implying a systemlifetime improvement by a factor around 2.

6. Incorporating energy-latency tradeoffs forcommunication activities

While DVS has been widely applied into various applicationsfor energy saving in computation activities, techniques for ex-ploring the energy-latency tradeoffs of communication activ-ities are gaining interest. An important observation [11] isthat in many channel coding schemes, the transmission en-ergy can be significantly reduced by lowering the transmis-sion power and increasing the duration of the transmission.Techniques such as modulation scaling [23] have been pro-posed for implementing such tradeoffs. Recently, algorithmsfor applying such techniques in the context of packet trans-missions or data gathering in wireless networks have beenstudied in [11,23,32].

Our approaches can be extended to incorporate the abovetradeoffs. In the following, we discuss through the exampleof modulation scaling that explores the tradeoffs by adaptingthe modulation level to match the traffic load.

For ease of analysis, we focus on the Quadrature Am-plitude Modulation (QAM) scheme [28]. The techniquespresented in this paper are extendible to other modulationschemes as well. Given a communication activity Ei with

a packet of si bits, assuming a fixed symbol rate Ri , the trans-mission time can be calculated as [23]:

τi = si

biRi

, (1)

where bi is the modulation level in terms of the constellationsize (number of bits per symbol). The corresponding energydissipation can be modeled as a function of τi , denoted asfi(τi). We have [23]

fi(τi) = [Ci

(2si/(τiRi ) − 1

) + Di

]τiRi, (2)

where Ci is determined by the quality of transmission (interms of bit error rate) and the noise power, and Di is adevice-dependent parameter that determines the power con-sumption of the electronic circuitry of the sensor nodes. Theenergy-latency tradeoffs for transmitting 1 bit is plotted in fig-ure 10. The settings for Ci , Di and Ri are extracted from [23].Also, we may estimate the energy dissipation for receiving thepacket as

f ′i (τi) = DiRiτi . (3)

In practice, the value of bi is typically set to positive evenintegers, resulting in discrete values of τi . For any communi-cation activity Ei ∈ E, let t ′ij denote the time cost with bi setto the j th modulation level. Also, let es

ij and erij denote the

corresponding sending and receiving energy costs. We cancalculate the values of t ′ij ’s, es

ij ’s, and erij ’s based on equa-

tions (1)–(3).To modify our ILP formulation, a set of 0–1 variables {uij }

are needed to indicate the modulation level of the communica-tion activities. Specifically, uij equals one iff the modulationlevel of Ei is set to the j th level. Moreover, we replace theconstraint set marked with * in figure 1 with the followingone, which states that the transmission time of Ei = (a, b)

depends on the modulation level for Ei and the locations ofTa and Tb:

∀Ei = (a, b) ∈ E, δ(i) = γ (i) +∑

j

(uij t

′ij

)(1 − rab).

Moreover, we change the constraint on the auxiliary vari-able E in figure 3 as follows:

∀PEk∑

Ti∈T {xik

∑j (yij eij )}

Rk

+∑

Ei=(a,b)∈E{xak(1 − xbk)∑

j (uij esij ) + (1 − xak)xbk

∑j (uij e

rij )}

Rk

� E .

For the 3-phase heuristic, we assume that both voltage andmodulation levels of the system are set to the highest optionsin phases 1 and 2. We then slightly modify phase 3, suchthat the energy savings achieved by lowering the modulationlevels of communication activities are also examined. Themodified pseudo code is shown in figure 11. One concern


(a) (b)

(c) (d)

(e)

Step i edge examined L if clustering clustering? Li

0 1451 (T4, T6) 135 yes 1352 (T1, T2) 120 yes 1203 (T3, T6) 100 yes 1004 (T6, T7) 100 no 1005 (T2, T7) 80 yes 806 (T5, T6) no 807 (T1, T3) no 80

(f)

Figure 9. Clustering steps for the application in figure 8.

is that to decrease the transmission energy at the sender, weactually increase the receiving energy at the receiver. Thus,in lines 13 and 14 of figure 11, we ensure that the modulationscaling is performed only when the increase in the reception

energy does not cause the value of E to increase. By doing so,our heuristic can handle the situation in highly dense WSNs,where the receiving energy is comparable with the sendingenergy.

124 YU AND PRASANNA

Figure 10. Energy-latency tradeoffs for transmitting one bit of data.

1. For each PEi , sort EDi in a non-increasing order

2. Do

3. i ← 1

4. Let PEr denote the critical sensor node

5. While i � |EDr | Do

6. Select the ith component in EDr ; let a denote thecorresponding activity

7. If L + tda > P , i = i + 1

8. Else

9. L ← L + tda

10. If a is a computatin activity

11. Lower the voltage level of a to the next availableoption

12. Else

13. If to lower the modulation level of a to the nextavailable option does not increase E

14. Lower the modulation level of a to the nextavailable option

15. Else i ← i + 1

16. If any voltage or modulation scaling is performed

17. Update eda and tda ; resort EDr if necessary

18. Find the new critical sensor node, PEr ′ ; update E

19. If r �= r ′

20. r ← r ′; i ← 1


22. Until E can not be reduced any more

Figure 11. Pseudo-code for the modified phase 3 that incorporates modulca-tion scaling.

7. Simulation results

A simulator based on the system and application models pre-sented in section 3 was developed to evaluate the performanceof our approach using application graphs from both a syn-thetic approach and real world problems. The goals of oursimulations are (1) to measure and compare the performanceof the 3-phase heuristic against the ILP-based approach; and(2) to evaluate the impact of the variations in several key sys-tem parameters on the performance of the heuristic, includ-ing the tightness of the latency constraint, the relative timeand energy costs of communication activities compared withcomputation activities, and the number of voltage levels.

The evaluation metrics are based on the system lifetimeobtained by different approaches. Let LT ILP and LTheu de-note the system lifetime obtained by the ILP-based approachand the 3-phase heuristic, respectively. In addition, let LT rawdenote the system lifetime obtained by assuming that no volt-age or modulation scaling is available (i.e., every sensor noderuns and transmits data at the highest speed). Since we do nothave a stand alone approach to obtain LT raw, LT raw was cal-culated based on the value of E obtained after phase 2 of the3-phase heuristic.

Unless otherwise stated, all the data presented in thissection is averaged over more than 100 instances so that a95% confidence interval with a 10% (or better) precision isachieved.

7.1. Synthetic application graphs

Simulation setup. The structure of the application graph wasgenerated using a method similar to the one described in [8].The only difference is that we enforce multiple source tasksin the generation of the DAG.

According to Rockwell’s WINS node [27], the powerconsumption of an Intel StrongARM 1100 processor with150 MIPS is around 200 mW. This implies that the time andenergy costs per instruction are around 5 nsec and 1 nJ. Also,the power of the radio module used in WINS is 100 mW at100 Kbps, implying that the time and energy costs for trans-mitting a bit are around 10 µsec and 1 µJ. In the following,we set the parameters for our simulator such that the time andenergy costs for computation and communication activitiesroughly follow the above data.

We set the maximum computation speed of each sensornode to 102 Mcps (million cycles per second) and the mini-mum speed to 0.3 × 102 Mcps. It is assumed that other levelsof computation speed are uniformly distributed between themaximum and minimum speeds. The computation require-ments of the tasks followed a gamma distribution with a meanvalue of 2 × 105 and a standard deviation of 105. The powerfunction of task Ti , gi(SP), was of the form ai · (SP/108)bi ,where ai and bi were random variables with uniform distrib-ution between 2 and 10, and 2 and 3 [20], respectively. Forexample, suppose ai = bi = 2. Then, to execute a task of2 × 105 instructions costs 2 msec and 4 mJ in the highestspeed, and 6.7 msec and 1 mJ in the lowest speed.


The time and energy costs of communication activities aredetermined by the number of data units to transmit and thevalues of τ and ε. Based on the data for WINS, we setτ = 10 µsec and ε = 1 µJ. To focus on the main issues,we set the startup energy dissipation of the radio to be zero.To study the effect of different communication load with re-spect to the computation load, the number of bits per com-munication activity follows a uniform distribution between200 CCR (1 ± 0.2), where CCR (communication to compu-tation ratio) is a parameter indicating the ratio of the averageexecution time of the communication activities to that of thecomputation activities. Intuitively, a larger value of CCR im-plies a relatively heavier communication loads compared withthe computation loads. Note that by varying CCR, we abstractnot only the variations in the amount of transmitted data, butalso the variations in the relative speed of computation andcommunication devices. In our simulations, CCR was variedwithin [0, 20].

The period of the application, P , was generated in the fol-lowing way. We first define the distance of a node in the appli-cation DAG as the number of edges in the longest path fromthe source to the node. Nodes are then divided into layers,with nodes in each layer having the same value of distance.Since the average time to execute a task in the highest speedis 2 msec, the computation time required for a layer is esti-mated as 2p/m msec, where p is the number of tasks inthe layer. By doing so, we implicitly assume full parallelismin executing the tasks at each layer. In addition, the expectednumber of communication activities initiated by a task is es-timated as its out-degree subtracted by 1. Assuming there arein total q communication activities requested by all the tasksin a specific layer, the corresponding time cost is estimatedas 2 CCRq/K msec. P is then set to the sum of the com-putation and communication time cost of all layers over u,where u ∈ [0, 1] is a parameter that approximates the overallutilization of the system. The setting of u is important as itdetermines the latency laxity for trading against energy. Intu-

itively, a larger value of u implies a tighter latency constraintand hence less latency laxity.

The remaining energy of sensor nodes follows a uniformdistribution between Emean(1 ± 0.3), where Emean is a fairlylarge number.

Small scale problems. We first conducted simulations forsmall scale problems, with 3 sensor nodes, 3 voltage levels,2 channel, and 7–10 tasks. The number of source tasks in theapplication graph is set to 2, while the maximal in-degree andout-degree for each node are set to 3. A commercial softwarepackage, LINDO [26], was used to solve the ILP problems.Due to the large running time for solving some problem in-stances, LINDO was interrupted after two hours of executionif the optimal solution was not yet found. Then, the best so-lution obtained so far was returned. We observed that in mostcases, LINDO was able to find the optimal solution withintwo hours.

The data shown in figure 12 is averaged over more than70 instances so that each data point has a 95% confidence in-terval with a 10% precision. In figure 12(a), we illustrate thelifetime improvement achieved by the ILP-based approach,which is calculated as LT ILP/LT raw − 1. We can see an im-provement around 3x–5x. Figure 12(b) shows the perfor-mance ratio of the 3-phase heuristic over the ILP-based ap-proach, i.e., LTheu/LT ILP. We can see that the 3-phase heuris-tic achieved up to 63% of the solution obtained by the ILP-based approach for the conducted simulations.

While the running time of the heuristic is around zero, theaverage running time of the ILP-based approach ranges from550 sec (n = 7, u = 0.5) to 5900 sec (n = 10, u = 0.8)on a Sun Blade1000 machine with a UltraSparc III 750 MhzCPU.

Large scale problems. A set of simulations were conductedfor evaluating the performance of the 3-phase heuristic forproblems with 10 sensor nodes, 8 voltage levels, 4 channels,60–100 tasks, CCR ∈ [0, 20], and u ∈ [0, 1]. The number of

(a) (b)

Figure 12. Lifetime improvement of our approaches for samll scale problems (3 sensor nodes, 3 voltage levels, 2 channels, CCR = 1): (a) lifetime improve-ment achieved by the ILP-based approach; (b) performance comparison of the ILP-based approach and the 3-phase heuristic.

126 YU AND PRASANNA

(a) (b)

Figure 13. Lifetime improvement of the 3-phase heuristic for large scale problems (10 sensor nodes, 8 voltage levels, 4 channels, 60–100 tasks): (a) lifetimeimprovement vs. system utilization (u) and communication to computation ratio (CCR); (b) lifetime improvement vs. number of tasks (CCR = 4).

source tasks in the application graph is set to 6. The maximalin-degree and out-degree for each node are set to 5. Due to thelarge size of the problems, it is impractical to obtain the opti-mal solutions by using the ILP-based approach. Thus, we usethe lifetime improvement achieved by the 3-phase heuristic asthe evaluation metric, which is calculated as LTheu/LT raw−1.The simulation results are shown in figure 13.

An improvement up to 3.5x in the system lifetime can beobserved from figure 13(a). We can see that the improvementincreases when u decreases, as the latency laxity increasesaccordingly. The lifetime improvement saturates when u ap-proaches 0, i.e., the latency constraint approaches ∞. Thecurve with u = 0.0 gives the upper bound of the improve-ment that can be achieved by our heuristic with respect tovariations in CCR.

The effect of CCR is more complicated. For example,when u = 0.5, the lifetime improvement increases whenCCR � 6 and decreases when CCR is beyond 6. Thisis because when CCR is small, the computation activitiesdominate the overall energy costs of the application. Byincreasing CCR, we actually increase the latency constraintwithout increasing the computation load, which in turn canbe traded for lifetime improvement. However, when CCRreaches some threshold value, the communication energy costbecomes more significant than that of the computation activ-ities. Thus, the lifetime improvement achieved by reducingcomputation energy becomes limited. We shall see later thatthis shortcoming can be overcome by incorporating modula-tion scaling into our heuristic.

Figure 13(b) shows the lifetime improvement with numberof tasks, n varying from 60 to 100. We can see that the per-formance of our approach is quite stable with respect to thevariation in n.

The miss rate (defined as the ratio of the number of in-stances that an approach fails to find a feasible solution to

Figure 14. Miss rate of the 3-phase heuristic (10 sensor nodes, 8 voltagelevels, 4 channels, 60 tasks, CCR = 0).

the total number of instances) of a heuristic is another keyissue. Note that in our simulations, not all instances are guar-anteed to have feasible solutions. We observed that the missrate of the 3-phase heuristic is significant only when CCR isclose to zero. Thus, we show the miss rate with CCR = 0in figure 14. Also, the running time of the heuristic is around0.5 msec on a Sun Blade1000 machine with a UltraSparc III750 Mhz CPU.

Impact of the number of voltage levels. We also studiedthe impact of the variations in the number of voltage levels.Simulations were conducted with 10 sensor nodes, 60 tasks,4 channels, CCR = 2, u ∈ {0.2, 0.5, 0.8, 1.0} and 1 to 10voltage levels. The results are demonstrated in figure 15.

The plots show that when u > 0.2, the performance ofthe heuristic can be significantly improved by increasing thenumber of voltage levels from 1 to 4. Further increase in thenumber of voltage levels does not improve the performance


much. This is understandable since the energy behaves as amonotonically increasing and strictly convex function of thecomputation speed. The first derivative of the energy functiontends to ∞ when the speed tends to ∞. Thus, the most portionof energy saving is obtained by changing the speed from thehighest option to some lower options, which can be efficientlyachieved with 4 voltage levels per sensor node.

When u = 0.2, the latency laxity is so large that the volt-age level of most tasks can be set to the lowest option. Thus,there is almost no improvement by increasing the number ofvoltage levels beyond 2.

Incorporating modulation scaling. We used modulationscaling to illustrate the energy-latency tradeoffs for commu-nication activities. We assume that all sensor nodes have theidentical settings for parameters Ci , Di and Ri . From [23],we set Di = 10−7 [23]. To investigate the impact of differentenergy/time ratio for data transmission, we set Ci to 10−7 and10−6 for different instances. The modulation level, bi , was set

Figure 15. Impact of variation in number of voltage levels (10 sensor nodes,4 channels, 60 tasks, CCR = 2).

to even numbers between 2 and 6. We set Ri = 1.7 · 104 sothat when bi = 6, it roughly takes 10 µsec and 1 µJ to trans-mit a bit (as shown in figure 10).

The simulations were conducted with 10 sensor nodes,8 voltage levels, 3 modulation levels ({2, 4, 6}), 60 tasks,u ∈ {0.0, 0.2, 0.5, 0.8, 1.0}, and CCR ∈ [0, 20]. Com-pared with figure 13, we can observe a significant amount ofperformance improvement in figure 16. For example, whenu = 0.5, the highest lifetime improvement increases from3x in figure 13(a) to 6x in figure 16(a) and even 10x in fig-ure 16(b). The difference in performance improvement of fig-ures 16(a) and (b) is because that a larger Ci leads to largerenergy/time ratio of communication activities, which in turngives more advantage in reducing the communication energyby utilizing modulation scaling.

Similar to figure 13, larger improvement is observed whenu becomes smaller. In addition, the miss rate of the heuristicexhibits a similar trend as the cases with DVS only.

7.2. Application graphs from real world problems

In addition to synthetic application graphs, we also consid-ered application graphs of two real world problems: LU fac-torization algorithm [5] and Fast Fourier Transformation [7].These two algorithms are widely used as kernel operations forvarious signal processing, such as beamforming [21].

LU factorization. Figure 17(a) gives the sequential programfor the LU factorization without pivoting, where s denotesthe dimension of the matrix. The application graph of thealgorithm for the special case of s = 5 is given in figure 17(b).Each Tk,k represents a pivot column operation and each Tk,j

represents an update operation. The total number of tasks inthe application graph equals (s2 + s − 2)/2. Also, we assumethe input matrix is available at the sensor node where task T1,1is assigned.

We performed simulations with 10 sensor nodes, 8 voltagelevels, 4 channels, 3 modulation levels, and the matrix di-

(a) (b)

Figure 16. Lifetime improvement of the 3-phase heuristic incorporated with modulation scaling (10 sensor nodes, 8 voltage levels, 4 channels, 3 modulationlevels, 60 tasks): (a) small energy/time ratio for communication activities (Ci = 10−7); (b) large energy/time ratio for communication activities (Ci = 10−6).

128 YU AND PRASANNA

MatrixFactorization(a)1. For k = 1 to s − 1 Do2. For i = k + 1 to s Do // Tk,k

3. aik = aik/akk

4. For j = k + 1 to s Do5. For i = k + 1 to s Do // Tk,j

6. aij = aij − aik/akj

(a)

(b)

Figure 17. Matrix factorization algorithm: (a) sequential algorithm; (b) ex-ample application graph with a 4 × 4 matrix.

mension, s, varying from 5 to 20. Regarding the energy/timeratio for data transmission, we set Di = 10−6. It is easy toverify that the computation requirement of any task Tk,j iss − k ALU operations. Further, for any task, Tk,j , the sizeof data transmitted by any communication activity to the taskis s − k units in the matrix. We examined two cases with u

set to 0.5 and 0.8. In both cases, CCR was selected from{1.0, 3.0, 5.0, 8.0, 10.0}.

The lifetime improvement achieved by our 3-phase heuris-tic for the LU factorization algorithm is shown in figure 18.It can be observed that the performance of the heuristic im-proves when CCR increases or u decreases. The lifetime im-provement approaches 8x when CCR = 10.0. Also, veryfew improvement was observed during our simulations bysetting CCR beyond 10.0. The least amount of lifetime im-provement is around 15% when u = 0.8, CCR = 1.0, ands = 20.

Fast Fourier Transformation (FFT). The recursive, one-dimensional FFT algorithm is given in figure 19(a). In thefigure, A is an array of length l which holds the coefficientsof the polynomial and array Y is the output of the algorithm.The algorithm consists of two parts: recursive calls (lines 3, 4)and the butterfly operation (lines 6, 7). For an input vector ofsize l, there are 2 × l − 1 recursive call tasks and l × log l but-terfly operation tasks (we shall be assuming l = 2k for someinteger k). For example, the application graph with four datapoints is given in figure 19(b). The 7 tasks above the dashedline are the recursive call tasks, while the 8 tasks below theline are butterfly operation tasks.

We performed simulations used 10 sensor nodes, 8 volt-age levels, 4 channels, 3 modulation levels. Regarding theenergy/time ratio for data transmission, we set Di = 10−6.The vector size was varied from 4 to 64 incrementing by thepower of 2. We also examined two cases with u set to 0.5 and0.8. In both cases, CCR was selected from {1.0, 3.0, 5.0, 8.0}.

(a) (b)

Figure 18. Lifetime improvement for the matrix factorization algorithm (10 sensor nodes, 8 voltage levels, 4 channels, 3 modulation levels): (a) u = 0.5;(b) u = 0.8.


FFT(A,w)1. Set l = length(A)

2. If l = 1, return A

3. Y (0) = FFT((A[0], A[0], . . . , A[l − 2]),w2)

4. Y (1) = FFT((A[1], A[3], . . . , A[l − 1]),w2)

5. For i = 0 to l/2 Do6. Y [i] = Y (0)[i] + wi × Y (1)[i]7. Y [i + l/2] = Y (0)[i] − wi × Y (1)[i]8. Return Y

(a)

(b)

Figure 19. Fast Fourier Transformation (FFT) algorithm: (a) sequential algo-rithm; (b) example application graph with 4 points.

The lifetime improvement achieved by our 3-phase heuris-tic for the FFT algorithm is shown in figure 20. Again, theperformance of the heuristic improves when CCR increases oru decreases. The lifetime improvement is close to 10x whenCCR = 8.0 and l = 64. The least amount of lifetime im-provement is around 75% when u = 0.8, CCR = 1.0, andl = 4.

Note that the above two example applications have exactlyone source task that initially holds the entire data set, im-plying that data dissemination within the cluster is required.However, our technique is also applicable to applicationswhere data are locally sensed or gathered at each individualsensor node. For example, in figure 19(b), input data can begenerated by tasks T4–T7 through local sensing. Thus, therecursive calls above the dashed line to disseminate the databecome unnecessary.

8. Concluding remarks

In this paper, we have investigated the problem of allocatingan epoch-based real-time application to a single-hop cluster ofhomogeneous sensor nodes with multiple wireless channels.A new performance metric has been proposed to balance theenergy dissipation among all the sensor nodes. We have pre-sented both an ILP formulation and a polynomial time heuris-tic. Also, we have incorporated techniques that explore theenergy-latency tradeoffs of communication activities.

We have demonstrated through simulations that for smallscale problems, a lifetime improvement up to 5x is achievedby the ILP-based approach, compared with the case where noDVS is used. Also, the performance of the 3-phase heuris-tic achieves up to 63% of the system lifetime obtained by theILP-based approach. For large scale problems, a lifetime im-provements up to 10x was observed when both voltage andmodulation scaling were used. Simulations were also con-ducted for application graphs from LU factorization and FFT.The 3-phase heuristic achieves a lifetime improvement of up

(a) (b)

Figure 20. Lifetime improvement for the FFT algorithm (10 sensor nodes, 8 voltage levels, 4 channels, 3 modulation levels): (a) u = 0.5; (b) u = 0.8.

130 YU AND PRASANNA

to 8x for the LU factorization algorithm and an improvementof up to 9x for the FFT algorithm.

In the future, we would like to validate our approaches us-ing real systems. We are particularly interested in advancedapplications for WSNs, where systematic methodologies fortask allocation are mostly required for rapid and automatedsystem design.

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Yang Yu is a Ph.D. degree candidate in the Depart-ment of Electrical Engineering at the University ofSouthern California (USC). He received both BS andMS degrees in computer science from Shanghai Jiao-Tong University in China. His research interests in-clude energy-aware resource management for wire-less sensor networks, especially in algorithmic so-lutions for energy-aware communication schedulingand task allocation. He is a student member of theIEEE.



Viktor K. Prasanna is a Professor of Electrical En-gineering and Computer Science at the Universityof Southern California (USC). He is also a mem-ber of the NSF supported Integrated Media Sys-tems Center (IMSC) and an associate member of theCenter for Applied Mathematical Sciences (CAMS)at USC. His research interests include high perfor-mance computing, parallel and distributed systems,network computing, and embedded systems. He re-ceived his BS in electronics engineering from Ban-

galore University, MS from the School of Automation, Indian Institute ofScience, and Ph.D. in computer science from the Pennsylvania State Univer-

sity. He has published extensively and consulted for industries in the aboveareas. He is the steering committee co-chair of the International Paralleland Distributed Processing Symposium (IPDPS) (merged IEEE InternationalParallel Processing Symposium (IPPS) and Symposium on Parallel and Dis-tributed Processing (SPDP)). He is the steering committee chair of the In-ternational Conference on High Performance Computing (HiPC). He serveson the editorial boards of the Journal of Parallel and Distributed Computing.He is the editor-in-chief of the IEEE Transactions on Computers. He wasthe founding chair of the IEEE Computer Society’s Technical Committee onParallel Processing. He is a fellow of the IEEE.E-mail: [email protected]


Efficient and Robust Protocols for Local Detection andPropagation in Smart Dust Networks ∗,∗∗

IOANNIS CHATZIGIANNAKIS, SOTIRIS NIKOLETSEAS and PAUL SPIRAKISComputer Technology Institute (CTI) and Patras University, P.O. Box 1122, 261 10 Patras, Greece

Abstract. Smart Dust is a set of a vast number of ultra-small fully autonomous computing and communication devices, with very restrictedenergy and computing capabilities, that co-operate to quickly and efficiently accomplish a large sensing task. Smart Dust can be very usefulin practice, i.e., in the local detection of a remote crucial event and the propagation of data reporting its realization. In this work we makean effort towards the research on smart dust from an algorithmic point of view. We first provide a simple but realistic model for smart dustand present an interesting problem, which is how to propagate efficiently information on an event detected locally. Then we present varioussmart dust protocols for local detection and propagation that are simple enough to be implemented on real smart dust systems, and perform,under some simplifying assumptions, a rigorous average case analysis of their efficiency and energy consumption (and their interplay). Thisanalysis leads to concrete results showing that our protocols are very efficient and robust. We also validate the analytical results by extensiveexperiments.

Keywords: wireless sensor networks, algorithms, data propagation, stochastic processes, simulation

1. Introduction

Networked sensors (or Smart Dust) are very large systems,comprised of a vast number of homogenous ultra-small fullyautonomous computing and communication devices that co-operate to achieve a large sensing task. Each device has oneor more sensors, embedded processors and low-power radios,and is normally battery operated. Examining each such sin-gle device individually, might appear to have small utility.The realization of Smart Dust, however, lies in using and co-coordinating a vast number of such devices.

Smart Dust is a useful case of dynamic environments ofnetworked sensors that are spread over a global system andtry to communicate and compute efficiently and quickly, hav-ing only partial knowledge of the global conditions and hav-ing poor energy and computing resources. Typically, thesenetworked sensors coordinate to perform a common task. De-signing protocols to coordinate such systems (i.e., create a dy-namic and efficient network of these sensors) and monitoringtheir behavior as they operate in complex and dynamic globalenvironments is of great importance for information gatheringand processing in many practical situations.

As an example, [11] points that integrated low-power sens-ing devices will permit remote object monitoring and track-ing in inhospitable physical environments such as remotegeographic regions or toxic urban locations. They will alsoenable low maintenance sensing in the field (vehicles, equip-ment, personnel), the office buildings (projectors, furniture,

∗ Preliminary versions of this work have appeared in the 2nd ACM Work-shop on Principles of Mobile Computing (POMC, 2002 [9]) and the 3rdWorkshop on Mobile and Ad-hoc Networks (WMAN, 2003 [6]).

∗∗ This work has been partially supported by the IST Program of the Eu-ropean Union under contract numbers IST-1999-14186 (ALCOM-FT) andIST-2001-33116 (FLAGS).

books, people), the factory floor (motors, small robotic de-vices).

There are many possible models for such networked sen-sors. In this work, we consider networked sensors where(a) all nodes in the network are homogenous and constrainedby low availability of resources (energy, communication) and(b) the data being sensed by the nodes must be transmittedto a fixed control center located far away from the sensors.Thus direct communication between the sensor nodes and thecontrol center is impossible and/or expensive, since there areno “high-energy” nodes through which communication canproceed. This is the general framework for MIT’s µAMPSproject [20], which focuses on innovative energy-optimizedsolutions at all levels of the system hierarchy, from the phys-ical layer and communication protocols up to the applicationlayer.

To motivate the challenges in designing such sensor net-works, we can consider the following scenario where localdetection and fast propagation to the authorities of the real-ization of a crucial event can be achieved using smart dust.Think about thousand of disposable sensors scattered (e.g.,thrown from an aircraft) over a forest. Each of these sensorscan monitor the temperature at a single, very small geograph-ical area. The sensors coordinate to establish an efficient, dy-namic and short-lived communication network, dividing thetask of monitoring the terrain and offering continuous moni-toring of the environment in order to alert the authorities assoon as possible after a forest fire is detected by some sensor.

Several aspects of such systems of autonomous networkedentities emerge, which are quite different from those posedby standard computer networks. Such aspects include thevery poor and highly restricted resources (e.g., very low bat-tery power, low computing capabilities, total absence of syn-chrony and anonymity). Network protocols must be designed

134 CHATZIGIANNAKIS, NIKOLETSEAS AND SPIRAKIS

to achieve fault tolerance in the presence of individual nodefailure while minimizing energy consumption. Another im-portant aspect is the scalability to the change in network size,node density and topology. The network topology changesover time as some nodes may die, or possibly because newnodes join later.

This work, continuing our line of research on communica-tion in ad-hoc mobile networks [7,8], is an attempt towardscapturing the underlying foundational and algorithmic issuesin the design of such systems, abstracting accurately enoughthe real technological specifications involved and providingsome first concrete results for the efficiency of a variety ofsmart dust protocols using an average case analysis. We focusin this paper on the efficient use of smart dust in local detec-tion and propagation protocols. We first provide an abstractmodel for smart dust systems which is simple enough to allowan analysis to develop, being however at the same time quiterealistic, in terms of the technological specifications of realsmart dust systems it captures. Then we define the problemof local detection and propagation using smart dust and alsopropose some concrete performance and robustness measuresfor the average case analysis of protocols for this problem.

Our resultsFor the local detection and propagation problem using smartdust, we provide three protocols. All protocols are simpleenough to be implemented in real smart dust systems despitethe severe energy and computing power limitations of suchsystems. Furthermore, we give a rigorous average case analy-sis for the efficiency of these protocols. We consider a varietyof performance and robustness criteria, such as propagationtime, number of particle to particle transmissions (which alsocharacterizes energy consumption and time efficiency, assum-ing an efficient MAC protocol) and fault-tolerance:

1. Our first protocol, which we call the “local target protocol”(LTP), uses a fast and cheap search phase which is assumedto always return a nearby particle towards the authorities,uniformly in some range. We show that LTP is efficient,in the sense that it achieves a propagation time and an en-ergy consumption whose expected ratio over the optimalsolutions is at most π/2 ≈ 1.57.

2. Our second protocol, the “min-two uniform targets” proto-col (m2TP), applies the simple idea of getting at least twoparticles towards the authorities and selecting the best interms of propagation progress. It is, in fact, an optimizedand more efficient version of the local target protocol, andhas an expected time and energy ratio over the optimal so-lutions which is at most π2/8 ≈ 1.24.

3. Next we provide tight upper bounds to the distribution ofthe number of particle to particle data transmissions (andthus the efficiency) of a generalized target protocol.

4. We propose a new protocol which we call the “Sleep–Awake” protocol (SWP), that explicitly uses the energysaving characteristics, such as the alteration of sleep andawake time periods, of smart dust particles. By using both

analytic and extensive experimental means, we investigatethe relation between (a) the success probability and (b) thetime efficiency of the protocol, to the maximum sleepingtime period, for various values of other parameters, suchas particle density, particle distribution and angle α. Weinterestingly note that the new protocol is efficient, despitethe fact that the particles are allowed to enter a sleepingmode in order to save energy.

All protocols mentioned above are shown to be robust inthe following sense: (a) the protocols use the search and thebacktrack phases to explore the active (non-faulty) “next” par-ticles. Thus, the fact (demonstrated both by analysis and sim-ulation) that the protocols succeed with high probability, ex-hibits also fault-tolerance properties of the protocols; (b) ourfindings showing that the protocols succeed even in the caseof low densities also implies robustness.

Discussion of selected related workIn the last few years, Sensor Networks have attracted a lot ofattention from researchers at all levels of the system hierarchy,from the physical layer and communication protocols up tothe application layer.

At the MAC level, many researchers have done researchwork in an effort to minimize the power consumption. [27]presents a contention-based protocol that tries to minimize en-ergy consumption due to node idle listening, by avoiding theoverhearing among neighboring nodes. A recent work [30]exploits a similar method for energy savings, and further re-duce idle listening by avoiding any use of out-of-channel sig-naling. Additionally, their protocol trades off per-node fair-ness for further energy savings.

For establishing communication and routing informationto the control center, mobile ad-hoc routing protocols [24]may be used in sensor networks. However, although proto-cols for mobile ad-hoc networks take into consideration en-ergy conservation issues, most of them are not really suitablefor sensor networks. [19] presents a routing protocol suitablefor sensor networks that makes greedy forwarding decisionsusing only information about a node’s immediate neighborsin the network topology. This approach achieves high scala-bility as the density of the network increases. [14] presents aclustering-based protocol that utilizes randomized rotation oflocal cluster heads to evenly distribute the energy load amongthe sensors in the network. [21] introduces a new energy ef-ficient routing protocol that does not provide periodic datamonitoring (as in [14]), but instead nodes transmit data onlywhen sudden and drastic changes are sensed by the nodes. Assuch, this protocol is well suited for time critical applicationsand compared to [14] achieves less energy consumption andresponse time.

A family of negotiation-based information disseminationprotocols suitable for wireless sensor networks is presentedin [15]. Sensor Protocols for Information via Negotiation(SPIN) focus on the efficient dissemination of individual sen-sor observations to all the sensors in a network. However,in contrast to classic flooding, in SPIN sensors negotiate with

EFFICIENT AND ROBUST PROPAGATION PROTOCOLS FOR SMART DUST 135

each other about the data they possess using meta-data names.These negotiations ensure that nodes only transmit data whennecessary, reducing the energy consumption for useless trans-missions.

A data dissemination paradigm called directed diffusionfor sensor networks is presented in [18], where data-generatedby sensor nodes is named by attribute–value pairs. An ob-server requests data by sending interests for named data; datamatching the interest is then “drawn” down towards that nodeby selecting a single path or through multiple paths by us-ing a low-latency tree. [17] presents an alternative approachthat constructs a greedy incremental tree that is more energy-efficient and improves path sharing.

We note that, as opposed to the work presented in thispaper, the above research focuses on energy consumptionwithout examining the time efficiency of their protocols. Fur-thermore, these works contain basically protocol design andtechnical specifications, while quantitative aspects are onlyexperimentally evaluated and no theoretical analysis is given.Note also that our protocols are quite general in the sensethat (a) do not assume global network topology informa-tion, (b) do not assume geolocation information (such as GPSinformation) and (c) use very limited control message ex-changes, thus having low communication overhead.

Finally, our third protocol is using a similar approach tothe recent work of [26], where a new technique called SparseTopology and Energy Management (STEM) is proposed thataggressively puts nodes to sleep. Interestingly, the analysisand experiments of STEM show improvements of nearly twoorders of magnitude compared to sensor networks withouttopology management.

Some recent work

In [4] the authors present a new protocol for data propaga-tion that avoids flooding by probabilistically favoring certain(“close to optimal”) data transmissions. As shown by a geom-etry analysis, the protocol is correct, since it always prop-agates data to the sink, under ideal network conditions (nofailures). Using stochastic processes, they show that the pro-tocol is very energy efficient. Also, when part of the networkis inoperative, the protocol manages to propagate data veryclose to the sink, thus in this sense it is robust. They finallypresent and discuss large-scale experimental findings validat-ing the analytical results.

In [5], the authors have implemented and experimentallyevaluated two variations of LTP, under new, more general andrealistic modelling assumptions. They comparatively studyLTP to PFR, by using extensive experiments, highlightingtheir relative advantages and disadvantages. All protocols arevery successful. In the setting considered there, PFR seemsto be faster while the LTP based protocols are more energyefficient.

In [12], Euthimiou et al. study the problem of energy-balanced data propagation in wireless sensor networks. Theenergy balance property guarantees that the average per sen-sor energy dissipation is the same for all sensors in the net-

work, during the entire execution of the data propagation pro-tocol. This property is important since it prolongs the net-work’s lifetime by avoiding early energy depletion of sen-sors. They propose a new algorithm that in each step decideswhether to propagate data one-hop towards the final destina-tion (the sink), or to send data directly to the sink. This ran-domized choice balances the (cheap) one-hop transimssionswith the direct transimissions to the sink, which are more ex-pensive but “bypass” the sensors lying close to the sink. Notethat, in most protocols, these close to the sink sensors tend tobe overused and die out early.

In [1], the authors propose a new energy efficient and faulttolerant protocol for data propagation in smart dust networks,the Variable Transmission Range Protocol (VTRP). The basicidea of data propagation in VTRP is the varying range of datatransmissions, i.e., they allow the transmission range to in-crease in various ways. Thus data propagation in the protocolexhibits high fault-tolerance (by bypassing obstacles or faultysensors) and increases network lifetime (since critical sensors,i.e., close to the control center are not overused). They im-plement the protocol and perform an extensive experimentalevaluation and comparison to a representative protocol (LTP)of several important performance measures with a focus onenergy consumption. The findings indeed demonstrate thatthe protocol achieves significant improvements in energy ef-ficiency and network lifetime.

In [23], Nikoletseas et al. (a) propose extended versionsof two data propagation protocols: the Sleep–Awake Prob-abilistic Forwarding Protocol (SW-PFR) and the Hierarchi-cal Threshold sensitive Energy Efficient Network protocol(H-TEEN). These non-trivial extensions aim at improving theperformance of the original protocols, by introducing sleep–awake periods in the PFR protocol to save energy, and in-troducing a hierarchy of clustering in the TEEN protocol tobetter cope with large networks areas; (b) they have imple-mented the two protocols and performed an extensive exper-imental comparison (using simulation) of various importantmeasures of their performance with a focus on energy con-sumption; (c) they investigate in detail the relative advantagesand disadvantages of each protocol and discuss and explaintheir behavior; (d) in the light above they propose and discussa possible hybrid combination of the two protocols towardsoptimizing certain goals. Efficient collision avoidance proto-cols, particularly useful for multipath data propagation, havebeen proposed in [10].

A brief description of the technical specifications of state-of-the-art sensor devices, a discussion of possible modelsused to abstract such networks and a presentation of somecharacteristic protocols for data propagation in sensor net-works, along with an evaluation of their performance analy-sis, can be found in the recent book chapter of Boukerche andNikoletseas [3].

2. The model

Smart dust is comprised of a vast number of ultra-small ho-mogenous sensors, which we call “grain” particles. Each


Figure 1. A smart dust cloud.

smart-dust grain particle is a fully-autonomous computingand communication device, characterized mainly by its avail-able power supply (battery) and the energy cost of computa-tion and transmission of data. Such particles cannot move.

Each particle is equipped with a set of monitors (sensors)for light, pressure, humidity, temperature, etc. Each particlehas two communication modes: a broadcast (digital radio)beacon mode (for low energy – short signals) and a directedto a point actual data transmission mode (usually via a laserbeam). Also, in a variation of our model capturing energy sav-ing specifications, each particle may alternate (independentlyof other particles) between a sleeping and an awake mode.During sleeping periods grain particles cease any communi-cation with the environment, thus they are unable to listen,receive and propagate data transmitted by other particles. De-pending on the specific application the sensing part may ceaseor not during the sleeping mode. In the case where sensingis not ceased during sleeping mode, detection of the crucialevent wakes the particle up.

We adopt here (as a starting point) a two-dimensional(plane) framework: a smart dust cloud (a set of particles)is spread in an area (for a graphical presentation, see fig-ure 1). Note that a two-dimensional setting is also used in[14,15,17,18,21].

Definition 2.1. Let d (usually measured in numbers of par-ticles/m2) be the density of particles in the area. Let R bethe maximum (beacon/laser) transmission range of each grainparticle. A receiving wall W is defined to be an infinite line inthe smart-dust plane. Any particle transmission within rangeR from the wall W is received by W .

We assume that W has very strong computing power, ableto collect and analyze received data and has a constant powersupply and so has no energy constraints. The wall represents,in fact, the authorities (the fixed control center) who the real-ization of a crucial event should be reported to. Note that awall of appropriately big (finite) length suffices. We plan toconduct an analysis of the (expected and/or with high proba-bility) deviation of the transmitted data from the vertical to the

wall position in order to provide upper bounds on the wall’slength needed.

Furthermore, we assume that there is a set-up phase of thesmart dust network, during which the smart cloud is droppedin the terrain of interest, when using special control messages(which are very short, cheap and transmitted only once) eachsmart dust particle is provided with the direction of W . By as-suming that each smart-dust particle has individually a senseof direction (e.g., through its magnetometer sensor), and usingthese control messages, each particle is aware of the generallocation of W .

We feel that our model, although simple, depicts accu-rately enough the technological specifications of real smartdust systems. Similar models are being used by other re-searchers in order to study sensor networks (see [14,21]). Incontrast to [18,19], our model is weaker in the sense that nogeolocation abilities are assumed (e.g., a GPS device) for thesmart dust particles leading to more generic and thus strongerresults. In [16] a thorough comparative study and descriptionof smart dust systems is given, from the technological pointof view. In the following section we report some basic techni-cal characteristics which we took into account when definingthe model of smart dust we use here.

3. Technological specifications of smart dust devices

New technology is changing the nature of sensors and the waythey interface with data acquisition and control systems. Re-searchers have developed an open-source hardware and soft-ware platform that combines sensing, communications, andcomputing into a complete architecture. The first commer-cial generation of this platform was dubbed the Rene Mote,and several thousand of these sensors have been deployed atcommercial and research institutions worldwide to promotethe development and application of wireless sensor networks.

The platforms development community is based on theopen-source model, which has become well known with theincreasingly popular Linux operating system. Most develop-ment work is done in the public domain, and it includes thehardware design and software source code. Users of the tech-nology contribute their developments back to the communityso that the base of code and hardware design grows rapidly.

It is worth noting that currently, a number of research insti-tutions in the U.S. are working on centimeter-scale (and evensmaller) distributed sensor networks [2,29].

3.1. Hardware design of wireless sensors

The basic MICA hardware uses a fraction of a Watt of powerand consists of commercial components a square inch in size.The hardware design consists of a small, low-power radio andprocessor board (known as a mote processor/radio, or MPR,board) and one or more sensor boards (known as a mote sen-sor, or MTS, board). The combination of the two types ofboards form a networkable wireless sensor.

The MPR board includes a processor, radio, A/D converter,and battery. The processor is an ATMEL ATMEGA, but there


CPU speed 4 MHzMemory ROM: 128 Kb FLASH

SDRAM: 4 KbEEPROM: 4 Kb

Power supply 2 AA batteriesPower consumption 0.75 mWProcessor current draw 5.5 mA (active current)

< 20 µA (sleep mode)Radio current draw 12 mA (transmit current)

1.8 mA (receive current)< 1 µA (sleep mode)

Output device 3 LEDsI/O port Expansion connected (51 pin)

Serial port (proprietary 16-pin)Network Wireless 4 Kbits/s at 916 MHz (ISM band)

Radio range depends on antennae configuration

Figure 2. MPR300CB specifications.

are other processors that would meet the power and cost tar-gets. The processor runs at 4 MHz, has 128 Kb of flash mem-ory and 4 Kb of SDRAM. In a given network, thousands ofsensors could be continuously reporting data, creating heavydata flow. Thus, the overall system is memory constrained,but this characteristic is a common design challenge in anywireless sensor network.

The MPR modules contain various sensor interfaces,which are available through a small 51-pin connector thatlinks the MPR and MTS modules. The interface includes: an8-channel, 10-bit A/D converter; a serial UART port; and anI2C serial port. This allows the MPR module to connect to avariety of MTS sensor modules, including MTS modules thatuse analog sensors as well as digital smart sensors. The MPRmodule has a guaranteed unique, hard-coded 64-bit address.

The processors, radio, and a typical sensor load consumesabout 100 mW in active mode. This figure should be com-pared with the 30 µA draw when all components are in sleepmode. Figure 2 shows a synopsis of the MPR specs.

The MTS sensor boards currently include light/tempera-ture, two-axis acceleration, and magnetic sensors and 420 mAtransmitters. The wireless transmission is at 4 Kbps rate andthe transmission range may vary. Researchers are also devel-oping a GPS board and a multisensor board that incorporatesa small speaker and light, temperature, magnetic, accelera-tion, and acoustic (microphone) sensing devices. The MICAdevelopers community welcomes additional sensor board de-signs.

3.2. Software and the TinyOS

A considerable portion of the challenge faced by the devel-opers of MICA devices is in the software embedded in thesensors. The software runs the hardware and networkmakingsensor measurements, routing measurement data, and control-ling power dissipation. In effect, it is the key ingredient thatmakes the wireless sensor network produce useful informa-tion.

To this end, a lot of effort has gone into the design of asoftware environment that supports wireless sensors. The re-sult is a very small operating system named TinyOS, or Tiny

Software footprint 3.4 KbTransmission cost 1 µJ/bitInactive state < 25 µAPeak load 20 mATypical CPU usage < 50%Events propagate thru stack < 40 µs

Figure 3. TinyOS key facts.

Microthreading Operating System, which allows the network-ing, power management, and sensor measurement details tobe abstracted from the core application development. Theoperating system also creates a standard method of develop-ing applications and extending the hardware. Although tiny,this operating system is quite efficient, as shown by the smallstack handling time. Figure 3 lists the key points of TinyOS.

4. The problem

An adversary A selects a single particle, p, in the plane-cloudand allows it to monitor a local crucial event E . The generalpropagation problem P is the following:

“How can particle p, via cooperation with the rest of thecloud, propagate information about event E to the receiv-ing wall W”?

Definition 4.1. Let hopt(p,W) be the (optimal) number of“hops” (direct, vertical to W transmissions) needed to reachthe wall, in the ideal case in which particles always exist inpair-wise distances R in the vertical line from p to W . Let� be a smart-dust propagation protocol, using a transmis-sion path of length L(�,p,W) to send info about event E towall W . Let h(�,p,W) be the number of hops (transmis-sions) taken to reach W . The “hops” efficiency of protocol �

is the ratio

Ch = h(�,p,W)

hopt(p,W).

Clearly, the number of hops (transmissions) needed char-acterizes the energy consumption and the time needed topropagate the information E to the wall. Remark that hopt =�d(p,W)/R�, where d(p,W) is the (vertical) distance of p

from the wall W .In the case where protocol � is randomized, or in the case

where the distribution of the particles in the cloud is a randomdistribution, the number of hops h and the efficiency ratio Ch

are random variables and we study here their expected values.The reason behind these definitions is that when p (or any

intermediate particle in the information propagation to W)“looks around” for a particle as near to W as possible to passits information about E , it may not get any particle in the per-fect direction of the line vertical to W passing from p. Thisdifficulty comes mainly from three causes: (a) due to the ini-tial spreading of particles of the cloud in the area and becauseparticles do not move, there might not be any particle in thatdirection; (b) particles of sufficient remaining battery power


may not be available in the right direction; (c) particles maytemporarily “sleep” (i.e., not listen to transmissions) in orderto save battery power.

Remark. Note that any given distribution of particles in thesmart dust cloud may not allow the ideal optimal number ofhops to be achieved at all. In fact, the least possible numberof hops depends on the input (the positions of the grain par-ticles). We have chosen, however, to compare the efficiencyof our protocols to the ideal case. A comparison with the bestachievable number of hops in each input case will of coursegive better efficiency ratios for our protocols.

5. The local target protocol (LTP)

Let d(pi, pj ) the distance (along the corresponding verticallines towards W) of particles pi, pj and d(pi,W) the (verti-cal) distance of pi from W . Let info(E) the information aboutthe realization of the crucial event E to be propagated. In thisprotocol, each particle p′ that has received info(E) from p

(via, possibly, other particles) does the following:

• Search phase. It uses a periodic low energy broadcast ofa beacon in order to discover a particle nearer to W thanitself (i.e., a particle p′′ where d(p′′,W) < d(p′,W)).

• Direct transmission phase. Then, p′ sends info(E) to p′′via a direct line (laser) time consuming transmission.

• Backtrack phase. If consecutive repetitions of the searchphase fail to discover a particle nearer to W , then p′ sendsinfo(E) to p (i.e., to the particle that it originally receivedthe information).

Note that one can estimate an a-priori upper bound on thenumber of repetition of the search phase needed, by using theprobability of success of each search phase. This bound canbe used to decide when to backtrack.

Also note that the maximum distance d(p′, p′′) is R, i.e.,the beacon transmission range (for a graphical representationsee figures 4, 5).

To enable a first step towards a rigorous analysis of smartdust protocols, we make the following simplifying assump-tion. The search phase takes zero time and always finds a p′′

Figure 4. Example of the search phase.

(of sufficiently high battery) in the semicircle of center p, inthe direction towards W . Note that this assumption on al-ways finding a particle can be relaxed in the following ways:(a) by repetitions of the search phase until a particle is found.This makes sense if at least one particle exists but was sleep-ing during the failed searches; (b) we may consider, insteadof just the semicircle, a cyclic sector defined by circles of ra-diuses R − �R, R and also take into account the densityof the smart cloud; (c) if the protocol during a search phaseultimately fails to find a particle towards the wall, it may back-track.

In this analysis we do not consider the energy spent in thesearch phase. Note, however, that even the case where thisis comparable to the energy spent in actual data transmission,the number of hops accounts for both (total energy spent isupper bounded by a multiple of actual data transmission en-ergy).

We also assume that the position of p′′ is uniform in thearc of angle 2a around the direct line from p′ vertical to W .Each data transmission (one hop) takes constant time t (so the“hops” and time efficiency of our protocols coincide in thiscase). We also assume that each target selection is randomindependent of the others, in the sense that it is always drawnuniformly in the arc (−α, α).

We are aware of the fact that the above assumptions maynot be very realistic in practice, however, they allows us toperform a first effort towards providing some concrete analyt-ical results.

Lemma 5.1. The expected “hops” efficiency of the local tar-get protocol in the α-uniform case is E(Ch) � α/ sin α, forlarge hopt. Also 1 � E(Ch) � π/2 ≈ 1.57, for 0 � α � π/2.

Proof. Due to the protocol, a sequence of points is gener-ated, p0 = p,p1, p2, . . . , ph−1, ph where ph−1 is a particlewithin W’s range and ph is part of the wall. Let αi be the(positive or negative) angle of pi with respect to pi−1’s verti-

Figure 5. Example of a transmission.


cal line to W . It is:

h−1∑

i=1

d(pi−1, pi) � d(p,W) �h∑

i=1

d(pi−1, pi).

Since the (vertical) progress towards W is then �i =d(pi−1, pi) = R cos αi , we get:

h−1∑

i=1

cos αi � hopt �h∑

i=1

cos αi .

From Wald’s equation for the expectation of a sum of a ran-dom number of independent random variables (see [25]), then

E(h − 1)E(cos αi) � E(hopt) = hopt � E(h)E(cos αi).

Now, ∀i, E(cos αi) = ∫ α

−αcos x(1/2α) dx = sin α/α. Thus

α

sin α� E(h)

hopt= E(Ch) � α

sin α+ 1

hopt.

Assuming large values for hopt (i.e., events happeningfar away from the wall, which is the most interesting casein practice since the detection and propagation difficulty in-creases with distance) we have (since for 0 � α � π/2 it is1 � α/ sin α � π/2) we get the result. �

6. Local optimization – the “min two uniform targets”protocol (m2TP)

Note that the same basic framework holds for any situation inwhich the local (vertical) progress in the direction towards W(i.e., �i) is of the same, independent, distribution. I.e., italways holds (via the Wald’s equation) that

RE(�i)

� E(h)

hopt� R

E(�i)+ 1

hopt

⇒ E(Ch) ≈ RE(�i)

= 1

E(cos αi)(1)

for large h. To understand the power of this, let us assumethat the search phase always returns two points p′′, p′′′ eachuniform in (−α, α) and that the protocol selects the best ofthe two points, with respect to the local (vertical) progress.

Lemma 6.1. The expected “hops” efficiency of the “min twouniform targets” protocol in the α-uniform case is

E(Ch) ≈ α2

2(1 − cos α),

for 0 � α � π/2 and for large h.

Proof. Let αi1, αi2 the angles of the particles found and letαi = min{|αi1|, |αi2|}. Then, for any 0 � φ � α, it is:

P{αi > φ} = P{|αi1| > φ ∩ |αi2| > φ

}

=(

2α − 2φ

2α

)2

=(

α − φ

α

)2

.

Thus, the distribution function of αi , for any 0 � φ � α, is

Fαi (φ) = P{αi � φ} = 1 − (α − φ)2

α2 = 2αφ − φ2

α2

and the probability density function is, for any 0 � φ � α:

fαi (φ) = d

dφP{αi � φ} = 2

α

(1 − φ

α

).

The expected local progress is:

E(cos αi) =∫ α

0cos φfαi (φ) dφ = 2(1 − cos α)

α2. (2)

�

We remark that

limα→0

E(Ch) = limα→0

2α

2 sin a= 1

and

limα→π/2

E(Ch) = (π/2)2

2(1 − 0)= π2

8≈ 1.24.

Lemma 6.2. The expected “hops efficiency of the min-twouniform targets protocol is 1 � E(Ch) � π2/8 ≈ 1.24 forlarge h and for 0 � α � π/2.

We remark that, w.r.t. the expected hops efficiency of thelocal target protocol, the min-two uniform targets protocolachieves, because of the one additional search, a relative gainwhich is (π/2 − π2/8)/(π/2) ≈ 21.5%. We experimentallyinvestigate the further gain of additional (i.e., m > 2) searchesin section 10.

7. Tight upper bounds to the hops distribution of thegeneral target protocol

Consider the particle p (which senses the crucial event) at dis-tance x from the wall. Let us assume that when p searches inthe sector S defined by angles (−α, α) and radius R, anotherparticle p′ is returned in the sector with some probability den-

sity f (−→p′ ) dA, where

−→p′ = (xp′, yp′) is the position of p′

in S and dA is an infinitesimal area around p′.

Definition 7.1 (Horizontal progress). Let �x be the projec-tion of the line segment (p, p′) on the line from p verticalto W .

We assume that each search phase returns such a particle,with independent and identical distribution f (·).

Definition 7.2 (Probability of significant progress). Letm > 0 be the least integer such that P{�x > R/m} � p,where 0 < p < 1 is a given constant.

Lemma 7.1. For each continuous density f (·) on the sector S

and for any constant p, there is always an m > 0 as above.


Proof. Remark that f (·) defines a density function f (·) on(0,R] which is also continuous. Let F ( ) its distribution func-tion. Then we want 1 − F (R/m) � p, i.e., to find the first m

such that 1 − p � F (R/m). Such an m always exists sinceF is continuous in [0, 1]. �

Definition 7.3. Consider the (discrete) stochastic process P

in which with probability p the horizontal progress is R/m

and with probability q it is zero, where q = 1 − p.Let Q the actual stochastic process of the horizontal

progress implied by f (·).

Lemma 7.2. PP {h � h0} � PQ{h � h0}.

Proof. The actual process Q makes always more progressthan P . �

Now let t = �x/(R/m)� = �mx/R�. Consider the integerrandom variable H such that P{H = i} = qi(1 − q) for anyi � 0. Then H is geometrically distributed. Let H1, . . . , Ht

be t random variables, independent and identically distributedaccording to H . Clearly then

Lemma 7.3. PP {number of hops is h} = P{H1 + · · · +Ht = h}.

The probability generating function of H is

H(s) = P{H = 0} + P{H = 1}s + · · · + P{H = i}si + · · · ,i.e.,

H(s) = p(1 + qs + q2s2 + · · · + qisi + · · · ) = p

1 − qs.

But the probability generating function of∑

t = H1 + · · · +Ht is then just (p/(1 − qs))t by the convolution theorem ofgenerating functions. This is just the generating function ofthe t-fold convolution of geometric random variables, and it isexactly the distribution of the negative binomial distribution(see [13], vol. 1, p. 253). Thus,

Theorem 7.4.

PP {the number of hops is h} =(−t

h

)pt (−q)h

=(

t + h − 1

h

)ptqh.

Corollary 7.5. For the process P , the mean and variance ofthe number of hops are:

E(h) = tq

p, Var(h) = tq

p2.

Note that the method sketched above, finds a distributionthat upper bounds the number of hops till the crucial event is

reported to the wall. Since for all f (·) it is h � x/R = hoptwe get that

EP (h)

hopt� �mx/R�q/p

x/R � (m + 1)q

p.

Theorem 7.6. The above upper bound process P estimatesthe expected number of hops to the wall with a guaranteedefficiency ratio (m + 1)/(1 − p)p at most.

Example. When for p = 0.5 we have m = 2 and the effi-ciency ratio is 3, i.e., the overestimate is 3 times the optimalnumber of hops.

8. The “sleep–awake” protocol (SWP)

We now present a new protocol for smart dust networks whichwe call the “sleep–awake” protocol. In contrast to the previ-ous protocols, we now assume that we can explicitly use theperiods that a particle is in awake mode or in sleeping mode.During sleeping periods grain particles cease any communi-cation with the environment, thus they are unable to listen,receive and propagate data transmitted by other particles.

The procedures of search transmission and backtrack arethe same as in the LTP.

In the above procedure, propagation of info(E) is done intwo steps; (i) particle p′ locates the next particle (p′′) andtransmits the information and (ii) particle p′ waits until thenext particle (p′′) succeeds in propagating the message furthertowards W . In both steps particle p′ will remain awake. Thisis done to speed up the backtrack phase in case p′′ does notsucceed in discovering a particle nearer to W . Note, however,that as soon as p′′ succeeds to propagate data, p′ resumes itssleep–awake mode.

Propagation protocols for such energy-restricted systemsshould at least guarantee that the wall eventually receives themessages that report a crucial event. The success of suchprotocols depends on the density d of grain particles/m2 andtheir distribution, the distribution of sleeping and awake timeperiods and, of course, on the angle α of the search beacon.

We below provide some first results on the interplay be-tween these parameters. In particular, we focus on the rela-tion between the maximum sleeping time period and the otherparameters, thus allowing to program the smart-cloud energysaving specifications accordingly.

To simplify the analysis we assume that the grain particlesare uniformly distributed on the smart-dust plane. Thus, inthe area inspected during a search phase of beacon angle α

between R and R + �R, the number of grain particles is

N = d

(α

ππR2 − α

ππ(R − �R)2

)

� α(2R�R − �R2)d � 2αdR�R. (3)

Now, we assume that the sleeping/awake time durations al-ternate independently in each particle and have lengths s, w,respectively (this can be easily achieved if during the start-up


phase, the first awake period w is set using a random bit gen-erator, or is hardcoded into the particle by the manufacturer).Thus, the probability that at least one of the N particles in thesender’s beacon search area is awake is:

P1 = P{at least one particle is awake}= 1 −

(s

s + w

)N

= 1 −(

s

s + w

)2αdR�R.

Thus the probability that the event report eventuallyreaches the wall is:

P{success} =∞∑

h0=1

Ph1 P{h = h0},

where P{h = h0} is the probability density function of therandom variable h.

Let now β = s/w, i.e., β represents the energy savingspecifications of the smart dust particles (a typical value for β

may be 100). Then,

Definition 8.1. The energy saving specification is:

en = s

s + w= 1 − 1

1 + β.

By taking d such that 2α�Rd � n(1 + β) we get P1 =1 − e−n. Then, by the Bernoulli inequality, we have P

E(h)1 �

1 − E(h)e−n. This probability is non-zero when

n > ln E(h).

This final condition allows to set the technical specifica-tions and the propagation time accordingly in order to guar-antee that the crucial event is eventually reported to the wall.

9. Implementation aspects and details

We now proceed by providing a more detailed description ofthe protocols implementation in our simulation environment.We also discuss implementation aspects of our protocols incurrent technology wireless sensor networks.

We assume that the particles are equipped with TinyOS,an event driven operating system suitable for smart dust [28].The pseudo-code presented in figures 6–8 demonstrates howto implement the SWP protocol. Note that the implementa-tion of LTP is very similar.

At every particle, we use a Boolean variable HOLDER todenote the status of the particle. It is set to true only if thesite holds info(E) (the information about the realization of thecrucial event E to be propagated). A variable PREVIOUSrecords the particle from which info(E) was received. A setOUTf is used to store any particle that failed to propagate amessage towards the wall (this set is used for backtrackingpurposes).

msgHandler rcvInfo(msg) {Timer.stop( );HOLDER = true;PREVIOUS = sender(msg);

}

msgHandler rcvReqBeacon(msg) {initiator = sender(msg);send(initiator)[BeaconMsg];

}

msgHandler rcvBeacon(msg) {remember(sender(msg), power(msg));

}

msgHandler rcvSuccess(msg) {Timer.start(PERIODs);PowerDisable( );

}

msgHandler rcvFail(msg) {Timer.stop( );HOLDER = true;OUTf = OUTf ∪ {sender(msg)};

Figure 6. The Message Handler procedures.

eventHandler SensorEvent {PowerEnable( );Timer.stop( );HOLDER = true;

}

eventHandler Timer.fired( ) {if (power == Enable) {

Timer.start(PERIODs);PowerDisable( );

} else {Timer.start(PERIODw);PowerEnable( );

}

Figure 7. The Event Handler procedures.

In addition, for SWP we use a decreasing clock timerCLOCK that can be explicitly activated, deactivated and set toa given value, and two constant variables PERIODw, PERI-ODs provided by the implementer that indicate the length ofthe awake and sleeping periods of the particle. For example,in TinyOS, to create a timer that expires every PERIODw mswe use the statement Timer.start(TIMER_REPEAT,PERIODw); and Timer.stop(); terminates the timer.Each time the timer expires, a Timer.fired() eventis triggered that invokes a function implemented by theuser.

Initially, the Boolean variables HOLDER and EXECUTINGare set to false, the variable PREVIOUS is set to itself andthe set OUTf is empty. For SWP, each particle is at awakemode with its CLOCK set to a period chosen randomly inthe range (0, PERIODw + PERIODs]. In TinyOS this canbe implemented using the method Random.rand() of thebuilt-in 16-bit Linear Feedback Shift Register pseudo-randomnumber generator.


The protocols use five types of messages: info(E), fail,success, requestBeacon, beacon. The first message type isused to propagate the actual information on the crucial event,while the next two (fail and success) are special control mes-sages used to signify a failure or a success in the attemptto propagate info(E) towards the receiving wall W . TherequestBeacon and beacon messages are used by the searchphase. Interestingly, in TinyOS, radio communication followsthe Active Message (AM) model, in which each packet on thenetwork specifies a handler ID that will be invoked on recip-ient nodes. When a message is received, the receive eventassociated with that handler ID is signaled. Thus we onlyneed to define one message handler per message type. Fig-ure 6 depicts the five message handlers implemented by theprotocols.

Remark that the beacon message handler assumes that thecommunication module is capable of measuring the signalstrength of the message received by executing the functionpower(msg). Similarly, the function sender(msg) isused to extract the originator of a message, assuming thatthe message structure maintains such kind of information.Also, the function remember(. . .) adds the informationto a temporary buffer. Using these primitives by sending arequestBeacon message the particle initiates the search phase

task main {if (HOLDER == true) {

next = SenseNeighbours( );if (next == nil)

BackTrack( );else {

send(next)[info(E)];send(PREVIOUS)[success];HOLDER = false;

}}

post main( );}

procedure SenseNeighbors {send( )[reqBeacon];

Set tempSet = DetectNeighbors( );if (tempSet == empty)

return nil;

Set out = tempSet − OUTf ;if (out == empty)

return nil;

return out.first( );}

procedure Backtrack {send(PREVIOUS)[fail];HOLDER = false;Timer.start(PERIODs);

PowerDisable( );}

Figure 8. The Main task and the SenseNeighbors, Backtrack procedures.

and then, after waiting for a sufficient period of time (so thatall neighbors respond to the request by sending a beacon),the DetectNeighbors() procedure processes the tem-porary buffer and returns a set containing those particles thatresponded to the broadcast of the search beacon, ordered bythe distance of the particles (i.e., d(p′, p′′)).

Apart from the message event handlers, the protocols usetwo additional types of events: (i) SensorEvent createdby the sensors of the particle when a crucial event is real-ized (i.e., when the particle is selected by the adversary A)and (ii) Timer.fired() created by the Timer when thecounting has finished. Figure 7 depicts the two generic eventhandlers implemented by our protocols. Remark that thePowerDisable() and PowerEnable() will force theparticle to enter a “snooze” mode where only the Timer is ac-tive.

Based on the above event driven functionality, particle p

executes continuously the Main task, shown in figure 8.TinyOS provides a two-level scheduling hierarchy consistingof tasks and hardware event handlers. Tasks are used to per-form longer processing operations, such as background dataprocessing, and can be preempted by hardware event handler.Remark that the post operation places the task on an internaltask queue which is processed in FIFO order.

10. Experimental evaluation

In this section we report on four sets of experiments that aimto validate the theoretical analysis of the previous sections.We have implemented the three protocols using C++ and thedata types for two-dimensional geometry of LEDA [22]. Eachclass is installed in an environment that generates sensor fieldsgiven some parameters (such as the area of the field, the dis-tribution function used to drop the particles), and performs anetwork simulation for a given number of repetitions, a fixednumber of particles and certain protocol parameters. Afterthe execution of the simulation, the environment stores theresults on files so that the measurements can be representedin a graphical way. Each experiment was conducted for morethan 10,000 times in order to achieve good average results.

In the first set of experiments, we investigate (a) the impactof the angle α and (b) the number of targets found during thesearch phase, on the hops efficiency of the Local target pro-tocol when considering the ideal case where the search phasealways finds a particle (of sufficiently high battery) in (−α, α)

(we call the measured efficiency, the ideal hops efficiency). Infigure 9 we observe that for both protocols, as α → 0, theideal hops efficiency Ch → 1. Actually, the ideal Ch initiallydecreases very fast with increasing α, while having a limitingbehavior of no further significant improvement when α � 40.Figure 10 shows the effect of finding more than one target dur-ing the search phase; as the number of targets increases, theideal hops efficiency Ch → 1. We note a similar thresholdbehavior, for a total number of 4 targets.

In the second set of experiments we study the performanceof the LTP and m2TP protocols in more realistic cases by gen-erating a variety of sensor fields in a 100 m × 100 m square.


In these fields, we drop n ∈ [100, 5000] particles uniformlydistributed on the smart-dust plane, i.e., 0.01 � d � 0.5.Each smart dust particle has a radio range of R = 5 m. Forcarrying out identical repetitions on our experiments we ex-plicitly place a particle at position (x, y) = (0, 50) and weassume that this particle detects the event. The wall is locatedat x = 100. In this set of experiments, the particle p′ dis-covered in the search phase can be located anywhere withinthe cyclic sector defined by circles of radiuses 0,R and an-gles (−α, α). Note that this experimental setup is based onthat used in [14,17,21]. Also, remark that the efficiency ismeasured over the successful tries, i.e., we do not take intoaccount those runs that backtracked, however we keep trackof the total number of times that the protocol was required tobacktrack.

In figure 11 we observe that opposed to the ideal case (i.e.,when the search phase always returns a particle on the semi-circle), we do not get significant improvement in the hops ef-ficiency as the angle α is reduced. This is basically becausethe discovered particle p′ might be close to p and thus thelocal improvement made is of limited significance. Note thatthe min-two uniform targets protocol (m2TP) achieves betterefficiency compared to the local target protocol (LTP).

Figure 12 depicts the effect of density d on the hops ef-ficiency of the two protocols. Interestingly, we observe thateven for quite low density of particles (i.e., d � 0.2) the hops

Figure 9. Ideal hops efficiency for angles α ∈ [5, 90].

Figure 10. Ideal hops efficiency for different number of targets.

efficiency remains unaffected. This is a result of our choicenot to include the failed searches in our measurements, thatis, the measurements include only the search phases that re-sulted in finding a particle p closer to W . To get a more com-plete view on the effect of density, figure 13 shows the failurerate (i.e., the number of times that the protocols backtracked)for different values of d . We observe that for low density(i.e., d � 0.1) both protocols almost always use the backtrack

Figure 11. Hops efficiency for angles α ∈ [5, 90] for d = 0.3.

Figure 12. Hops efficiency for density d ∈ [0.01, 0.5] and α = 90.

Figure 13. Failure rate for density d ∈ [0.01, 0.5] and α = 90.


Figure 14. Probability of success (P{success}) over particle density d = [0.01, 0.3] for various angles α = {45, 60, 90}, and random distribution.

Figure 15. Average hops efficiency (Ch) over particle density d = [0.01, 0.3] for various angles α = {45, 60, 90}, and random distribution.

mechanism, while when d � 0.2 the failure rate drops veryfast to zero. This can be justified by taking into account theaverage degree of each particle for various density d .

In the third set of experiments, we evaluate the perfor-mance of the SWP protocol in the case when all the particlesremain awake (i.e., en = 0). We consider this a first step to in-vestigate (a) the impact of the angle α and (b) the effect of theparticles density d on the probability of success (P{success}),hops efficiency (Ch) and average number of backtracks. Theparticle density was 0.01 � d � 0.3 and used three different

angles, α = {45, 60, 90} (in degrees). The reported experi-ments for the three different performance measures we con-sidered are illustrated in figures 14–16.

Examining figure 14, that shows the probability of suc-cess (P{success}), we first observe that for particle densityd < 0.05 (i.e., throwing a small number of particles) theprotocol fails to propagate the critical event (i.e., the suc-cess probability is zero). However, the probability of suc-cess increases very fast exhibiting a threshold-like behaviouras the particle density increases, and the protocol almost al-


Figure 16. Average number of backtracks over particle density d = [0.01, 0.3] for various angles α = {45, 60, 90}, and random distribution.

Figure 17. Probability of success (P{success}) over en (where en = s/(s + w)) for various particle densities d = {0.15, 0.2, 0.25}, fixed angle α = 90 andrandom distribution.

ways succeeds to propagate the critical event when d > 0.15.As expected (due to equation (3)), setting a smaller angleα reduces the probability of success in each density casesince the number of particles that respond to the search phasegets smaller. So for α = 60 the P{success} gets close to 1when d > 0.2 while for α = 45, P{success} → 1 whend > 0.25.

Regarding the average hops efficiency (Ch) we interest-ingly observe in figure 15 that even for a small particle den-sity, the hops efficiency is close to the optimal. Actually, as

the particle density crosses d = 0.05 (i.e., when P{success} >

0) the hops efficiency gets close to 2.6. In fact, when d = 0.1,Ch = 1.74 while no further gain is achieved if we throw moreparticles (i.e., increase d). This is because of a sufficientlylarge density leads to many particles found in the search, ofwhich particles already some are close to the vertical line.Similar results hold for α = 60 and α = 45, although at highparticle densities, the hops efficiency is slightly better.

Finally, in figure 16 we can see the average number ofbacktracks performed by the protocol in the attempt to prop-


Figure 18. Average hops efficiency (Ch) over en (where en = s/(s + w)) for various particle densities d = {0.15, 0.2, 0.25}, fixed angle α = 90 and randomdistribution.

Figure 19. Average number of backtracks over en (where en = s/(s + w)) for various particle densities d = {0.15, 0.2, 0.25}, fixed angle α = 90 and randomdistribution.

agate the critical event to the wall. We observe that for lowdensity (i.e., d � 0.1) the protocols almost always uses thebacktrack mechanism, while when d � 1.5 the number ofbacktracks performed drops very fast to zero. Furthermore,we observe that the number of backtracks is initially highbut decreases with a fast rate as the particle density increases.This can be justified by taking into account the average degreeof each particle for various density d . More specifically, whenα = 90 and d = 0.15 the protocol almost always succeeds inpropagating the crucial event without the need to backtrack.

The last set of experiments aims to evaluate the impactof the energy saving specification en on the performance ofthe protocol. Again, we measure the probability of success(P{success}), hops efficiency (Ch) and average number ofbacktracks over energy saving specification (en), for three dif-ferent particle densities (d = 0.15, 0.2, 0.25) and three differ-ent angles α = {45, 60, 90} (in degrees). We have set theawake period w = 2 and the sleeping period s ∈ [0, 15]thus making en ∈ [0, 0.88] (recall that en = s/(s + w)).Figures 17–19 show the measured performance for the dif-


Figure 20. Probability of success (P{success}) over en (where en = s/(s + w)) for various angles α = {45, 60, 90}, fixed particle density d = 0.2 and randomdistribution.

Figure 21. Average hops efficiency (Ch) over en (where en = s/(s + w)) for various angles α = {45, 60, 90}, fixed particle density d = 0.2 and randomdistribution.

ferent particle densities d and figures 20–22 for the differentangles α.

In figures 17 and 20, that show the probability of suc-cess for different particle densities d and angles α, we ob-serve that the protocol experiences a threshold behavior whenen = 0.75: when en � 0.75 the probability of successis close to 1 while for en > 0.75, P{success} drops veryfast to zero. In other words, even if we set the particles tobe awake only the 25% of each sleep–awake cycle, it doesnot affect the success of the protocol to propagate the in-formation to W . However, in figures 18 and 20 we ob-serve a similar threshold behavior for the average hops ef-

ficiency when en = 0.5: the hops efficiency remains unaf-fected when en � 0.5 while it decreases very fast (i.e., Ch

increases) when en > 0.5. Interestingly, figures 19 and 21show that for the same threshold value (en = 0.5) the pro-tocol almost always succeeds without the need to backtrack,while for en > 0.5 the number of backtracks increases veryfast with en. Thus, although for en � 0.8 the probabilityof success is close to 1, setting the energy saving specifica-tion to en = 0.5 seems to be more reasonable. This leads tothe conclusion that by setting the particles to be active onlythe 50% of the overall period for which the protocol is exe-cuted, we manage to decrease the energy requirements while


Figure 22. Average number of backtracks over en (where en = s/(s + w)) for various angles α = {45, 60, 90}, fixed particle density d = 0.2 and randomdistribution.

keeping the hops efficiency (and thus time efficiency) unaf-fected.

11. Conclusions and future work

We presented here a model for Smart Dust and three basicprotocols (and their average case performance) for local de-tection and propagation. We plan to investigate protocols thattrade-off hops efficiency and time, as well as study the fault-tolerance of protocols as a function of smart dust parameters(such as density of the cloud, the energy saving character-istics, etc.). We also intend to investigate alternative back-track mechanisms and study their effect on the efficiency andfault-tolerance of the protocol. Also, we are currently work-ing towards the design of local protocols than can monitorthe spreading of a time-sequence of events (i.e., tracking pro-tocols). We plan to provide performance comparisons withother protocols mentioned in the related work section. Fi-nally, we plan to also explicitly introduce sensor faults andstudy the performance (efficiency, fault-tolerance) of our pro-tocols in this case.

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Ioannis Chatzigiannakis is a Researcher of Re-search Unit 1 (“Foundations of Computer Science,Relevant Technologies and Applications”) at theComputer Technology Institute (CTI), Greece. Hehas received his B.Eng. degree from the Universityof Kent, UK in 1997 and his Ph.D. degree from theComputer Engineering and Informatics Departmentof Patras University, Greece in 2003, under the su-pervision of Prof. Paul Spirakis. His research in-terests include distributed computing, mobile com-

puting and algorithmic engineering. He has served as an external reviewer

in major international conferences. He has participated in several EuropeanUnion funded R&D projects, and worked in the private sector.


Sotiris E. Nikoletseas is currently a Lecturer Pro-fessor at the Computer Engineering and InformaticsDepartment of Patras University, Greece and also aSenior Researcher and Director of Research Unit 1(“Foundations of Computer Science, Relevant Tech-nologies and Applications”) at the Computer Tech-nology Institute (CTI), Greece. His research in-terests include probabilistic techniques and randomgraphs, average case analysis of graph algorithmsand randomized algorithms, algorithmic applications

of probabilistic techniques in distributed computing (focus on ad-hoc mobilenetworks and wireless sensor networks), algorithmic applications of combi-natorial and probabilistic techniques in fundamental aspects of modern net-works (focus on network reliability and stability), approximation algorithmsfor computationally hard problems. He has published over 80 scientific ar-ticles in major international conferences and journals and has co-authoreda book on probabilistic techniques, a chapter in the Handbook of Random-ized Computing (Kluwer Academic) and several chapters in books of inter-national circulation in topics related to distributed computing. He has beeninvited speaker in international scientific events and Universities and he hasdelivered several tutorials and keynote talks. He has been a reviewer forimportant computer science journals and has served in the Program and Or-ganizing Committees of International Conferences and Workshops. He hasparticipated in many European Union funded R&D projects.E-mail: [email protected]

Paul G. Spirakis born in 1955, got his Ph.D. fromHarvard University in 1982. He became an AssistantProfessor at NYU (the Courant Institute) the sameyear. He was then elected as a tenured AssociateProfessor at Patras University, Greece, in the De-partment of Computer Engineering and Informatics.He became a Full Professor in the same Departmentin 1990. He served as the Chairman of the Depart-ment of Computer Engineering and Informatics forsix years. Since 1996 he is the Director of the Re-

search and Academic Computer Technology Institute (RACTI) of Greece.His research interests include algorithms, probabilistic techniques, distrib-uted and parallel computing and average case analysis. Recently P. Spirakisis active in the area of algorithmic aspects of game theory and also in algo-rithmic aspects of ad-hoc and sensor networks. Paul Spirakis has won the topprize of the Greek Mathematical Society in 1973 as a student. Since then hewon several awards and many competitive grants. He was appointed a dis-tinguished Visiting Researcher of Max Planck Informatik. He served as theNational Representative of Research in Informatics in the EU for four yearsand is now a Member of the ISTAG group, a high level EU group respon-sible for Research Planning in Informatics. He is currently one of the twoVice Chairs of the European Association for Theoretical Computer Science(EATCS). He is a high level consultant of the Greek State and Industry inInformatics. Paul Spirakis has published extensively in many journals andconferences of his field (more than 150 publications currently). The journalsand conferences where his work appears are among the most competitiveworldwide. He coauthored two books with Cambridge University Press andseven books in Greek. Paul Spirakis serves as an Editor in many scientificjournals of computer science and usually in the scientific commitees of someof the most prestigeous computer science conferences. He also serves peri-odically as a high level research evaluator for the EU and Greece.E-mail: [email protected]


Training a Wireless Sensor Network ∗

A. WADAA, S. OLARIU and L. WILSONDepartment of Computer Science, Old Dominion University, Norfolk, VA 23529-0162, USA

M. ELTOWEISSYDepartment of Computer Science, Virginia Tech, Falls Church, VA 22043, USA

K. JONESNASA Langley Research Center, Hampton, VA 23681, USA

Abstract. The networks considered in this paper consist of tiny energy-constrained commodity sensors massively deployed, along with oneor more sink nodes providing interface to the outside world. Our contribution is to propose a scalable energy-efficient training protocol fornodes that are initially anonymous, asynchronous and unaware of their location. Our training protocol imposes a flexible and intuitive coor-dinate system onto the deployment area and partitions the anonymous nodes into clusters where data can be gathered from the environmentand synthesized under local control. An important by-product of the training protocol is a simple and natural data fusion protocol as wellas an energy-efficient protocol for routing data from clusters to the sink node. Being energy-efficient, our training protocol can be run oneither a scheduled or ad-hoc basis to provide robustness and dynamic reconfiguration. We also outline a way of making the training protocolsecure by using a parameterized variant of frequency hopping.

Keywords: wireless sensor networks, self-organization, dynamic coordinate system, training, clustering, security, energy-efficient protocols

1. Introduction

Recent advances in nano-technology have made it possibleto develop a large variety of Micro Electro-Mechanical Sys-tems (MEMS) – miniaturized low-power devices that inte-grate sensing, special-purpose computing and wireless com-munications capabilities [18–20,44]. It is expected that thesesmall devices, referred to as sensor nodes, will be mass-produced and deployed, making their production cost negli-gible. Individual sensor nodes have a small, non-renewablepower supply and, once deployed, must work unattended.For most applications, we envision a massive deployment ofsensor nodes, perhaps in the thousands or even tens of thou-sands [23,40].

Aggregating sensor nodes into sophisticated computa-tional and communication infrastructures, called wireless sen-sor networks, will have a significant impact on a wide arrayof applications ranging from military, to scientific, to indus-trial, to health-care, to domestic, establishing ubiquitous wire-less sensor networks that will pervade society redefining theway in which we live and work [28,32]. The novelty ofwireless sensor networks and the tremendous potential fora multitude of application domains has triggered a flurry ofactivity in both academia and industry. We refer the readerto [1,2,21,30,35,38] for a summary of recent applications ofwireless sensor networks.

The fundamental goal of a sensor network is to produce,over an extended period of time, globally meaningful infor-

∗ This work was supported, in part, by a grant from the Commonwealth ofVirginia Technology Research Fund (SE 2001-01) through the Common-wealth Information Security Center.

mation from raw local data obtained by individual sensornodes. Importantly, this goal must be achieved in the contextof prolonging as much as possible the useful lifetime of thenetwork and ensuring that the network remains highly avail-able and continues to provide accurate information in the faceof security attacks and hardware failure. The sheer number ofsensor nodes in a sensor network combined with the uniquecharacteristics of their operating environment (anonymity ofindividual sensors, limited power budget and a possibly hos-tile environment), pose unique challenges to the designers ofprotocols. For one thing, the limited power budget at theindividual sensor node level mandates the design of ultra-lightweight data gathering, fusion, and communication pro-tocols. An important guideline in this direction is to performas much local data processing at the sensor level as possible,avoiding the transmission of raw data through the sensor net-work. Recent advances in hardware technology are makingit plain that the biggest challenge facing the sensor networkcommunity is the development of ultra-lightweight commu-nication protocols ranging from training, to self-organization,to network maintenance, to security, to data collection andfusion, to routing, among many others [28,31,37].

There are several possible techniques that can be used tointerface sensor networks to the outside world and, in par-ticular, to harvest the information they produce. Perhaps thesimplest involves using one or several special sink nodes de-ployed alongside with the sensor nodes. In this scenario, theraw data collected by individual sensor nodes is fused, instages, and forwarded to the sink nodes that provide the in-terface to the outside world. However, in some applications,it is impossible or impractical to deploy sink nodes within the

152 WADAA ET AL.

sensor network. In such cases the task of harvesting the infor-mation produced by the sensor network and that of providingan interface to the outside world may be performed by aircraftand/or helicopters over-flying the sensor network, or by lasertransmission to a satellite constellation. In this latter case,the bulk of the inter-sensor communications is by radio, sincesuch communications are point to multi-point, while special-ized sensors acting as local sinks communicate with the satel-lite constellation using laser beams, for example.

1.1. Securing wireless sensor networks

It is anticipated that in most application domains, sensor net-works will constitute an information source that is a mission-critical system component and will, thus, require commensu-rate security protection. If an adversary can thwart the workof the network by perturbing the information produced, stop-ping production, or pilfering information, then the usefulnessof sensor networks will be drastically curtailed. Thus, secu-rity is a major issue that must be resolved in order for thepotential of wireless sensor networks to be fully exploited.The task of securing wireless sensor networks is compli-cated by the fact that the sensors are mass-produced anony-mous devices with a severely limited energy budget, and,initially no knowledge of their location in the deploymentenvironment. Security must be provided even though sensornodes are unattended and vulnerable to a vast array of attacks[3,4,10,22,30,42].

Wireless sensor networks are sufficiently different fromad-hoc networks that security solutions designed specificallyfor the former do not apply to the latter [10,32,42]. Indeed,in was recently noted that the ultra-lightweight protocols im-posed by the stringent energy limitations may leave not muchroom for advanced encryption schemes. Consequently, pro-tection against overhearing in military applications and pri-vacy protection in personal systems needs to be inherentlybuilt into the concepts underlying sensor network models andprotocols from the beginning. Reliability is expected to be aresult of the large number of sensors deployed for a specifictask. However, this can only be obtained if defective sensorscan be excluded from the communication, and the sensors arecalibrated – either individually or collectively, either beforedeployment or continuously in their environment.

1.2. Our contributions

We view our main contribution at several levels:

• First we propose a virtual infrastructure – a dynamic co-ordinate system – for a massively deployed collection ofanonymous sensor nodes. This coordinate system yields,at no extra cost, a clustering scheme: two nodes are in thesame cluster only if they have the same coordinates.

• We then go on to show that training the sensor nodes – theprocess through which nodes learn their coordinates – canbe performed by a protocol that is at the same time light-weight and secure. Indeed, we outline a way of making the

training protocol secure by using a parameterized variantof frequency hopping.

• Next, we show that in a trained wireless sensor networkrouting and data fusion can be performed by very simpleand energy-efficient protocols.

• Finally, we show how to design the coordinate system suchas to minimize the power expended in collecting and rout-ing data.

The remainder of this paper is organized as follows. Sec-tion 2 discusses the sensor node model used throughout thework. In particular, it discusses “genetic” material with whichsensor nodes are endowed prior to deployment and which willbe key in securing sensor networks. Section 3 discusses wire-less sensor networks, as a conglomerate of individual sensornodes that have to self-organize and self-govern. In particu-lar, therein we discuss interfacing sensor networks with theoutside world, a work model for sensor networks, as well as abrief preview of the training process. Next, section 4 proposesrouting and data fusion algorithms in a trained sensor net-work. Section 5 is the backbone of the entire paper, presentingthe theoretical underpinnings of the training process. We notethat within this section we discuss the details of a lightweightsynchronization protocol for sensor networks. Section 6 dis-cusses the longevity of sensor networks in terms of a numberof system parameters. Section 7 takes a look at the problemof evaluating the energy expenditure per sensor node. Sec-tion 8 shows how to choose the coronas in such a way that theenergy expended for conveying the results to the sink node isminimized. Finally, section 9 offers concluding remarks andmaps out areas for future investigations.

2. The sensor node model

We assume a sensor node to be a device that possesses threebasic capabilities: sensory, computation, and wireless com-munication as illustrated in figure 1. The sensory capability isnecessary to acquire data from the environment; the computa-tional capability is necessary for aggregating data, processingcontrol information, and managing both sensory and commu-nication activity. Finally, the wireless communication capa-bility is necessary for sending (receiving) aggregated data andcontrol information to (from) other sensors or the sink.

We assume that individual sensor nodes operate subject tofollowing fundamental constraints.

(a) Sensor nodes are anonymous – they do not have fabrica-tion-time identities.

(b) Sensor nodes are tiny, commodity devices that are mass-produced in an environment where testing is a luxury.

(c) Each sensor has a non-renewable energy budget; whenthe on-board power supply is exhausted, the sensor nodeis expired.

(d) In order to save energy, each sensor node is in sleep modemost of the time, waking up at random points in time forshort intervals under the control of an internal timer.

TRAINING A WIRELESS SENSOR NETWORK 153

Figure 1. The anatomy of a sensor node.

(e) Each sensor has a modest transmission range, perhaps afew meters. This implies that out-bound messages sentby a sensor can reach only the sensors in its proximity,typically a small fraction of the sensors deployed.

(f) Individual sensor nodes must work unattended – once de-ployed it is either infeasible or impractical to devote at-tention to individual sensor nodes.

At any point in time, a sensor, will be engaged in perform-ing one of a finite set of possible operations, or will be asleep.Three basic operations are sensing (to collect raw measure-ments), data fusion and/or aggregation (to derive target datafrom raw measurements), routing (to communicate raw mea-surements, target data, and control data). We assume eachoperation performed by a sensor consumes a known fixedamount of energy and that a sleeping sensor performs no op-eration and consumes, essentially, no energy.

It is worth mentioning that while the energy budget cansupply short-term applications, sensors dedicated to workover years may need to scavenge energy from the specific en-vironment they are placed into, employing light, temperature,vibration, kinetics, magnetic fields, etc.

2.1. Genetic material

The node’s genetic material plays a key role in driving thefunctionality of different node protocols. To illustrate this, weconsider protocols implementing our proposed security solu-tion for the sensor network.

We assume that at pre-deployment time the sensor nodesare injected, in a secure environment, with the following ge-netic material:

• a standard pseudo-random number generator (one ofpublic-domain algorithms available);

• a set of secret seeds to be used as parameters for the ran-dom number generator;

• an initial time (at this point all the sensor nodes are syn-chronous to the sink node).

It is important to note that immediately after deploymentall the clocks are synchronous. In time, however, clocks willdrift and periodic re-synchronization becomes necessary. Forreasons of simplicity, we assume that synchronization is al-ways done to the master clock running at the sink. As we willshow in detail later in this section, one of our main contribu-tions is a light-weight re-synchronization protocol.

Classical frequency hopping mechanisms have been usedas a means of combating jamming both hostile and non-hostile and of implementing frequency diversity and interfer-ence averaging in a non-hostile context [13,45]. Typicallythese mechanisms offered little cryptographic value. Cryp-tographic techniques such as encryption, on the other hand,are customarily used to address security problems in all butthe physical layer in the network. The key idea behind ourproposed security solution is that by extending classical fre-quency hopping techniques using symmetric key cryptogra-phy, security problems in the physical layer, as well as inother layers in the network can be uniformly addressed in aunified framework; we call this framework randomized fre-quency hopping.

We are now in a position to show how the genetic materialis used in support of secure communications in a sensor net-work. For this purpose, it is useful to imagine three sequencesof random numbers as follows:

• an infinite sequence of t1, t2, . . . , ti , . . . of time epochlengths;

• an infinite sequence n1, n2, . . . , ni , . . . of frequency setsdrawn from a large universe, e.g., the ISM band;

• for every ni (i � 1), an infinite permutation f i1 , f i

2 , . . . offrequencies from ni .

Importantly, these sequences can be generated locally by eachsensor node using the injected genetic material and, therefore,do not need to be communicated after deployment.

We assume that time is ruled into epochs. During the ithtime epoch, of length ti , a frequency set ni will be used sub-ject to a hopping pattern described by the hopping sequencef i

1 , f i2 , . . . . Thus, as long as a sensor node is synchronous

to the sink, it knows the current time epoch, the offset intothe epoch, the set of frequencies in use during the epoch, aswell as the hopping pattern in force during the epoch. Toan outside observer, however, successive epoch lengths, hop-ping sets, and hopping patterns appear as the product of anunknown random process. Given that techniques are knownto discover a hopping sequence by monitoring transmissions,security can only be provided if the design modifies the hop-ping sequence in less time than is required to discover thesequence. The choice of frequency hopping parameters de-termines the time required to discover the sequence (the mag-nitude of the challenge to an adversary).

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2.2. Making sensor nodes tamper-resistant

The most obvious tamper resistance strategies are hardware-based and involve special hardware circuits within the sen-sor node to protect sensitive data, special coatings or tamperseals. However, hardware solutions to the tampering prob-lem require extra circuitry that increases the cost and hard-ware complexity of sensor nodes. Worse yet, the additionalhardware is very likely to consume valuable energy, alreadyin short supply. Also special coatings and seals may offerprotection against some but, certainly, not all tampering at-tempts. Indeed, it is assumed that a sufficiently capable adver-sary can extract confidential information, thus compromisingthe sensor node. Thus, not surprisingly, tamper resistance ortamper protection is not found in present-day sensor nodes[3,4,10]. Since wireless sensor networks must function unat-tended, the potential for physical tampering attacks is signif-icant. It is worth noting that while pre-deployment tamperdetection may be worthwhile, post-deployment tamper de-tection is of little use in wireless sensor networks since, inthe vast majority of applications, inspecting individual sensornodes is not an option. Also, physical tampering may com-promise only the node attacked (ideal), the immediate neigh-borhood of the node attacked, or the entire network. To copewith these conditions, our solution subscribes to the notion of‘self-guarding’ in that each sensor node should be able to de-tect, independently, physical tampering and should react suchthat the impact of the attack is minimal. Our solution to en-dow individual sensor nodes with tamper resistance does notrequire additional or more sophisticated hardware.

In order to set the stage for discussing our solution, we notethat the tampering threat model assumes that the adversary is

• either trying to force open an individual sensor node in-situ; or

• is physically removing the sensor node from the deploy-ment area.

We guard against the first threat by blanking out the mem-ory, triggered by a simple switch. We guard against the sec-ond threat by relying on local data that the sensor can collect,thus establishing a unique signature of its neighborhood thatis difficult to forge. To be more specific, immediately afterdeployment each sensor transmits, during its wake time, ona specified sets of frequencies, using a frequency hoppingsequence established prior to deployment. This allows in-dividual sensor nodes to collect an array of signal strengthsfrom the sensors in their locale. It is important to recall thatsensors do not have identities and that, consequently, the ar-ray of signal strengths is the only data available to the sensornode. This array, establishes, in the obvious way, a signatureof the neighborhood of the node. For this reason the arraywill be referred to as the node’s neighborhood signature array(NSA, for short). If the node is removed from the area of de-ployment, it will notice changes in the signals received whencompared to its NSA and erase its own memory to prevent thetampering agent from gaining access to information secret to

the sensor network. Note also that tampering attempts that in-volve the removal of several sensor nodes simultaneously willalso be defeated since some node in the set of removed nodeswill notice changes in its NSA and can alert the others.

3. Structure and organization of a wireless sensornetwork

We envision a massive deployment of sensor nodes, perhapsin the thousands or even tens of thousands. The sensor nodesare aggregated into sophisticated computational and com-munication infrastructures, called wireless sensor networks,whose goal is to produce globally meaningful informationfrom data collected by individual sensor nodes. However,the massive deployment of sensors nodes in a sensor net-work, combined with anonymity of individual sensors, lim-ited power budget and – in many applications – a hostile envi-ronment, pose daunting challenges to the design of protocolsfor sensor networks. For one thing, the limited power bud-get at the individual sensor node level mandates the design ofultra-lightweight communication protocols. Likewise, issuesconcerning how the data collected by individual sensor nodescould be queried and accessed and how concurrent sensingtasks could be executed internally are of particular signifi-cance. An important guideline in this direction is to performas much local data processing at the sensor level as possible,avoiding the transmission of raw data through the sensor net-work. Indeed, it is known that it costs 3 J of energy to transmit1 Kb of data a distance of 100 meters. Using the same amountof energy, a general-purpose processor with the modest speci-fication of 100 million instructions/Watt executes 300 millioninstructions [31,37].

As a consequence, the sensor network must be multi-hopand only a limited number of the sensor nodes count the sinkamong their one-hop neighbors. For reasons of scalability,it is assumed that no sensor node knows the topology of thenetwork.

Our work focuses on the design of ultra-lightweight orga-nization and communication protocols for a class of wirelesssensor networks consisting of a single sink node and a largenumber of sensors nodes randomly deployed in the transmis-sion range of the sink.

A basic management problem in wireless sensor networksis to balance the utility of the activity in the network againstthe cost incurred by the network resources to perform thisactivity. The scarce resource in the network that is of primaryconcern is energy.

3.1. Interfacing sensor networks

We assume that the sensor network is connected to the outsideworld (e.g., point of command and control, the Internet, etc.)through a gateway node. The gateway node may or may notbe collocated with the sensor nodes in the deployment area.Referring to figure 2, we note that the interface with the out-side world may be achieved by a helicopter or aircraft over-flying the sensor network, and collecting information from aselect group of reporting nodes. In such scenarios communi-cation between individual sensor nodes is by radio, while the


Figure 2. A sensor network with a mobile external gateway.

Figure 3. A sensor network with a central sink node.

reporting nodes are communicating with the external gatewayby laser.

One can easily have a mobile sink, or collection of mobilesinks for fault tolerance, assume the role of the gateway inthe network. In case the sink is collocated with the sensornetwork, it can also be in charge of performing any necessarytraining and maintenance operations.

A somewhat complementary view, illustrated in figure 3 isto have a sink node collocated with the sensor nodes play therole of the gateway. In this case, the sink node has a full rangeof computational capabilities, can send long-range directionalbroadcasts to all sensors, can receive messages from nearbysensors, and has a steady power supply. However, since thesink is a single point of failure in this model, we envisionthat in practice multiple (backup) sink nodes will exist in thenetwork.

3.2. A work model for wireless sensor networks

We take the view that the sensor network performs the tasksmandated by an end-user (perhaps a point of command and

Figure 4. Illustrating the transaction-based network management.

control) that is remote from the network itself. Assumingthe sensor network model depicted in figure 3, the sink nodeserves as the interface between the end user and the network.We characterize the work activity in the network in terms ofan event model. Under the event model, the utility of the sen-sor network is measured by the time period during which itguarantees a specific Quality of Service (QoS) for detectionand notification of event types of interest to the application.Based on this work model, we propose a hierarchical multi-level network management approach, as illustrated in figure 4.The hierarchy involves the following layers:

• application layer: high-level consumers of informationproduced by the sensor network;

• event layer: provides the interface between the sensor net-work and the application layer.

We now discuss each layer in detail. Referring to figure 4,the application layer issues high-level requests, of a coarsesemantic granularity defined in terms of application-level ab-stractions, referred to as Application events (A-events, forshort) to be performed by the sensor network. The A-event isa task that takes the form of a tuple consisting of a high-levelaction, along with a desired level of QoS. As an example, theA-event (Fire, p) requires that the occurrence of fire be de-tected in the area of interest with probability at least p. Here,of course, p specifies the requested QoS.

The event layer provides the interface between the ap-plication layer and the sensor network. This layer receivesA-events, i.e., high-level tasks and QoS requests from the ap-plication layer, considers the current state of the sensor net-work and its capabilities including the remaining energy bud-get both globally and within the individual clusters, and thennegotiates a contract with the application layer before com-mitting the network. Due to this negotiation, the network willnot squander resources needlessly by attempting to carry outan A-event that it does not currently have the resources to pro-vide. Also a set of A-events queueing for service in the eventlayer will be prioritized in order to get the greatest benefitsfrom the sensor network. After a contract has been agreedupon, the event layer translates the corresponding A-eventinto individual tasks, termed primitive events (P-events, forshort), assigned to individual clusters. The clusters must thenperform these tasks at the QOS level required and send thedata back to the sink for further consolidation and analysis

156 WADAA ET AL.

in the event layer. The polished information from this ef-fort is provided to the application layer for proper dissemi-nation.

To continue our example, assume that the event layer de-termines that the A-event (Fire, p) is feasible for the sensornetwork. Assuming that the occurrence of fire is predicatedon high temperature, low humidity and the presence of smoke,the event layer will then translate (Fire, p) into the following(P-events):

• (Temperature, t0, q): detect with probability larger thanq whether the temperature reading is higher than thresh-old t0.

• (Smoke, q ′): detect with probability larger than q ′ thatthere is smoke.

• (Humidity, h0, q′′): detect with probability higher than q ′′

whether the humidity is lower than threshold h0.

On the other hand, if the A-event (Fire, p) is infeasible forthe sensor network, the event layer will negotiate with theapplication layer for a new task, for example, (Fire, p′) withp′ < p.

3.3. Training a wireless sensor network

It was recognized that some applications require sensory datawith some location awareness, encouraging the developmentof communication protocols that are location aware and per-haps location dependent. The practical deployment of manysensor networks will result in sensors initially unaware oftheir location: they must be trained in this vital information.Further, due to limitations in form factor, cost per unit andenergy budget, individual sensor nodes are not expected to beGPS-enabled. Moreover, many probable application environ-ments limit satellite access.

The localization problem is for individual sensor nodes todetermine, as closely, as possible their geographic coordinatesin the area of deployment. Prominent solutions to the localiza-tion problem are based on multilateration [7–9,12,16,29,33].Most of these solutions assume the existence of several an-chor nodes that are aware of their location (perhaps by endow-ing them with a GPS-like devices). Sensor nodes receivinglocation messages from at least three sources can approxi-mate their own locations. For a good survey of localizationprotocols for wireless sensor networks we refer to [25].

In some other applications, exact geographic location isnot necessary: all that the individual sensor node need iscoarse-grain location awareness. There is an obvious trade-off: coarse-grain location awareness is lightweight but theresulting accuracy is only a rough approximation of the ex-act geographic coordinates. Figure 5 illustrates a possibleway of inducing such a coarse-grain location awareness byan overflying aircraft or helicopter. All that the individualsensor nodes need is to determine their approximate distanceto three different positions of the training agent. We omit thedetails.

Our approach is different: we obtain this coarse-grainlocation awareness by the training protocol that imposes a

Figure 5. Acquiring coarse-grain location awareness.

coordinate system onto the sensor network. An interestingby-product of our training protocol is that it provides a par-titioning into clusters and a structured topology with naturalcommunication paths. The resulting topology will make itsimple to avoid collisions between transmissions of nodes indifferent clusters, between different paths and also betweennodes on the same path. This is in contrast with the major-ity of papers that assume routing along spanning trees withfrequent collisions.

Clustering was proposed in large-scale networks as ameans of achieving scalability through a hierarchical ap-proach. For example, at the medium access layer, clusteringhelps increase system capacity by promoting the spatial reuseof the wireless channel; at the network layer, clustering helpsreducing the size of routing tables and striking a balance be-tween reactive and proactive routing. It is intuitively clear thatwireless sensor networks benefit a great deal from clustering;indeed, separating concerns about inter-cluster managementand the intra-cluster management can substantially decrease,and load balance the management overhead. Given the im-portance of clustering, a number of clustering protocols forwireless sensor networks have been proposed in the recentliterature [5,11,15]. However, virtually all clustering proto-cols for wireless sensor networks assume tacitly or explicitlythat individual sensor nodes have identities.

As it turns out, our clustering protocol has the followingdesirable features:

• lightweight as a by-product of training;

• organizes anonymous asynchronous nodes;

• a cluster is the locus of all nodes having the same coordi-nates; and

• individual nodes need not know the identity of other nodesin their cluster.


Figure 6. A trained sensor network.

In the remainder of this work we assume a wireless sensornetwork that consists of a sink and a set of sensors randomlydeployed in its broadcast range as illustrated in figure 3. Forsimplicity, we assume that the sink node is centrally placed,although this is not really necessary. The task of trainingrefers to imposing a coordinate system onto the sensor net-work in such a way that each sensor belongs to exactly onesector.

The coordinate system divides the sensor network area intoequiangular wedges. In turn, these wedges are divided intosectors by means of concentric circles or coronas centeredat the sink and whose radii are determined to optimize thetransmission efficiency of sensors-to-sink transmission as willbe discussed later. Sensors in a given sector map to a cluster,the mapping between clusters and sectors is one-to-one.

Referring to figure 6, the task of training a sensor networkinvolves establishing:

Coronas: The deployment area is covered by k coronas de-termined by k concentric circles of radii r1 < r2 < · · · <rkcentered at the sink node.

Wedges: The deployment area is ruled into a number of an-gular wedges centered at the sink node.

As illustrated in figure 6, at the end of the training periodeach sensor node has acquired two coordinates: the identity ofthe corona in which it lies, as well as the identity of the wedgeto which it belongs. Importantly, the locus of all the sensornodes that have the same coordinates determines a cluster.

4. Routing and data fusion in a trained sensor network

The main goal of this section is to show that once a wirelesssensor network has been trained, both routing and data fusionbecome easy and straightforward.

4.1. Routing

The routing problem in sensor networks differs rather sub-stantially from routing in other types of wireless networks.

Figure 7. Illustrating communication paths to the sink.

For one thing, individual sensor nodes do not have uniqueidentifiers; thus, standard addressing methods do not work di-rectly. For another, the stringent energy limitations present insensor network render the vast majority of conventional rout-ing protocols impractical.

Given the importance of routing, it is not surprising to seethat a number of routing protocols specifically designed forwireless sensor networks were proposed in the literature. Forexample, in [21] Intanagonwiwat et al. describe directed dif-fusion and a companion routing protocol based on interest ta-bles at the expense of maintaining a cache of information in-dexed by interest area at each node. Shah and Rabaey [34]responds to client requests by selecting paths that maximizethe longevity of the network rather than minimize total powerconsumed by a path with path options established by localflooding. The protocols of Kulik et al. [24] are based ona push-pull system where the nodes send metadata first us-ing routing that is optimal for point-to-point communication,but does not benefit from established predefined paths. Otherrouting protocols include rumor routing [6], and multi-pathrouting [14], among others. As we are about to demonstrate,our training protocol provides a novel solution to the routingproblem by yielding energy-efficient paths based routing.

Recall that sensor networks are multi-hop. Thus, in orderfor the sensing information to be conveyed to the sink node,routing is necessary. Our cluster structure allows a very sim-ple routing process as described below. The idea is that theinformation is routed within its own wedge along a virtualpath joining the outermost sector to the sink, as illustratedin figure 7. The collection of all the virtual paths (one perwedge) defines a tree. In this tree, each internal node, exceptfor the root, has exactly one child, largely eliminating MAClevel contention in sending sensor information to the sink.

Recently, a number of MAC layer protocols for wire-less sensor networks have been proposed in the literature[36,41,43]. It’s worthwhile to note that in our routing schemeby appropriately staggering transmissions in neighboringwedges, collision and, therefore, the need for retransmissions

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is completely eliminated. Thus, our training protocol impliesan efficient MAC protocol as well.

4.2. Data fusion

Once sensory data was collected by a multitude of sensornodes, the next important task is to consolidate the data inorder to minimize the amount of traffic to the sink node. Weplace the presentation in the context of our work model. To bemore specific, we assume that the cluster identified by (i, j)

– that is, the set of sensor nodes located in sector Ai,j , wherei is the corona identifier, and j is the wedge identifier, areto perform a certain task T . A number of sensors in sec-tors A1,j , A2,j , . . . , Ai−1,j are selected to act as routers ofthe data collected by the sensors in Ai,j to the sink. Collec-tively, these sensors are the support sensors of task T .

It is, perhaps, of interest to describe the process by whichthe sensors associated with T are selected. To begin, dur-ing a time interval of length � the sink will issue a call forwork specifying the identity j of the wedge in which the taskis to be performed, as well as the identity i of the corona inwhich data is to be collected. The sensor nodes in wedge j

that happen to wake up during the interval � and that havean appropriate energy level stay awake and will participate inthe task either as either data collectors or as routers dependingon their respective position within the wedge. It is intuitivelyclear that by knowing the number of sensors, the density ofdeployment and the expected value of sleep periods, one canfine-tune � in such a way that a suitable number of routerswill be awake in wedge j in support of T . Likewise, we canselect the set D of data collecting sensors in Ai,j . Let S de-note the set of support sensors for T . It is appropriate to recallthat a by-product of the call for work is that all the sensors inS are synchronized. In order to make the task secure the sen-sors in S will share a secret key that allows them access toa set of time epochs, a set of frequencies to be used in eachtime epoch, and a hopping sequence to be used within eachepoch. For details we refer the reader to the description of therandomized frequency hopping security framework proposedin section 2.

Assume that the results of the data collection specific totask T can be partitioned into 2m (m � 0), disjoint groups.Thus, each sensor performing data collection will encode itsdata in a string of m bits.

Since, typically, D contains a large number of sensors, itis important to fuse individual results into a final result thatwill be sent to the sink node. We now outline two possiblesolutions to the data fusion problem. Using the algorithm ofNakano and Olariu [26] that does not require sensors to haveidentities, the sensors in D acquire temporary identities rang-ing from 1 to |D|. Using their newly acquired identities, indi-

vidual data values are being transmitted to the sensor whoseidentity is 1 who will perform data fusion and will send thefinal result to the sink node as discussed in section 7. Theadvantage of this data fusion scheme is that there is no dataloss and all the collected values will be correctly fused. Thereare, however, many disadvantages. For one thing, the initial-ization algorithm of [26] requires every sensor in D to expendan amount of energy proportional with log |D|. For another,the final result of the data collection is concentrated in a sin-gle sensor (i.e., the sensor with temporary identity 1), who isa single point of failure.

We now propose a much simpler data fusion scheme thatinvolves some data loss but that is fault tolerant and does notrequire the sensors in D to have unique identities. The ideais that the sensors in D transmit the data collected bit by bitstarting, say, left to right as follows: a value of 0 is not trans-mitted, while a 1 will be transmitted. The sensors in Ai−1,j

that have been elected as routers in support of transaction Tpick up the values transmitted. The following disambiguationscheme is used:

• No bit is received – in this case a 0 is recorded;

• A bit of 1 is received – in this case a 1 is recorded;

• A collision is recorded – in this case a 1 is recorded.

It is clear that as a result of this disambiguation scheme,every sensor in Ai−1,j that is in support of T stores the logicalOR of the values stored by sensors in D. Note also that whilethere was loss of information in the process of fusing data,no further loss can occur in traversing the path from Ai−1,j

to the sink: this is because all routers in Ai−1,j transmit thesame bit string.

4.3. An example

For an example of data fusion consider a sensor network thatis tasked to monitor and report the temperature in cluster Ai,j .Referring to table 1, for the application at hand temperaturesbelow 111 F are considered to be non-critical and if such atemperature is reported no specific action is to be taken. Bycontrast, temperatures above 111 F are considered to be criti-cal and they trigger a further monitoring action. The encodingfeatured in table 1 is specifically designed to reflects the rela-tive importance of various temperature ranges. For example,the temperature ranges in the non-critical zone are twice aslarge as those in the critical zone. Also, notice that the left-most bit differentiates critical from non-critical temperatures.Thus, if the sink nodes receives a reported temperature whoseleftmost bit is a 1, then further action is initiated; if, on theother hand, the leftmost bit is 0, then no special action is nec-essary.

Table 1Illustrating temperature ranges and their encoding.

Temp 51–60 61–70 71–80 81–90 91–100 101–110 111–115 116–120 121–125 126–130 131–135 136–140 141–145 146–150

Code 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111


Figure 8. Illustrating data fusion: (a) trading loss in data aggregation/re-porting for energy; (b) trading energy for lossless data aggregation/reporting.

Let us see how our data fusion works in this context. Re-ferring to figure 8(a) assume that a group of three sensors d0,d1, and d2 in Ai,j have collected data and are about to transmitit to the sensors s0 and s1 in Ai−1,j . The values collected areencoded, respectively, as 0110, 0101 and 0110. Thus, none ofthe values indicates a critical situation. After transmission anddisambiguation, the sensors in Ai−1,j will store 0111 which isthe logical OR of the values transmitted. Notice that althoughthe data fusion process involves loss of information, we donot loose critical information. This is because the logical ORof non-critical temperatures must remain non-critical. Con-versely, if the logical OR indicates a critical temperature, oneof the fused temperatures must have been critical and thus ac-tion must be initiated. It is also interesting to note that whenthe sensors in Ai−1,j transmit to those in Ai−2,j no furtherloss of information occurs.

There is an interesting interplay between the amount ofloss in data aggregation (fusion) and the amount of energyexpended to effect it. As we are about to show, if we are will-ing to expend slightly more energy, lossless data aggregationcan be achieved.

The corresponding tradeoff is interesting in its own rightbeing characteristic of choices that present themselves in thedesign of protocols for wireless sensor networks. For illus-tration purposes, assume that it is necessary to determine themaximum of the bit codes stored by the sensors in Ai,j .

To solve this problem, all the sensors in Ai,j that have col-lected relevant information engage in the following protocolthat is guaranteed to aggregate the values into the maximum.Assume that each sensor stores an n-bit code for the range.Starting with the highest significant bit to the lowest:

1. Sensors in Ai,j that have a 0 in position p listen for twotime slots; if in any of these slots a 1 or a collision mes-sage is received, they terminate their participation in theprotocol.

2. Sensors that have a 1 in position d transmit in the first timeslot and sleep in the second.

3. Sensors in Ai−1,j do the following:

3.1. Any sensor that has received a 1 or a collision in thefirst slot, echoes a 1 in the second.

3.2. Any sensor that has not received a transmission in thefirst slot sleeps in the second slot.

Figure 8(b) illustrates how the maximum of the values col-lected by sensors d0, d1, and d2 in Ai,j is correctly communi-cated to the support sensors s0, and s1 in Ai−1,j . In this case,we assume d0, and d1 are not in direct communication rangeof each other. Note that s0 receives a collision correspondingto the third most significant bit; consequently it echoes a 1,thereby enabling d1 to terminate the protocol. Similarly, s1receives a collision, and echoes a 1 for the same bit position(not shown in the figure). It is easy to confirm that by exploit-ing the associatively of the maximum, the simple protocol thatwe just outlined correctly forwards to the sink the maximumof the values stored by sensors in Ai,j .

5. Our lightweight training protocol

Our proposed model for a sensor network assumes that afterdeployment the sensor nodes must be trained before they canbe operational in the network. Recall that sensor nodes donot have identities and are initially unaware of their location.It follows that untrained nodes are not addressable and can-not be targeted to do work in the network. The main goal ofthis section is to present, in full detail, our lightweight highlyscalable training protocol for wireless sensor networks. Thekey advantage of this protocol is that each node participatingin the training incurs an energy cost that is logarithmic in thenumber of clusters and wedges defined by the protocol. Beingenergy efficient, this training can be repeated on a scheduledor ad-hoc basis providing robustness and dynamic reorgani-zation.

After deployment nodes sleep until wakened by their in-dividual timers. Thus, each node sleeps for a random periodof time, wakes up briefly and if it hears no messages of in-terest, selects a random number x and returns to sleep x timeunits. Clocks are not synchronized but over any time inter-val [t, t + �t] a percentage directly proportional to �t of thenodes are expected to wake up briefly. During this time inter-val the sink continuously repeats a call to training specifyingthe current time and a rendezvous time. Thus, in a probabilis-tic sense a certain percentage of nodes will be selected fortraining. The time interval �t can be adjusted to control thepercentage of nodes that are selected. Using the synchroniza-tion protocol we describe in section 5.1 the selected sensorsnodes reset their clocks and set their timer appropriately be-fore returning to sleep.

5.1. The synchronization protocol

It is natural to assume that, just prior to deployment, the sen-sor nodes are synchronized. However, due to natural clock

160 WADAA ET AL.

drift, re-synchronization is necessary. Re-synchronization isdone with respect to the master clock running at the sink.

Suppose that the sink dwells τ micro-seconds on each fre-quency in the hopping sequence. For the purpose of show-ing how synchronization is effected, assume that time is ruledinto epochs as discussed before. For every i (i � 1), we let listand for �ti/τ�; thus, epoch ti involves a hopping sequenceof length li . We can think of the epoch ti as being partitionedinto li slots, each slots using its own frequency selected byvirtue of the hopping sequence out of the set ni of frequen-cies associated with epoch ti . It is clear that determining theepoch and the position of the sink in the hopping sequencecorresponding to the epoch is sufficient for synchronization.

Our synchronization protocol is predicated on the assump-tion that clock drift is bounded. Specifically, assume thatwhenever a sensor node wakes up during its local time epochti the master clock is in one of the time epochs ti−1, ti , or ti+1.

Using its genetic information, the sensor node knows thelast frequencies λi−1, λi and λi+1 on which the sink willdwell in the time epochs ti−1, ti , and ti+1, respectively. Itsstrategy, therefore, is to tune in, cyclically, to these frequen-cies, spending τ/3 time units on each of them. It is clear that,eventually, the sensor node meets the sink node on one of the-ses frequencies. Assume, without loss of generality, that thenode meets the sink on frequency λ in some (unknown) slot s

of one of the epochs ti−1, ti , or ti+1. To verify the synchro-nization, the node will attempt to meet the sink in slots s + 1,s + 2 and s + 3 at the start of the next epoch. If a match isfound, the node declares itself synchronized. Otherwise, thenode will repeat the above process.

We note that even if the sensor node declares itself syn-chronized with the sink, there is a slight chance that, it is not.The fact that the node has not synchronized will be discov-ered quickly and it will again attempt to synchronize. Thereare ways in which we can make the synchronization proto-col deterministic. For example, the hopping sequence can bedesigned in such a way that the last frequency in each epochis unique and it is not used elsewhere in the epoch. How-ever, this entails less flexibility in the design of the hoppingsequence and constitutes, in fact, an instance of a differentialsecurity service where the level of security is tailored to suitthe application or the power budget available.

5.2. The corona training protocol

The main goal of this subsection is to present the details ofthe corona training protocol. The wedge training protocol issimilar and will not be discussed further.

Let k be an integer1 known to the sensor nodes and let thek coronas be determined by concentric circles of radii r1 <

r2 < · · · < rk centered at the sink node.The idea of the corona training protocol is for each individ-

ual sensor node to learn the identity of the corona to which itbelongs. For this purpose, each individual sensor node learnsa string of log k bits from which the corona number can be de-termined easily. To see how this is done, it is useful to assume

1 For simplicity we shall assume that k is a power of two.

Figure 9. Illustrating corona training.

time ruled into slots s1, s2, . . . , sk−1 and that the sensors cansynchronize2 to the master clock running at the sink node.

In time slot s1 all the sensors are awake and the sink trans-mits with a power level corresponding to rk/2. In other words,in the first slot the sensors in the first k/2 coronas will re-ceive the message above a certain threshold, while the otherswill not. Accordingly, the sensors that receive the signal setb1 = 0, the others set b1 = 1.

Consider a k-leaf binary tree T and refer to figure 9. In thefigure the leaves are numbered left to right from 1 to k. Theedges of T are labeled by 0’s and 1’s in such a way that anedge leading to a left subtree is labeled by a 0 and an edgeleading to a right subtree is labeled by a 1. Let l (1 � l � k),be an arbitrary leaf and let b1, b2, . . . , blog k be the edge labelsof the unique path leading from the root to l. It is both wellknown and easy to prove by a standard inductive argumentthat

l = 1 +log k∑

j=1

bj 2logk−j (1)

(for example, applying equation (1) to leaf 7 we have: 7 =1 + 0 · 23 + 1 · 22 + 1 · 21 + 0 · 20).

Referring again to figure 9, let the interior nodes of thetree be numbered in preorder from 1 to k −1 and let T ′ be thetree consisting of the interior nodes only. Let u be an arbitrarynode in T ′, and let b1, b2, . . . , bi−1 be the edge labels on theunique path from the root to u. We take note of the followingtechnical result.

Lemma 5.1. Let p(u) be the preorder number of u in T ′.Then, we have

p(u) = 1 +i−1∑

j=1

cj ,

where

cj =

1 if bj = 0,k

2jif bj = 1.

2 See section 5.1.


Proof. The proof is by induction on the depth i of node u inT ′. To settle the basis, note that for i = 1, u must be the rootand p(u) = 1, as expected.

For the inductive step, assume the statement true for allnodes in T ′ of depth less that u. Indeed, let v be the parent ofu and consider the unique path of length i −1 joining the rootto u. Clearly, nodes u and v share b1, b2, . . . , bi−2 and, thus,c1, c2, . . . , ci−2. By the inductive hypothesis,

p(v) = 1 +i−2∑

j=1

cj . (2)

On the other hand, since v is the parent of u, we can write

p(u) = p(v) +{ 1 if u is the left child of v,

k

2i−1 otherwise.(3)

Notice that if u is the left child of v we have bi−1 = 0 andci−1 = 1; otherwise bi−1 = 1 and ci−1 = k/2i−1. Thisobservation, along with (2) and (3) combined, allows us towrite

p(u) = 1 +i−2∑

j=1

cj + ci−1 = 1 +i−1∑

j=1

cj

completing the proof of the lemma. �

For further reference we also need the following technicalresult.

Lemma 5.2. Let u be an arbitrary node of the tree T ′ and letn(u) denote its inorder number in T ′. Let m be the left-to-right rank among the leaves of T of the rightmost leaf of theleft subtree of T rooted at u. Then, n(u) = m.

Proof. We proceed by induction on the inorder number ofa node in T ′. Indeed, if n(u) = 1, then u must be the left-most leaf in T ′ and, thus, its left subtree in T consists of theleftmost leaf of T t, settling the base case.

Assume that the statement true for all nodes of T ′ withinorder number smaller than that of u. we shall distinguishbetween the following two cases.

Case 1. v is an ancestor of u in T ′. Let T ′(v) be the subtreeof T ′ rooted at v. In this case u must be the leftmost leaf in theright subtree of T ′(v). Let q be the left-to-right rank amongthe leaves of T of the rightmost leaf of the left subtree ofT ′(v). By the inductive hypothesis n(v) = q . Since u is aleaf in T ′ it has exactly two children in T , namely the leavesof ranks q+1 and q+2. Thus, in this case, n(u) = n(v)+1 =q + 1, as claimed.

Case 2. u is an ancestor of v in T ′. Let T ′(u) be the subtreeof T ′ rooted at u. In this case v must be the rightmost leaf inthe left subtree of T ′(u). Assume that n(v) = r . Observe thatv has exactly two leaf children T . By the induction hypothesisthese children have ranks r and r + 1. Thus, in this case,n(u) = n(v) + 1 = r + 1, as claimed.

This completes the proof of lemma 5.2. �

With these technicalities out of the way, we now return tothe corona training protocol. In our setting, the preorder andinorder numbers of internal nodes in T correspond, respec-tively, to time slots in the training protocol and to the trans-mission ranges used by the sink. More precisely, consider anarbitrary integer i (2 � i � log k − 1), and assume that atthe end of time slot s a sensor node has learned the leftmosti − 1 bits b1, b2, . . . , bi−1. The following important result isimplied by lemma 5.1 and lemma 5.2.

Corollary 5.3. Having learned bits b1, b2, . . . , bi−1 a sensornode must wake up in time slot z = 1 + ∑i−1

j=1 cj to learn bitbi . Moreover in time slot z the sink node uses a transmissionrange of rinorder(z).

(To illustrate corollary 5.3, refer again to figure 9 where theinternal nodes are labeled by their preorder numbers. Con-sider the node labeled 2. It is easy to verify that its inordernumber is 4. Thus, all the nodes in the subtree rooted at 2 willbe awake in slot 2 and the sink node will transmit with a rangeof r4. Consequently, the sensor nodes at a distance from thesink not exceeding r4 will receive the signal, while the otherswill not.)

It is also worth noting that only the sensor nodes that needto be awake in a given time slot will stay awake, the oth-ers will sleep minimizing the power expenditure. Yet anotherinteresting feature of the training protocol we just describedis that individual sensor nodes sleep for as many contiguousslots as possible before waking up, thus avoiding repeatedwake–sleep transitions that are known to waste energy.

Securing the training protocol is especially important sincetraining is a prerequisite for subsequent network operations.As argued in [22] our parameterized frequency hoppingscheme guarantees that the physical layer of wireless com-munications is secure.

At the same time, in case the corona training process hasto be aborted before it is complete, corollary 5.3 guaranteesthat if the training process re-starts at some later point, everysensor node knows the exact time slots when it has to wakeup in order to learn its missing bits.

6. Reasoning about the longevity of the sensor network

The main goal of this section is to explore the energy require-ments of the sensor network in terms of a model of work. In-deed, we adopt a transaction-based model whereby the sensornetwork is subjected to a set T of transactions. Each transac-tion involves the nodes in a sector (i.e., a cluster) and involvesperforming local sensing by the sensors, data fusion and send-ing the resulting information to the sink. Recall that, as dis-cussed in section 4.1, one of the key benefits of our trainingis that transmitting the result of the transaction from a sectorto the sink node amounts to routing the information along apath lying within the same wedge (see also figure 7). Thus,we associate each transaction with such a path. We will nowanalyze the energy expended by sensor nodes to fulfill theirpath-related duties.

162 WADAA ET AL.

Figure 10. Illustrating a wedge W and the associated sectors.

Table 2Summary of system parameters.

Parameter Description

rk Radius of circle of deploymentρ Deployment densitytx Maximum transmission range of a sensor nodeE Total energy budget packed by a sensor nodeθ Angle subtended by wedge W

n Number of sensor nodes in wedge W

N Number of sector-to-sink paths that W sees during the lifetimeof the network

T Total number of transactions that W can handle during thelifetime of the network

Throughout the remainder of this work we assume a sen-sor network deployed in a circular area and a co-located sinknode placed at its center. Consider a wedge W subtended byan angle of θ and refer to figure 10. W is partitioned into k

sectors A1, A2, . . . , Ak by its intersection with k concentriccircles, centered at the sink node, and of monotonically in-creasing radii r1 < r2 < · · · < rk . It is important to note thatrk , the deployment radius, is a system parameter and, thus,a constant for a particular sensor network. The system para-meters are summarized in table 2.

For convenience of notation we write r0 = 0 and interpretA0 as the sink node itself. We assume the following regularitycondition:

for all i, 2 � i � k,ri + ri−2

2� ri−1 � √

riri−2 (4)

which, essentially, specifies the way coronas relate to eachother. Specifically,

• the condition (ri + ri−2)/2 � ri−1 is equivalent to ri −ri−1 � ri−1 − ri−2, confirming our intuitive idea thatcorona widths are non-decreasing.

• Similarly, the condition ri−1 � √riri−2 implies that

rk

rk−1� rk−1

rk−2� · · · � ri

ri−1� ri−1

ri−2� · · · � r2

r1

which can be interpreted as limiting the growth of consec-utive coronas.

Let n denote the total number of sensor nodes deployed inwedge W . We assume a uniform deployment with density ρ.In particular, with A standing for the area of wedge W , wecan write

n = ρA = ρθ

2r2k . (5)

Let n1, n2, n3, . . . , nk stand for the number of nodes de-ployed in the sectors A1, A2, A3, . . . , Ak , respectively. Sincethe deployment is uniform, it is easy to confirm that forevery i (1 � i � k),

ni = ρAi = ρθ

2

(r2i − r2

i−1

). (6)

Let N denote the number of sector-to-sink paths (hence-forth, simply denoted by paths) that the wedge W sees duringthe lifetime of the sensor network. By our previous discus-sion there is a one-to-one map between paths and transac-tions. Thus, N equals the total number T of transactions thatthe wedge can handle during the lifetime of the network.

We make the following assumptions motivated by the uni-formity of the deployment:

• each sensor node in W is equally likely to be the source ofa path to the sink;

• for 2 � i � k, each sensor in sector Ai−1 is equally likelyto serve as the next hop for a path that involves a nodein Ai .

By virtue of the first assumption, the expected number ofpaths originating at a node in W is

N

n. (7)

Consider sector A1. Since the N paths have the sink nodeas their destination, the nodes in sector A1 must collectivelyparticipate in all the N paths. Since A1 contains n1 nodes, theexpected number of transmissions per node is N/n1. Assum-ing a quadratic power degradation factor, the energy expendedby a node in A1 per path served is cr2

1 for some positive con-stant c. Thus, the total energy E1 consumed by a node in A1to fulfill its path-related duties is

E1 = N

n1cr2

1

which, by (6), can be written as

E1 = N

n1cr2

1 = 2Nc

ρθr21

r21 = 2Nc

ρθ. (8)


Quite surprisingly, (8) asserts that the total energy ex-pended by a node in A1 is independent of the value of r1. LetT denote the total number of transactions performed by theentire sensor network (not just wedge W ) during its lifetimeand let N be the corresponding number of node-to-sink paths.Assuming that the T transactions are uniformly distributedthroughout the sensor network, we can write

N

2π= N

θ. (9)

By (8) and (9) combined, the total energy needed by a nodein A1 to handle its path-related duties is

E1 = 2Nc

ρθ= Nc

ρπ. (10)

Let E denote the total energy budget of a sensor node. Sincethe sensor nodes in A1 must have sufficient energy to handletheir path-related duties, by using (10) we can write

Nc

ρπ< E.

Recalling that in our work model there is a one-to-one cor-respondence between transactions and sector-to-sink paths,this inequality can be written in its equivalent form

T c

ρπ< E. (11)

Equation (11) tells us that for a given energy budget E, inorder to guarantee a network longevity of T transactions, thedeployment density ρ must satisfy the inequality

ρ >T c

Eπ. (12)

7. Energy constraints

We now turn to the task of evaluating the energy expenditureper node in an arbitrary sector Ai with i � 1. Since the casei = 1 was handled in the previous section, we now assumei � 2. Observe that nodes in a generic sector Ai (2 � i � k),are called upon to serve two kinds of paths:

• paths originating in a sector Aj with i < j � k; and

• paths originating at a node in Ai .

It is easy to confirm that the number of paths involving nodesin Ai includes all paths except those originating in one ofthe sectors A1, A2, . . . , Ai−1. Therefore, the total number ofpaths that the nodes in Ai must handle is

N − N

n(n1 + n2 + · · · + ni−1).

By (5) and (6) combined with elementary manipulations, thisexpression can be written as

N

[1 − r2

1 + (r22 − r2

1 ) + (r23 − r2

2 ) + · · · + (r2i−1 − r2

i−2)

r2k

]

= N

[1 − r2

i−1

r2k

]. (13)

Recall that sector Ai contains ni nodes. This implies that eachnode in Ai must participate in

N

ni

[1 − r2

i−1

r2k

]

paths. Using (6), the number of paths handled by each nodein Ai can be written as

2N

ρθ

[1 − r2

i−1

r2k

]1

r2i − r2

i−1

. (14)

Observe that the width of sector Ai is ri − ri−1. It followsthat the transmission range needed to send information be-tween Ai and Ai−1 is ri − ri−1. Thus, in a quadratic powerdegradation model, we shall assume that the energy expendedby a node in Ai to send information to sensors in Ai−1 is

c(ri − ri−1)2.

Let the total amount of energy expended by a node in Ai beEi . By (9) and (14), we have

Ei = Nc

πρ

[1 − r2

i−1

r2k

]1

r2i − r2

i−1

(ri − ri−1)2.

Simple manipulations show that

Ei = Nc

πρ

[1 − r2

i−1

r2k

]ri − ri−1

ri + ri−1. (15)

Observing that

1 − r2i−1

r2k

< 1

and thatri − ri−1

ri + ri−1< 1

equation (15) implies that

Ei <Nc

πρ. (16)

Inequality (16) captures a very important conclusion: theenergy expended by a sensor node in sector Ai to fulfill itspath-related duties is strictly less than the energy expendedby a node in sector A1. More generally, it is instructive tocompare the energy expended for path-related duties by in-dividual sensor nodes in sectors Ai and Ai−1. Intuition sug-gests that Ei < Ei−1. This is because as we move awayfrom the sink node, by the regularity condition (4) the sec-tors become larger and larger and, as a result, more energy isneeded for intra-sector communication and governance tasks.This implies that less energy should be spent on path-related

164 WADAA ET AL.

duties. The next result shows that, in fact, this intuition iscorrect.

Theorem 7.1. Consider a sequence of radii r1 < r2 <

· · · < rk satisfying the regularity condition. Then for every i,2 � i � k, Ei < Ei−1.

Proof. Recall that by (15) the energy Ei spent by each nodein Ai is

Ei = Nc

πρ

[1 − r2

i−1

r2k

]ri − ri−1

ri + ri−1.

Likewise, the energy Ei−1 spent by each node in Ai−1 is

Ei−1 = Nc

πρ

[1 − r2

i−2

r2k

]ri−1 − ri−2

ri−1 + ri−2.

Since ri−2 < ri−1 it is clear that

1 − r2i−1

r2k

< 1 − r2i−2

r2k

.

Moreover, in order to show that

ri − ri−1

ri + ri−1� ri−1 − ri−2

ri−1 + ri−2

we only need show that

1 − ri − ri−1

ri + ri−1� 1 − ri−1 − ri−2

ri−1 + ri−2

or, equivalently,

ri−2

ri−1 + ri−2� ri−1

ri + ri−1

which boils down to r2i−1 � riri−2, which is implied directly

by our regularity condition. This completes the proof of thetheorem. �

8. Crafting the coronas

The main goal of this section is to show how to select the radiir1, r2, . . . , rk in such a way that the energy spent per sector-to-sink path is minimized. For this purpose, let Ei denote thetotal amount of energy expended by the nodes along a genericpath transferring data from sector Ai to the sink. Write r0 = 0and assume that A0 is the sink node itself; since in transmit-ting from Aj to Aj−1 (2 � j � i), the amount of energyspent is c(rj − rj−1)

2, it follows that

Ei = c

i∑

j=1

(rj − rj−1)2. (17)

Recall the Lagrange identity [17, p. 64]

∑

1�p<q�i

(apbq − aqbp)2 =i∑

p=1

a2p

i∑

p=1

b2p −

(i∑

p=1

apbp

)2

.

For every j (1 � j � i), write aj = rj − rj−1 and bj = 1.Noticing that

• ∑ip=1 a2

p = Ei/c,

• ∑ip=1 b2

p = i,

• (∑i

p=1 ap)2 = r2i ,

and substituting in Lagrange’s identity, we obtain

∑

1�p<q�i

(ap − aq)2 = i

i∑

p=1

a2p −

(i∑

p=1

ap

)2

= iEi

c− r2

i .

Thus, we can write

Ei = c

i

(r2i +

∑

1�p<q�i

(ap − aq)2)

. (18)

Clearly, the left-hand side of the above equality is minimizedwhenever

∑

1�p<q�i

(ap − aq)2 = 0

which occurs if and only if

a1 = a2 = a3 = · · · = ai.

Thus, for some positive constant d we have

for every j (1 � j � i), rj − rj−1 = d. (19)

It is worth noting that equation (19) satisfies the regularitycondition (4). It is easy to see that equation (19) implies

ri = id (20)

and so, substituting in (18) we obtain

Ei = icd2.

Let tx be the maximum transmission range of a sensor node.The way our coronas are organized, suggests setting

d = tx

yielding

Ei = i(ct2

x

). (21)

To summarize, in order to minimize the total amount ofenergy spent on a path originating at a sensor node in coronaAi down to the sink node, all the coronas must have the samewidth tx and the optimal amount of energy is i times the en-ergy needed to send the desired information between adjacentcoronas.

8.1. A synopsis of power expenditure

Recall that by equation (10) and theorem 7.1, for every choiceof radii r1 < r2 < · · · < rk subject to the regularity condition,we have the following energy monotonicity property:

E1 > E2 > · · · > Ek.


Table 3Illustrating various energy ratios.

Corona Energy ratio

2 0.3333 . . .

3 0.24 0.1428 . . .

5 0.1111 . . .

6 0.0909 . . .

7 0.0769 . . .

Figure 11. Illustrating the energy ratio Ei/E1.

It is instructive to get a feel for how fast the energy expen-ditures drop as we move away from the sink node. For thispurpose, we rewrite equations (10) and (15) as

Ei = E1

[1 − r2

i−1

r2k

]ri − ri−1

ri + ri−1< E1

ri − ri−1

ri + ri−1.

By using (20), the inequality above can be written as

Ei

E1<

1

2i − 1.

Table 3 summarizes the energy ratio Ei/E1 for various valuesof i.

As shown by table 3 and also in the companion figure 11,the energy expended by a sensor node in support of its path-related obligations drops significantly as we move away fromthe sink. For example, in the 6th corona, the power expendedby a node is less than 10% of the energy expended by a nodein the first corona. This leaves plenty of energy for othertasks.

9. Concluding remarks

In this work we have proposed a virtual infrastructure – a dy-namic coordinate system – for a massively-deployed collec-tion of anonymous sensor nodes. This coordinate systemprovides, at no extra cost, an interesting clustering schemeaccording to which two nodes are in the same cluster only ifthey have the same coordinates. Notice that this clusteringscheme works for anonymous sensor nodes. As a corollary,

sensor nodes do not know the identity of the other nodes in thesame cluster. Our second contribution was to show that train-ing the sensor nodes – the process of learning their coordi-nates – can be performed by a protocol that is at the same timelightweight and secure. Being energy efficient, this trainingcan be repeated on either a scheduled or ad-hoc basis to pro-vide robustness and dynamic reorganization. We also showedthat in a trained wireless sensor network the tasks of routingand data fusion can be performed by very simple and energy-efficient protocols. Finally, we showed how to design the co-ordinate system such as to minimize the power expended incollecting and routing data.

In this paper we addressed the problem of training a sensornetwork in a two-dimensional plane. In practice, however, thenetwork training problem manifests itself in three dimensions,for example, because of irregularities in a rugged deploymentterrain. To extend our work we have developed solutions forthe three-dimensional training problem where the majority ofthe nodes are assumed to reside in one logical base plane,while the remaining nodes are dispersed over other parallelplanes. The goal was to mimic the case of minor terrain ir-regularities. However, training a sensor network of nodes ar-bitrarily dispersed in a three-dimensional space remains anopen problem.

In spite of these encouraging results, more work has to bedone. We are currently looking at security-related problemin wireless sensor networks. This is an extremely importantproblem as the information provided by the sensor networkmay be used for decision making in military or civilian en-vironments where human life is at stake. Looking at variousissues related to securing sensor networks promises to be anexciting are for future work.

Acknowledgements

The authors wish the thanks three anonymous referees fortheir constructive comments that have lead to an improvedpresentation. A preliminary version of this paper [39] hasappeared in Proceedings of 3rd International Workshop onWireless, Mobile and Ad Hoc Networks (WMAN’03), Nice,France, April 2003.

Appendix

Our proposed training protocol consists of a call for trainingprotocol, and a corona and wedge training protocol.

A.1. Call for training protocol

This protocol enables the sink node to issue calls for trainingto a random subset of nodes of an arbitrary size post deploy-ment. Each call communicates the next rendezvous time forcorona and wedge training along with the local time at thesink node. We use the notations from table 4.

166 WADAA ET AL.

Table 4Parameters of the training pseudocode.

Parameter Description

k Total number of coronas in the coordinate systemw Total number of wedges in the coordinate systemrT The sink commences transmission of training mes-

sages at this timeclock Local clock on board the sensor nodeclock.set(t) Set the local clock to time t

time() Returns the local clock time in a sensor node or thesink node

Timer A local timer in a sensor node or the sink nodeTimer.set(t) Set the timer to expire at time t

sleep() Sleep until a timer interrupt occursradio The radio device on board a sensor noderadio.receive() Put radio in receive mode for one message time if a

message is being transmitted it return the messageX An ordered tuple (X(i) is the ith component of the

tuple)message(X) Construct a message that has ordered tuple X as its

contentsradio.transmit(r,m) Transmit message m, using a transmission radius of r

� Time interval during which calls for training mes-sages are transmitted

λ The time slot size. For simplicity, we assume that isalso the time required to transmit (receive) a protocolmessage

Timer.interrupt() Blocks until a timer interrupt occursR(i) Radius of concentric circle i, 1 � i � k

inorder(k, i) Inorder number for the node in a full binary tree ofheight log k whose preorder number is i

Call for training (trainer (sink node))CFTtrainer(�){deadline = time() + �

while (time() < deadline){m = message((time(), rT , k,w))

radio.transmit(rk,m)

}}

Call for training (trainee (sensor node))CFTtrainee(�){while (true){

Timer.set(time() + random())

sleep()

if (m = radio.receive()){clock.set(m.X(1) + σ)

Timer.set(m.X(2))

sleep()cTrainMe()

}}

}

Note that cTrainMe() invokes the corona training protocol(sensor node side) that is described below. The parameter σ is

used to calibrate the time attributes in the message received bythe sensor node to account for propagation and computationdelays.

A.2. Corona and wedge training protocol

Since training for wedges uses the same protocol used intraining for coronas in an analogous way, we limit our dis-cussion here to corona training.

Corona training (trainer (sink node))int cTrainThem(int k){

for (i = 1; i < k; + + i){m = message((i))radius = R(inorder(k, i))

radio.transmit(radius,m)

}}

Corona training (trainee (sensor node))int cTrainMe(int k){

start = time()messageNumber = 1coronaNumber = 0bitCounter = 0offset = k/2

while (bitCounter < log k){if(m = radio.receive()){

(coronaNumber � 1)+ = 0+ + messageNumberoffset/ = 2+ + bitCounterTimer.set(start + (messageNumber − 1)∗λ)

Timer.interrupt()}else{

(coronaNumber � 1)+ = 1messageNumber+ = offsetoffset/ = 2+ + bitCounterTimer.set(start + (messageNumber − 1)∗λ)

sleep()

}}return (coronaNumber)

}

It should be noted that a sensor node remains awake andtunes in to receive the protocol message j + 1, if it succeedsin receiving message j . If the node tunes in but fails to receivemessage j , the node sleeps until the time of the next messageto which it must listen.


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Ashraf Wadaa is a Lecturer of Computer Scienceand a Ph.D. candidate at Old Dominion University.Wadaa’s dissertation is on wireless sensor networks.His research interests include wireless sensor, and adhoc networks, network security, database systems,and distributed computing. He published in refer-eed journals and conference proceedings. He alsoserved on a number of technical committees for con-ferences, and workshops. His funding record in-cludes a grant from the Commonwealth Information

168 WADAA ET AL.

Security Center (CISC) in Virginia to pursue research in sensor network se-curity. Wadaa earned a B.S. (1986 with top rank) in Computer Science andAutomatic Control from Alexandria University, Egypt. He was nominated byOld Dominion University for an outstanding teacher award in 2002. Wadaais a member of the honor society of Phi Kappa Phi.E-mail: [email protected]

Stephan Olariu received the M.Sc. and Ph.D. de-grees in computer science from McGill University,Montreal, Canada in 1983 and 1986, respectively. In1986 he joined the Computer Science Departmentat Old Dominion University where he is now a fullprofessor. He has published extensively in variousjournals, book chapters, and conference proceedings.His research interests include wireless networks andmobile computing, parallel and distributed systems,performance evaluation, and medical image process-

ing. Prof. Olariu serves on the editorial board of several archival journalsincluding IEEE Transactions on Parallel and Distributed Systems, Journalof Parallel and Distributed Computing, International Journal of Foundationsof Computer Science, Journal of Supercomputing, International Journal ofComputer Mathematics, VLSI Design, and Parallel Algorithms and Applica-tions.E-mail: [email protected]

Larry Wilson is an Associate Professor of ComputerScience at Old Dominion University. He receivedhis Ph.D and M.Sci. degrees from the University ofTexas at Austin. His B.Sci. is from Midwestern Uni-versity. Dr. Wilson is currently working in the areaof wireless sensor networks. Previous areas of re-search include design and analysis of parallel algo-rithms, software reliability and Mikusinski OperatorFunctions.E-mail: [email protected]

Mohamed Eltoweissy is a (visiting) Professor ofComputer Science at Virginia Tech. He is also aProfessor of Computer Science at James MadisonUniversity. Eltoweissy’s research interests includeinformation security and privacy, wireless sensorand ad hoc networks, network security, computer-supported cooperative work, and distributed com-puting. He published extensively in books, refer-eed journals and conference proceedings. He alsoserved on numerous technical committees for con-

ferences, workshops, seminars, and NSF panels. He has an aggressive recordof funding (over $10 million). Eltoweissy founded the Commonwealth In-formation Security Center (CISC) in Virginia and was a founding memberof the award-winning Virginia Alliance for Secure Computing and Network-ing (VA SCAN). Eltoweissy earned a Ph.D. in computer science (1993) fromOld Dominion University, and M.S. (1989) and B.S. (1986 with top rank) incomputer science and Automatic Control from Alexandria University, Egypt.He was nominated by JMU for the Virginia State-wide Outstanding FacultyAwards in 2003. Eltoweissy is a member of ACM, ACM SIGSAC, and thehonor societies of Phi Kappa Phi and Upsilon Pi Epsilon.


Kennie Jones received the B.S. degree in Biologyfrom Christopher Newport College, Newport News,VA, in 1977 and the M.S. degree in computer sci-ence from Old Dominion University (ODU), Nor-folk, VA, in 1981. He is currently pursuing a Ph.D.degree in computer science from ODU. After teach-ing in secondary schools for 4 years, he joined Com-puter Sciences Corporation in 1981 as a contractorto the National Aeronautics and Space Administra-tion (NASA). In 1990, he became an employee of

NASA. In that capacity, he has made significant contributions in the areas ofcomputer graphics and data management. He is currently researching sensornetworks and their application to NASA missions as a member of ODU’sSensor Network Research Team.E-mail: [email protected]


Quorum-Based Asynchronous Power-Saving Protocols for IEEE802.11 Ad Hoc Networks ∗

JEHN-RUEY JIANGDepartment of Information Management, Hsuan-Chuang University, Taiwan

YU-CHEE TSENGDepartment of Computer Science and Information Engineering, National Chiao-Tung University, Taiwan

CHIH-SHUN HSUDepartment of Computer Science and Information Engineering, National Central University, Taiwan

TEN-HWANG LAIDepartment of Computer and Information Science, The Ohio State University Columbus, OH 43210, USA

Abstract. This paper investigates the power mode management problem for an IEEE 802.11-based mobile ad hoc network (MANET) thatallows mobile hosts to tune to the power-saving (PS) mode. There are two major issues that need to be addressed in this problem: (a) wakeupprediction and (b) neighbor discovery. The former is to deliver buffered packets to a PS host at the right time when its radio is turned on.The latter is to monitor the environment change under a mobile environment. One costly, and not scalable, solution is to time-synchronize allhosts. Another possibility is to design asynchronous protocols as proposed by Tseng et al. in [25]. In this paper, we adopt the latter approachand correlate this problem to the quorum system concept. We identify a rotation closure property for quorum systems. It is shown thatany quorum system that satisfies this property can be translated to an asynchronous power-saving protocol for MANETs. Thus, the resultbridges the classical quorum system design problem in the area of distributed systems to the power mode management problem in the areaof mobile ad hoc networks. We derive a lower bound for quorum sizes for any quorum system that satisfies the rotation closure property.We identify a group of quorum systems that are optimal or near optimal in terms of quorum sizes, which can be translated to efficientasynchronous power-saving protocols. We also propose a new e-torus quorum system, which can be translated to an adaptive protocol thatallows designers to trade hosts’ neighbor sensibility for power efficiency. Simulation experiments are conducted to evaluate and comparethe proposed protocols.

Keywords: IEEE 802.11, distributed system, mobile ad hoc network (MANET), power management, quorum system, wireless communi-cation

1. Introduction

The mobile ad hoc network (MANET) has attracted a lotof attention recently. A MANET consists of a set of mo-bile hosts, and does not have the support of any base station.Hosts may communicate in a multi-hop manner. Applicationsof MANETs include communications in battlefields, disasterrescue operations, and outdoor activities.

Power saving is a critical issue for portable devices sup-ported by batteries. Battery power is a limited resource, andit is expected that battery technology is not likely to progressas fast as computing and communication technologies do.Hence, how to save the energy consumption in a MANET,which is all supported by batteries, has been intensively stud-ied recently (e.g., power control is studied in [8,9,17,26,28],

∗ Y.C. Tseng’s research is co-sponsored by the MOE Program for PromotingAcademic Excellence of Universities, Taiwan, under grant numbers A-91-H-FA07-1-4 and 89-E-FA04-1-4, by NSC of Taiwan under grant numbersNSC92-2213-E009-076 and NSC92-2219-E009-013, and by the Lee andMTI Center of NCTU. J.R. Jiang’s research is sponsored by NCS of Taiwanunder grant number NSC 92-2213-E-364-002.

power-aware routing in [6,18,19,24], and low-power modemanagement in [1,2,7,10,13,20,22,23,27,29]).

This paper investigates the power mode management prob-lem in an IEEE 802.11-based MANET, which is characterizedby multi-hop communication, unpredictable mobility, and noplug-in power. IEEE 802.11 [11] has defined its power-saving(PS) mode for single-hop (fully connected) MANETs basedon periodical transmissions of beacons. The protocol, whenapplied to a multi-hop MANET, may encounter several prob-lems, including costly clock synchronization and even incor-rect network partitioning [25].

There are two major issues that need to be addressed in thepower mode management problem in a multi-hop MANET:

• Wakeup prediction. Since a host entering the PS mode willreduce its radio activity, other hosts who intend to sendpackets to the PS host need to know when the host willturn its radio on so as to correctly deliver packets to it atthe right time.

• Neighbor discovery. Because hosts’ transmission/recep-tion activities are reduced under the PS mode, a host may

170 JIANG ET AL.

take longer time, or may be even unable, to detect thearrival and departure of other hosts in its radio coveredrange. Thus, hosts may become less sensitive to neigh-borhood change. Neighbor discovery is essential for routediscovery in a MANET. A host may incorrectly report thatanother host is unreachable if the route to this host has togo through some PS hosts that are not detectable by theirneighbors on the path.

One possible solution to the above problems is to always time-synchronize all hosts. This approach is adopted by IEEE802.11 under the ad hoc mode. However, 802.11 only con-siders single-hop MANETs. Time synchronization in a large-scale distributed environment is generally very costly. It iseven infeasible in a mobile environment since communica-tion delays are typically long and, worse, the MANET maybe temporarily partitioned at any time, making time synchro-nization impossible. Another solution is to develop asyn-chronous power-saving protocols. This is first investigatedin [25], where three solutions are proposed. Among them,the quorum-based protocol is probably the most interestingone. It has the merit of sending the fewest beacon signals(and is thus very energy-efficient). The central idea in thequorum-based protocol can be related to the grid quorum sys-tem [15]. This leads to a more general question: Can we applyother forms of quorum systems to this asynchronous power-saving problem? The result can potentially bridge the impor-tant quorum system concept in traditional distributed systemsto the area of mobile computing, which may in turn gener-ate more efficient asynchronous power-saving protocols. Thiswork does confirm such possibility.

In this paper, we correlate the asynchronous power-savingproblem to the concept of quorum systems, which are widelyused in the design of distributed systems [4,12,14,15]. A quo-rum system is a collection of sets such that the intersection ofany two sets is always non-empty. Not all quorum systemsare applicable to the power-saving problem. We identify a ro-tation closure property for quorum systems. It is shown that,through our mechanism, any quorum system satisfying thisproperty can be translated to an asynchronous power-savingprotocol for MANETs. We derive a lower bound for quorumsizes for any quorum system satisfying the rotation closureproperty. We identify a group of quorum systems that areoptimal or near optimal in terms of quorum sizes (the gridquorum system [15], the torus quorum system [12], the cyclicquorum system [14], and the finite projective plane quorumsystem [15]), which can be translated to efficient asynchro-nous power-saving protocols. We also propose a new e-torusquorum system, which can be translated to an adaptive proto-col that allows designers to trade hosts’ neighbor sensibilityfor power efficiency. A host can dynamically adjust its bea-con rate according to its mobility. Simulation experiments areconducted to evaluate and compare the proposed protocols interms of the survival ratio, the route establishment probability,and the power efficiency.

The rest of this paper is organized as follows. Prelimi-naries are given in section 2. Section 3 introduces the rota-

tion closure property. Section 4 shows several quorum sys-tems that satisfy this property. Section 5 presents our adap-tive power-saving protocol. Simulation results are presentedin section 6. Conclusions are drawn in section 7.

2. Preliminaries

2.1. Power-saving modes in IEEE 802.11

IEEE 802.11 supports two power modes: active and power-saving (PS). Under the PS mode, a host can reduce its radioactivity by only monitoring some periodical signals (such asbeacons) in the network. Tuning a host to the PS mode cansave a lot of energy. For example, table 1 summarizes thepower consumption of ORiNOCO IEEE 802.11b PC GoldCard [21]. However, PS mode should be used cautiously sothat the network throughput and delay do not get hurt.

Under the ad hoc mode, IEEE 802.11 divides the time axisinto equal-length beacon intervals, each of which starts withan ATIM (Ad hoc Traffic Indication Map) window. The ATIMwindow is relatively small compared to the beacon interval.PS hosts must remain active during the ATIM window so as tobe notified by those intending senders, and may go to doze inthe rest of the beacon interval if no one intends to send packetsto it. It is assumed that the ad hoc network is fully connected,so time synchronization is not an issue. In the beginning ofa beacon interval, each mobile host will contend to send abeacon frame. Any successful beacon serves the purpose ofsynchronizing mobile hosts’ clocks as well as inhibiting otherhosts from sending their beacons. To avoid collisions, eachbeacon is led by a random backoff between 0 and 2CWmin −1slots.

After the beacon, a host with buffered packets can senda direct ATIM frame to each of its intended receivers in thePS mode. ATIMs are transmitted by contention in accor-dance with the DCF (Distributed Coordination Function) ac-cess procedure. A receiver, on hearing the ATIM, should re-ply an ACK and remain active. After the ATIM window, hostshaving neither packets to send nor packets to receive can goback to the PS mode to save energy. The buffered unicastpackets are then sent based on the DCF access procedure af-ter the ATIM window. If the sender does not receive an ACK,it should retry in the next ATIM window. If a mobile host isunable to transmit its ATIM frame in the current ATIM win-dow or has extra buffered packets, it should retransmit ATIMsin the next ATIM window. To protect PS hosts, only RTS (re-quest to send), CTS (clear to send), ACK, Beacon, and ATIMframes can be transmitted during the ATIM window. An ex-ample is illustrated in figure 1.

Table 1Power consumption of the ORiNOCO IEEE 802.11b PC Gold Card

(11 Mbps).

Mode PS (doze) Transmit Receive Monitor

PowerConsumed 60 mW 1400 mW 950 mW 805 mW

QUORUM-BASED ASYNCHRONOUS POWER-SAVING PROTOCOLS 171

Figure 1. Transmission scenarios for PS hosts in a single-hop 802.11 MANET.

Figure 2. Structures of quorum intervals and non-quorum intervals.

2.2. Review: a quorum-based PS protocol

IEEE 802.11 only considers single-hop MANETs. For multi-hop MANETs, the following two issues have to be addressed:wakeup prediction and neighbor discovery. In [25], three so-lutions are proposed to solve these problems: the dominating-awake-interval, the periodically-fully-awake-interval, and thequorum-based protocols. Among them, the quorum-basedone has the merit of sending the fewest beacon signals. Be-low, we briefly review the quorum-based protocol proposedin [25]. Still, the time axis is divided evenly into beacon in-tervals. Hosts can be arbitrarily asynchronous in their clocks.Beacon intervals are classified into two types (refer to fig-ure 2):

• Quorum interval. It starts with a beacon window followedby a MTIM window. After the MTIM window, the hostremains active (in monitor mode) for the rest of the beaconinterval.

• Non-quorum interval. It starts with a MTIM window. Af-ter the MTIM window, the host may go to the PS mode ifit has no packets to send or receive.

Similar to IEEE 802.11, the beacon window is for hosts tocompete sending their beacons. The MTIM window is sim-ilar to the ATIM window – a host with buffered packets cancompete to send notifications to intended receivers in the PSmode to wake them up. It is named so to reflect that it isused for multi-hop ad hoc networks. We assume that beacon

windows are not longer than MTIM windows (the assumptionis practical considering these two window’s functionality; theassumption will also be used in our later proofs). With thesedefinitions, we say that a PS host is active when it is currentlyin a beacon window, a MTIM window, or in a quorum inter-val.

In [25], it is proposed that each host divides its beaconintervals into groups such that each group consists of n con-secutive intervals. Each group is organized as an

√n × √

n

array in a row-major manner. The host then picks intervalsalong an arbitrary row and an arbitrary column from the ar-ray as quorum intervals, and the remaining intervals as non-quorum intervals. Thus, there are 2

√n − 1 quorum intervals.

It is shown that no matter how asynchronous hosts’ clocksare, a PS host always has two or more beacon windows thatare fully covered by another PS host’s active period in everyn consecutive beacon intervals. Intuitively, this implies thattwo hosts can discover each other at least twice in every n

consecutive beacon intervals, if their beacon frames do notencounter collisions during transmission.1 Thus, the neigh-bor discovery problem is resolved. Further, by carrying clockinformation in beacon frames, the wake-up prediction prob-lem is also solved.

Figure 3 shows an example with n = 16. Host A picks in-tervals along the first row and the second column as its beaconintervals. Host B, which does not coordinate with A, picks thethird row and the third column. In the middle, we show thecase where A’s and B’s clocks are perfectly synchronized, inwhich case intervals 2 and 9 of A and B are fully covered byeach other. On the bottom, we show the case where A andB are asynchronous in clocks. The beacon windows of in-tervals 0 and 13 of A are fully covered by the duration whenB is active. On the contrary, the beacon windows of inter-vals 2 and 8 of B are fully covered by the duration when A isactive.

1 Collision is inevitable in any kind of contention-based MAC protocols.

172 JIANG ET AL.

Figure 3. Arrangement of quorum intervals based on the grid quorum systemin [25].

2.3. Problem statement

The arrangement of quorum intervals in [25] is in fact basedon the grid quorum system [15]. This leads to the follow-ing interesting question. Can one simply take any quorumsystem, which is a collection of pairwise non-disjoint sets,and apply it to solve the asynchronous power-saving prob-lem in MANET? The answer is negative, due to the followingcounterexample. Let’s number each host’s beacon intervalsby 0, 1, and 2 repeatedly, and let {{0}} be the quorum sys-tem. Hence, each host will pick interval 0 as its quorum in-terval. It is evident that two hosts whose clocks drift by 1 or 2beacon intervals will never be able to hear each other’s bea-cons. Now, an even more interesting question arises: Whatkind of quorum systems is applicable to solve the asynchro-nous power-saving problem in MANETs?

The quorum-based power-saving (QPS) problem is for-mally defined as follows. We are given a universal set U ={0, . . . , n − 1}, n � 2, which represents a set of consecutivebeacon intervals of mobile hosts. The goal is to determine un-der U a quorum system Q, which is a collection of pairwisenon-disjoint subsets of U , each called a quorum, such thateach mobile host has freedom to pick any quorum G ∈ Q tocontain all its quorum intervals (the beacon intervals not in G

are thus non-quorum intervals). The quorum system Q hasto guarantee that for any two arbitrarily time-asynchronoushosts A and B, host A’s beacon windows are fully covered byhost B’s active durations at least once in every n consecutivebeacon intervals, and vice versa.

3. Quorum systems for the QPS problem

Definition 1. Given a universal set U = {0, . . . , n − 1}, aquorum system Q under U is a collection of non-empty sub-sets of U , each called a quorum, which satisfies the intersec-tion property:

∀G,H ∈ Q: G ∩ H �= ∅.

For example, Q = {{0, 1}, {0, 2}, {1, 2}} is a quorum sys-tem under U = {0, 1, 2}.

Figure 4. Timing drift of clocks of two asynchronous hosts.

Definition 2. Given a non-negative integer i and a quorum H

in a quorum system Q under U = {0, . . . , n − 1}, we definerotate(H, i) = {(j + i) mod n | j ∈ H }.

Definition 3. A quorum system Q under U = {0, . . . , n− 1}is said to have the rotation closure property if

∀G,H ∈ Q, i ∈ {0, . . . , n − 1}: G ∩ rotate(H, i) �= ∅.

For instance, the quorum system Q = {{0, 1}, {0, 2}, {1,

2}} under {0, 1, 2} has the rotation closure property. How-ever, the quorum system Q′ = {{0, 1}, {0, 2}, {0, 3}, {1, 2, 3}}under {0, 1, 2, 3} has no rotation closure property because{0, 1} ∩ rotate({0, 3}, 3) = ∅.

Throughout the rest of this paper, we will assume that bea-con windows are not longer than MTIM windows. The fol-lowing theorem connects quorum systems to the QPS prob-lem.

Theorem 1. If Q is a quorum system satisfying the rotationclosure property, Q is a solution to the QPS problem.

Proof. Let A and B be two asynchronous PS hosts in aMANET which choose G and H ∈ Q to represent their quo-rum intervals, respectively. Without loss of generality, let A’sclock lead B’s clock by k × BI + �t , where BI is the lengthof one beacon interval, k < n is a non-negative integer, and0 � �t < BI. This is illustrated in figure 4. First, we showthat B’s beacon window is fully covered by A’s active dura-tions at least once every n beacon intervals. The pattern H

of B is, in fact, rotate(H, k) from A’s point of view, withan extra delay of �t . Note that in the following discussion,time always refers to A’s clock. By the rotation closure prop-erty of Q, G ∩ rotate(H, k) �= ∅. Let e be any element inG ∩ rotate(H, k) and let s be the starting time of A’s inter-val e. Also, let BW and MW be the lengths of one beaconwindow and one MTIM window, respectively. Taking intoaccount the next interval e + 1, we know that A is active froms to s+BI +MW . Since B’s beacon window falls in the range[s+�t, s+�t+BW] and BW � MW, it is easy to see that forany value of �t , [s +�t, s +�t + BW] ⊆ [s, s + BI + MW].So this part is proved.

Next, we show the reverse direction that A’s beacon win-dow is fully covered by B’s active durations at least onceevery n beacon intervals. We first observe that if 0 < �t <

BI, the pattern G of A is rotate(G, n − k − 1) from B’spoint of view, with an extra delay of BI − �t (note that


0 < BI − �t < BI). We also observe that if �t = 0, thepattern G is rotate(G, n − k) with 0 delay from B’s point ofview. Thus, a proof similar to that in the last paragraph can beapplied to prove the reverse direction by exchanging A and B

and substituting �t with BI − �t . �We comment that the above proof requires the property

that BW � MW; otherwise, the conclusion of [s + �t, s +�t + BW] ⊆ [s, s + BI + MW] may not be true.

It is important to note that the number of quorum inter-vals reflects the power consumption of PS hosts since quo-rum intervals are more energy-consuming (recall that a PShost needs to send a beacon and remains active in each quo-rum interval). Given a fixed n, the cost can be measured bythe sizes of quorums in the quorum system. It is desirablethat the quorum sizes are as small as possible. In the follow-ing theorem, we derive a lower bound on quorum sizes forany quorum system satisfying the rotation closure property.A quorum system is said to be optimal if the sizes of all itsquorums meet the lower bound.

Theorem 2. Let Q be a quorum system under {0, . . . , n−1}.If Q satisfies the rotation closure property, then any quorumin Q must have a cardinality � √

n.

Proof. Let H = {h1, . . . , hk} be any quorum in Q, where0 < k < n. There are two cases.

Case 1. H �= rotate(H, i) for any i �= n (mod n).Since h1, h2, . . . , hk are distinct elements, it is clear thath1 + i, h2 + i, . . . , hk + i (mod n) are also distinct forany i = 1..n − 1. So, |rotate(H, i)| = k. Let’s callrotate(H, i), i = 1..n − 1, the rotating quorums of H . Foreach element hj ∈ H , it belongs to exactly k − 1 rotatingquorums of H , namely rotate(H, (hj −hj ′ ) mod n) for everyhj ′ �= hj . By the rotation closure property, H must containat least one element from each of the n − 1 rotating quorumsof H . Since each element appears in exactly k − 1 rotat-ing quorums of H and there are k elements in H , we havek(k − 1) � n − 1, which implies k >

√n. Thus, the theorem

holds for case 1.Case 2. H = rotate(H, i) for some i �= n (mod n). Let

d be the smallest integer such that H = rotate(H, d). It is asimple result in number theory that n is a multiple of d . So itcan be concluded that H = rotate(H, d) = rotate(H, 2d) =rotate(H, 3d) = · · · = rotate(H, n − d). That is, when map-ping the quorum elements of H onto the time axis, H can beregarded as n/d equivalent segments, each of length d . Infact, from H , we can define a smaller quorum

H ′ = {j mod d | j ∈ H }under the universal set {0, . . . , d − 1}. Intuitively, on the timeaxis, H can be considered as a concatenation of n/d copiesof H ′. Since H ∩ rotate(H, i) �= ∅, we can conclude that H ′∩rotate(H ′, i) �= ∅ for any i under modulo-d arithmetic. So{H ′} is also a quorum system satisfying the rotation closureproperty under the universal set {0, . . . , d − 1}. We can applythe result in case 1 and infer that |H ′| �

√d. It follows that

|H | = (n/d)|H ′| � (n/d)√

d >√

n. �

Figure 5. Two quorums of the torus quorum system in a 3 × 6 torus.

4. Quorum systems with the rotation closure property

Although there are volumes of works devoted to quorum sys-tems, none of them discusses the rotation closure propertyto the best of our knowledge. In this section, we prove thatthe grid quorum system [15], the torus quorum system [12],the cyclic quorum system [14], and the finite projective planequorum system [15] are all optimal or near optimal quorumsystems (in terms of quorum sizes) satisfying the rotation clo-sure property.

4.1. The grid quorum system

The grid quorum system [15] arranges elements of the univer-sal set U = {0, . . . , n − 1} as a

√n × √

n array. A quorumcan be any set containing a full column plus a full row of ele-ments in the array. Thus, each quorum has a near optimal sizeof 2

√n − 1. As noted above, the work in [25] adopts the grid

quorum system. Below, we prove the rotation closure prop-erty for the grid quorum system. The theorem, when accom-panied with theorem 1, can simplify the lengthy correctnessproof of the work in [25], which needs to deal with compli-cated timing relation between quorum and non-quorum inter-vals among different asynchronous hosts.

Theorem 3. The grid quorum system satisfies the rotationclosure property.

Proof. Let Q be a grid quorum system. Let H ∈ Q, whichcontains all elements on the column c of the array, namelyc, c + √

n, . . . , c + (√

n − 1)√

n, where 0 � c < n (notethat we number columns from 0 to

√n − 1). Now observe

that rotate(H, i) must contain all elements on column (c + i)

(mod√

n). It follows that rotate(H, i) must have intersectionwith any quorum G ∈ Q because G must contain a full rowin the array. �

4.2. The torus quorum system

Similar to the grid quorum system, the torus quorum sys-tem [12] also adopts an array structure. The universal set isarranged as a t × w array, where tw = n. Following the con-cept of torus, the rightmost column (resp., the bottom row) inthe array are regarded as wrapping around back to the leftmostcolumn (resp., the top row). A quorum is formed by pickingany column c, 0 � c � w − 1, plus w/2� elements, eachof which falls in any position of column c + i, i = 1..w/2�.Figure 5 illustrates the construction of two torus quorums G

and H under U = {0, . . . , 17} with t = 3 and w = 6. G is

174 JIANG ET AL.

formed by picking the second column plus three elements,each from one of the third, fourth, and fifth columns. H isformed by picking the sixth column plus three elements, eachfrom one of the first, second, and third columns. G and H

intersect at element 7.As shown in [12], if we let t = w/2, the quorum size will

be ≈ √2tw = √

2n, which is near optimal. By equating n,the torus quorum size is about 1/

√2 that of the grid quorum

size. Below, we prove the rotation closure property for thetorus quorum system.

Theorem 4. The torus quorum system satisfies the rotationclosure property.

Proof. Let Q be a torus quorum system formed by a t × w

array and H ∈ Q be a quorum containing column c. By de-finition, H also contains another w/2� elements, each fromone of the w/2� succeeding columns of column c. Clearly,rotate(H, i) still has the torus quorum structure for an arbi-trary i. It follows that for any G ∈ Q, G ∩ rotate(H, i)

�= ∅. �

4.3. The cyclic quorum system

The cyclic quorum systems [14] are constructed from the dif-ference sets as defined below.

Definition 4. A subset D = {d1, d2, . . . , dk} of Zn is called adifference set under Zn if for every e �= 0 (mod n) there existelements di and dj ∈ D such that di − dj = e (mod n).

Definition 5. Given any difference set D = {d1, d2, . . . , dk}under Zn, the cyclic quorum system defined by D is Q ={G1,G2, . . . ,Gn}, where Gi = {d1 + i, d2 + i, . . . , dk + i}(mod n), i = 0, . . . , n − 1.

For example, D = {0, 1, 2, 4} ⊆ Z8 is a difference setunder Z8 since each e = 1..7 can be generated by taking thedifference of two elements in D. Given D, Q = {G0 ={0, 1, 2, 4}, G1 = {1, 2, 3, 5}, G2 = {2, 3, 4, 6}, G3 ={3, 4, 5, 7}, G4 = {4, 5, 6, 0}, G5 = {5, 6, 7, 1}, G6 ={6, 7, 0, 2}, G7 = {7, 0, 1, 3}} is a cyclic quorum system un-der Z8.

Given any n, a difference set as small as k can be foundwhen k(k − 1) + 1 = n and k − 1 is a prime power. Such adifference set is called the Singer difference set [3]. For ex-ample, the sets {1, 2, 4} under Z7 and {1, 2, 4, 9, 13, 19} un-der Z31 are Singer difference sets. Note that in this case thequorum size k meets the lower bound in theorem 2. So cyclicquorum systems defined by the Singer difference sets are opti-mal. Reference [14] had conducted exhausted searches to findthe minimal difference sets under Zn for n = 4..111. The re-sults are useful here to construct near-optimal cyclic quorumsystems.

Theorem 5. The cyclic quorum system satisfies the rotationclosure property.

Proof. Let H be a quorum in the cyclic quorum system Qgenerated from the difference set D = {d1, d2, . . . , dk}. Bydefinition, rotate(H, i) is also a quorum in Q for any i. Thenby the intersection property, the theorem holds. �

4.4. The finite projective plane quorum system

The finite projective plane (FPP) quorum system [15] arrangeselements of the universal set U = {0, . . . , n − 1} as ver-tices on a hypergraph called the finite projective plane, whichhas n vertices and n edges, such that each edge is connectedto k vertices and two edges have exactly one common ver-tex. (Note that the hypergraph is a generalization of typicalgraphs, where each edge is connected to only two vertices.)A quorum can be formed by the set of all vertices connectedby the edge, and thus has a size of k. It has been shown in [15]that a FPP can be constructed when n = k(k−1)+1 and k−1is a prime power. Otherwise, the FPP may or may not exist.In [14], the FPP construction is associated to the constructionof Singer difference sets, and it is shown that the FPP quorumsystem can be regarded as a special case of the cyclic quorumsystem when n = k(k − 1) + 1 and k − 1 is a prime power. Itfollows that FPP quorum systems also own the rotation clo-sure property, and are optimal, when existing.

4.5. Quorum systems with one quorum

In this subsection, we discuss the rotation closure property forthose quorum systems with only one quorum. The result hasstrong connection to the difference sets, and can help identifythe quorum systems that are solution to the QPS problem.

Theorem 6. Let Q = {H } be a quorum system under U ={0, . . . , n− 1}. Q satisfies the rotation closure property if andonly if H is a difference set of Zn.

Proof. For the “if” part, let H be a difference set of Zn. Forany i, there must exist two elements hx, hy ∈ H such thathx − hy = i. It follows that hx = hy + i ∈ rotate(H, i) ∩ H .So rotate(H, i) ∩ H �= ∅ for any i.

For the “only if” part, suppose for contradiction that H isnot a difference set of Zn. Then there exists an i �= 0 suchthat hx − hy �= i for all possible combinations of hx and hy

in H . Since rotate(H, i) = {(hy + i) mod n | hy ∈ H }, itfollows that H ∩ rotate(H, i) = ∅, a contradiction. �

Corollary 1. Let Q be a quorum system under U = {0, . . . ,

n − 1}. Q does not satisfy the rotation closure property if atleast one quorum in Q is not a difference set under Zn.

Theorem 6 says that if a quorum system has a differenceset being its sole quorum, it satisfies the rotation closure prop-erty and is thus a solution to the QPS problem. Such a quo-rum system has the practical advantage that it is very easy tomaintain since it has only one quorum to keep. For exam-ple, from each of the minimal difference sets found in [14](for n = 4..111), a solution to the QPS problem exists by


simply putting the difference set as the single quorum in thequorum system. On the contrary, when n is too large suchthat exhaustive searches (as in [14]) are prohibited, we canpick any quorum G in the quorum systems with the rotationclosure property. Then G is a difference set by the contra-position of corollary 1. For example, from the torus quorumsystem, we can quickly find a lot of near-optimal differencesets by arranging numbers from 0 to n − 1 as an array. Notethat in situations when n can not be divided into a product of t

and w, we can always add a “virtual element” on the array, asproposed in [15], to solve the problem. For example, whenn = 13, we can make a 2×7 array with the last position filledby 0 as the virtual element.

5. An adaptive QPS protocol

All the quorum systems discussed above ensure that given afixed n, two asynchronous mobile hosts picking any two quo-rums have at least one intersection in their quorums. It wouldbe desirable to have an adaptive solution in the sense that thenumber of intersecting elements can be dynamically adjusted.One of the main reasons to do so would be to adjust this valueto adapt to host mobility. Intuitively, the number of beaconsthat two hosts can hear from each other is proportional to thenumber of intersecting elements. Thus, a host with highermobility may like to have more intersections with its neigh-boring hosts so as to be more environment-sensitive. On thecontrary, a host with lower mobility may not need to intersectin so many elements with its neighbors so as to save moreenergy. The proposed solution is adaptive in this sense.

We assume that a host is able to calculate its mobility lev-els, either through attaching a GPS device, or simply by eval-uating the number of hosts that are detected to leave/enter thehost’s radio coverage. We leave this as an independent issue,and only focus on the design of adaptive quorum systems tomeet our goal.

The proposed solution is basically an extension of thetorus quorum system, and is thus called the extended torus(e-torus) quorum system. An e-torus quorum system is alsodefined based on two given integers t and w such that U ={0, 1, . . . , tw − 1} is the universal set. Elements of U arearranged in a t × w array. Below, we use [x, y] as an arrayindex, 0 � x < t and 0 � y < w.

Definition 6. On a t × w array, a positive half diagonalstarting from position [x, y], where 0 � x < t and 0 �y < w, consists of element [x, y] plus w/2� elements[(x + i) mod t, (y + i) mod w], for i = 1..w/2�. A negativehalf diagonal starting from position [x, y] consists of element[x, y] plus w/2�−1 elements [(x+i) mod t, (y−i) mod w],for i = 1.. w/2� − 1.

Intuitively, a positive (resp., negative) half diagonal is apartial diagonal on the array starting from the array index[x, y] with a length w/2�+1 (resp., w/2�). A positive diag-onal goes in the southeast direction, while a negative one goes

(a)

(b)

Figure 6. (a) the “Christmas tree” structure of an e-torus(4) quorum, and(b) the intersection of an e-torus(2) quorum and an e-torus(3) quorum.

in the southwest direction. The diagonal is slightly differentfrom typical “diagonal” in matrix algebra in that the array isnot necessarily square and that the torus has the wrap-aroundproperty.

Definition 7. Given any integer k � t , a quorum of ane-torus(k) quorum system is formed by picking any position[r, c], where 0 � r < t and 0 � c < w, such that the quo-rum contains all elements on column c plus k half diagonals.These k half diagonals alternate between positive and nega-tive ones, and start from the following positions:

[r +

⌊i × t

k

⌋, c

], i = 0..k − 1.

Intuitively, each quorum in the e-torus(k) quorum systemlooks like a Christmas tree with a trunk in the middle and k

branches, each as a half diagonal, alternating between posi-tive and negative ones. Figure 6(a) illustrates the conceptualstructure of an e-torus(4) quorum.

Theorem 7. The e-torus quorum system satisfies the rotationclosure property.

Proof. Since any e-torus quorum is a super set of a torusquorum, the theorem holds. �

Theorem 8. Let G be an e-torus(k1) quorum and H be ane-torus(k2) quorum derived from the same array. For any in-tegers i and j , |rotate(G, i) ∩ rotate(H, j)| � (k1 + k2)/2�.

Proof. This theorem can be easily observed from the geo-metric structure of the e-torus quorum system. The value of|rotate(G, i) ∩ rotate(H, j)| can be observed from how treebranches intersect with the trunks of “Christmas trees”. �

176 JIANG ET AL.

Figure 7. Analysis of neighbor sensibility of an e-torus(k1) and an e-torus(k2) quorum systems under a 7×14 torus.

For example, figure 6(b) shows how an e-torus(3) quorumand an e-torus(2) quorum intersect with each other. The in-tersecting elements are guaranteed to appear in the trunks ofthe “Christmas trees”. Note that two branches from two e-torus quorums may “cross with” each other, but intersectionis not necessarily guaranteed (from the geometric structuresof branches, it does look like that they are guaranteed to in-tersect). The reason is illustrated in the zoomed-in part infigure 6(b), where the two branches just miss each other onthe array. Also note that by our arrangement, the intersect-ing elements of two e-torus quorums are unlikely to concen-trated in certain areas of the array. Instead, they will be spreadevenly over the trunks. This is a desirable property because itimplies that the quorum intervals that two mobile hosts maydetect each other will be spread evenly over the time axis.

Based on the above features, we propose an adaptive QPSprotocol as follows. We can rank a host’s mobility into k-levels, where level 1 means the lowest mobility, and level k

means the highest mobility. Whenever a host determines thatits mobility falls within level i (1 � i � k), it adjusts its quo-rum intervals based on any e-torus(i) quorum. Consequently,a host can dynamically adjust its sensibility to the environ-ment change in its neighborhood.

6. Performance comparison and simulation results

6.1. Analytical comparison

In this subsection, we compare the proposed quorum-basedprotocols analytically. We evaluate the active ratio, which isdefined to be the number of quorum intervals over n (the sizeof universal set), and the neighbor sensibility (NS), which isthe worst-case delay for a PS host to detect the existence ofa newly approaching PS host in its neighborhood. The NSof the grid quorum system is BI × (n − √

n + 1), whichhappens when two quorums intersect at indices (i, j) and

(i + 1, j − 1) of the array. The NS of two e-torus quo-rum systems e-torus(k1) and e-torus(k2) under a t × w torusis discussed below (refer to figure 7 for illustration). When(k1 = k2 = 1), (k1 = 2 ∧ k2 = 1), or (k1 = 1 ∧ k2 = 2),the two quorums may intersect at only one interval, so NS isBI × n. For (k1 = 3 ∧ k2 = 1) or (k1 = 1 ∧ k2 = 3), the NSis BI × (n − 1), which happens when two quorums intersectat two consecutive quorum intervals. For (k1 = 1 ∧ k2 = 4)or (k1 = 4 ∧ k2 = 1), the NS is BI × (n − 2t/k1�), whichhappens when the intersections fall in one column. Table 2contains the NS of other cases. Table 2 also summarizes theactive ratio and neighbor sensibility of the proposed quorum-based protocols. Figure 8 further demonstrates the active ratioof the proposed protocols for n = 5..100. The cyclic quorumperforms the best in terms of active ratio. The FPP quorumsystem, when available, represents the optimal solution.

6.2. Simulation results

In this subsection, we compare the proposed power-savingprotocols through a simulator written in C. An area of size1000 m × 1000 m is simulated. Each host has an antennawith a transmission rate of 2 Mb/s and a transmission ra-dius of 250 m, and has an initial battery energy of 100 J. TheMAC part basically follows the IEEE 802.11 standard [11],except the power management part. Routes with randomsources/destinations are generated, and the AODV (ad-hocon-demand distance vector) routing protocol [16] is adopted.Four parameters are tunable in our simulations:

• Mobility. Host mobility follows the random way-pointmodel, with pause time of 20 sec. When moving, a host’sspeed can range in 0–20 m/sec.

• Traffic load. Routes are generated by a Poisson distribu-tion with rates between 1–4 routes/sec. For each route,10 packets, each of size 1 KB, are sent.


Table 2Active ratios and neighbor sensitivity of quorum-based protocols.

Quorum system Active ratio Neighbor sensibility

Grid ≈ 2√n

BI × (n − √n + 1)

Torus ≈√

2√n

BI × n

Cyclic ≈ 1√n

BI × n

Finite projective plane k/n, where k(k − 1) + 1 = n,and k − 1 is a prime power

BI × n

e-torus(k),(under t × w torus)

s/n, where s = t + wk/2� + r(w − 1)/2�,r = 0 if k is even and r = 1 if k is odd

For e-torus(k1) and e-torus(k2):• BI × n, for (k1 = k2 = 1), (k1 = 2 ∧ k2 = 1), or

(k1 = 1 ∧ k2 = 2)

• BI × (n − 1), for (k1 = 3 ∧ k2 = 1) or(k1 = 1 ∧ k2 = 3)

• BI × (n − 2t/k1�), for (k1 = 1 ∧ k2 = 4) or(k1 = 4 ∧ k2 = 1)

• less than BI × (n − 2t/k1�), for (k1 > 4 ∧ k2 = 1) or(k1 = 1 ∧ k2 > 4)

• BI × (n − w + 1), for (k1 = 2 ∧ k2 = 2)

• less than BI × (n − w + 1), for (k1 � 2 ∧ k2 > 2) or(k1 > 2 ∧ k2 � 2)

Figure 8. Active ratios of different quorum systems for n = 5..100.

• Beacon interval. The length of one beacon interval is 100–400 ms.

• Number of hosts. The total number of mobile hosts in theMANET is 50–200 hosts.

Three performance metrics are measured in the simula-tions:

• Survival ratio. The number of surviving hosts (with non-zero energy) over the total number of hosts.

• Neighbor discovery time. Average time to discover anewly approaching neighbor.

• Throughput. The average number of MAC-layer datapackets successfully received in the network per second.

Except the survival ratio, the above metrics are evaluated upto the time when 10% of the hosts run out of energy. A hostcan go to the PS mode when it does not serve as a source,destination, or relay of any route. A broadcast (such as theAODV route request message) may need to be sent multipletimes if the sending host finds that some of its neighbors arein the PS mode [25]. This is necessary because these PS hostsmay wake up at different times and we need multiple trans-

Table 3Power consumption parameters used in the simulation.

Unicast send 454 + 1.9 × L µJ/packetBroadcast send 266 + 1.9 × L µJ/packetUnicast receive 356 + 0.5 × L µJ/packetBroadcast receive 56 + 0.5 × L µJ/packetIdle 843 µJ/msDoze 27 µJ/ms

Table 4Traffic-related parameters used in the simulation.

Unicast packet size 1024 bytesBroadcast packet size 32 bytesBeacon window size 4 msMTIM window size 16 ms

missions to cover all of them. However, once a route is estab-lished (via the notification of a route reply message), all hostsin the route have to tune to the active mode.

Table 3 summarizes the power consumption parametersused in our simulations, which are obtained from real exper-iments using Lucent WaveLAN cards [5]. Sending/receivinga unicast/broadcast packet of L bytes has a cost Pbase + Pbyte

× L, where Pbase is the power consumption independent ofpacket length and Pbyte is the power consumption per byte.Unicast consumes more power than broadcast because it in-curs extra control frames (RTS, CTS, and ACK). Idle/dozerepresents the condition when a host has no send and receiveactivity and is in the active/PS mode, respectively. The traffic-related parameters are summarized in table 4.

Below, we show how mobility, beacon interval length, traf-fic load, and host density affect the performance of the pro-posed PS protocols. We mainly compare the cyclic quorumsystem (which has the lowest active ratio) and the e-torus quo-rum system (which is more adaptive). Below, C(n) standsfor the cyclic quorum system under {0, 1, . . . , n − 1}, and

178 JIANG ET AL.

Figure 9. Host survival ratio vs. mobility (beacon interval = 100 ms,100 hosts, traffic load = 1 route/sec).

Figure 10. Neighbor discovery time vs. mobility (beacon interval = 100 ms,100 hosts, traffic load = 1 route/sec).

E(t × w) the e-torus quorum system under a t × w torus.For the e-torus quorum system, four speed levels (1–4) are as-sumed. A host is said to be at speed level i if its speed is largerthan 5(i − 1) m/sec and less than or equal to 5i m/sec. Tomake comparison, we also simulate an “always-active (AA)”scheme in which all hosts are active all the time.

6.2.1. Impact of mobilityMobility has a negative impact on survival ratio. Figure 9compares the cases when all hosts are stationary and when allhosts’ moving speed = 20 m/sec. Mobility will incur higherenergy consumption because hosts may spend more energy inretransmitting packets. On the contrary, mobility has very lit-tle impact on AA. However, because hosts can tune to the PSmode, C(98) and E(7×14) still outperform AA significantlyin terms of survival ratio. The survival ratio of C(98) is betterthan that of E(7×14) because its active ratio is smaller.

Figure 10 shows the impact of mobility on the neighbordiscovery time. Mobility has a negative impact on neighbordiscovery time for C(98). On the contrary, E(7×14) can bet-ter adapt itself to mobility. We even see shorter neighbor dis-covery time when host mobility becomes higher (at the costof more beacon intervals).

Figure 11 shows the impact of mobility on throughput.Mobility has a negative impact on throughput for all schemesbecause more retransmissions are incurred as hosts movefaster. The results show that C(98) and E(7×14) will slightlydegrade throughputs compared to AA when we allow hoststo tune to the PS mode, which is reasonable. However, the

Figure 11. Throughput vs. mobility (beacon interval = 100 ms, 100 hosts,traffic load = 1 route/sec).

Figure 12. Survival ratio vs. beacon interval length (100 hosts, traffic load =1 route/sec, moving speed = 0–20 m/sec with mean = 10 m/sec).

Figure 13. Neighbor discovery time vs. beacon interval length (100 hosts,traffic load = 1 route/sec, moving speed = 0–20 m/sec with mean =

10 m/sec).

benefit is that the network can be used for much longer time,as reflected by the axis “throughput × lifetime”, where thelifetime of a network is counted up to the point when 10% ofhosts runs out of energy.

6.2.2. Impact of beacon interval lengthWe observe the impact of beacon interval (BI) length on hostsurvival ratio by varying the beacon interval length between100–400 ms. Figure 12 shows that a longer BI will slightlyshorten the lifetime of the network for C(98) and E(7×14).We believe that this is due to a higher transmission cost forbroadcasting route request packets. However, a longer BImakes hosts conserve more energy, which in turn prolongsthe lifetime of the hosts. This may explain the crossing pointsin figure 12.

A longer BI also hurts the neighbor discovery time. Asshown in figure 13, the neighbor discovery time will increase


Figure 14. Throughput vs. beacon interval length (100 hosts, traffic load =1 route/sec, moving speed = 0–20 m/sec with mean = 10 m/sec).

Figure 15. Survival ratio vs. traffic load (beacon interval = 100 ms,100 hosts, mobility = 0–20 m/sec with mean = 10 m/sec).

linearly as BI increases for both for C(98) and E(7×14). TheE(7×14) scheme, which can tune its quorum intervals adap-tively, has much shorter neighbor discovery time compared toC(98).

Figure 14 shows the impact of BI on throughput. Longerbeacon intervals do decrease throughputs. This is because ittakes longer time for a host to wake up its neighboring PShosts to help relay packets. The result shows that E(7×14)slightly outperforms C(98) in terms of throughput due toits adaptivity, and the gain will enlarge slightly as BI in-creases. However, C(98) outperforms E(7×14) in terms ofthroughput × lifetime.

6.2.3. Impact of traffic loadNext, we observe the effect of traffic load. We vary the trafficload in the range of 1–4 routes/sec in the simulations. Fig-ure 15 shows how traffic load decreases host survival ratios.Higher traffic loads do reduce host survival ratios of C(98)and E(7×14), which is reasonable. On the contrary, the im-pact of traffic load on AA is insignificant because anywayhosts have to stay awake all the time. Overall, the proposedschemes still outperform the AA scheme in terms of survivalratio significantly. The effect of traffic load on throughput andaccumulated throughput is shown in figure 16. The trend issimilar to the earlier observation. Traffic load does not influ-ence the neighbor discovery time much, so the related resultsare omitted.

6.2.4. Impact of host densityIn this experiment, we vary the number of hosts in the rangeof 50–200. Since the network area is fixed, this parameter

Figure 16. Throughput vs. traffic load (beacon interval = 100 ms, 100 hosts,mobility = 0–20 m/sec with mean = 10 m/sec).

Figure 17. Survival ratio vs. host density (beacon interval = 100 ms, trafficload 1 route/sec, mobility = 0–20 m/sec with mean = 10 m/sec).

Figure 18. Throughput vs. node density (beacon interval = 100 ms, trafficload 1 route/sec, mobility = 0–20 m/sec with mean = 10 m/sec).

reflects the host density of the network. Figure 17 showsthat a higher host density will bring down the network life-time. On the contrary, the AA scheme is almost unaffected.So a higher host density has a negative effect on survival ra-tio for our schemes. The reason can be explained as fol-lows. As the network becomes denser, when a route re-quest is issued, not only more hosts will help searching forroutes, but also the broadcast cost per individual host willincrease so as to wake up neighboring hosts (note that thetraffic load remains unchanged in this case). In terms of sur-vival ratio, C(98) outperforms E(7×14), which in turn out-performs AA.

As figure 18 shows, a higher node density has a nega-tive effect on throughput for quorum-based protocols, whileit does not influence the AA scheme much. When the nodedensity goes higher, broadcast cost will increase because of ahigher wake-up cost. The effect is an increased number of col-lisions and a lower probability of route establishment. Conse-quently, the throughput goes down. According to figure 18,

180 JIANG ET AL.

the throughput of the AA scheme is the highest, and thethroughputs of the E(7×14) and the C(98) schemes are veryclose. Again, when we consider “throughput × lifetime”, ourquorum-based protocols outperform the AA scheme signifi-cantly.

7. Conclusions

In this paper, we have addressed the asynchronous powermode management problem for an IEEE 802.11-basedMANET. We have correlated the problem to the conceptof quorum systems and identified an important rotation clo-sure property for quorum systems. We have proved that anyquorum system satisfying the rotation closure property canbe translated to an asynchronous power-saving protocol forMANETs. The purpose of the rotation closure property isto deal with asynchronism among hosts’ clocks. Under therotation closure property, we have derived a quorum sizelower bound for any quorum system. We have identified agroup of optimal or near optimal quorum systems. Optimalor near optimal quorum systems are preferable because ina quorum-based power-saving protocol, the number of bea-cons sent and the ratio of a host remaining active are bothproportional to the quorum size. We have shown that thegrid quorum system [15], the torus quorum system [12], thecyclic quorum system [14], and the finite projective planequorum system [15] are all optimal or near optimal quo-rum systems satisfying the rotation closure property. Wehave developed theorems to help identify good quorum sys-tems satisfying the rotation closure property, such as quo-rum systems with only one member, which are very easyto maintain. We have further proposed a new e-torus quo-rum system, which can be translated to an adaptive power-saving protocol allowing hosts to dynamically tune to differ-ent quorum systems according to their mobility, so as to tradeneighbor sensibility for power expenditure. Extensive sim-ulation results have been presented to evaluate these proto-cols.

References

[1] B. Chen, K. Jamieson, H. Balakrishnan and R. Morris, Span: Anenergy-efficient coordination algorithm for topology maintenance in adhoc wireless networks, in: Proc. of the International Conference onMobile Computing and Networking (2001) pp. 85–96.

[2] C.F. Chiasserini and R.R. Rao, A distributed power management policyfor wireless ad hoc networks, in: Proc. of IEEE Wireless Communica-tion and Networking Conference (2000) pp. 1209–1213.

[3] C.J. Colbourn and E.J.H. Dinitz, The CRC Handbook of CombinatorialDesigns (CRC Press, 1996).

[4] C.J. Colbourn, J.H. Dinitz and D.R. Stinson, Quorum systems con-structed from combinatorial designs, Information and Computation(2001) 160–173.

[5] L.M. Feeney and M. Nilsson, Investigating the energy consumption ofwireless network interface in an ad hoc networking environment, in:IEEE INFOCOM (2001) pp. 1548–1557.

[6] J. Gomez, A.T. Campbell, M. Naghshineh and C. Bisdikian, A distrib-uted contention control mechanism for power saving in random-access

ad-hoc wireless local area networks, in: Proc. of IEEE InternationalWorkshop on Mobile Multimedia Communications (1999) pp. 114–123.

[7] J.C. Haartsen, The Bluetooth radio system, IEEE Personal Communi-cations (February 2000) 28–36.

[8] L. Hu, Topology control for multihop packet radio networks, IEEETransactions on Communications 41 (October 1993) 1474–1481.

[9] C.F. Huang, Y.C. Tseng, S.L. Wu and J.P. Sheu, Increasing the through-put of multihop packet radio networks with power adjustment, in: Proc.of International Conference on Computer, Communication, and Net-works (2001).

[10] E.-S. Jung and N.H. Vaidya, An energy Efficient MAC protocol forwireless LANs, in: Proc. of INFOCOM 2002 (2002).

[11] LAN MAN Standards Committee of the IEEE Computer Society, IEEEStd 802.11-1999, Wireless LAN Medium Access Control (MAC) andPhysical Layer (PHY) specifications (IEEE, 1999).

[12] S.D. Lang and L.J. Mao, A torus quorum protocol for distributed mu-tual exclusion, in: Proc. of the 10th Internat. Conference on Paralleland Distributed Computing and Systems (1998) pp. 635–638.

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[19] J.H. Ryu, S. Song and D.H. Cho, A power-saving multicast routingscheme in 2-tier hierarchical mobile ad-hoc networks, in: Proc. of IEEEVehicular Technology Conference, Vol. 4 (2000) pp. 1974–1978.

[20] A.K. Salkintzis and C. Chamzas, An in-band power-saving protocolfor mobile data networks, IEEE Transactions on Communications 46(September 1998) 1194–1205.

[21] E. Shih, P. Bahl and M.J. Sinclair, Wake on wireless: An eventdriven energy saving strategy for battery operated devices, in: Proc. ofMOBICOM 2002 (2002).

[22] T. Simunic, H. Vikalo, P. Glynn and G.D. Micheli, Energy efficientdesign of portable wireless systems, in: Proc. of the International Sym-posium on Low Power Electronics and Design (2000) pp. 49–54.

[23] S. Singh and C.S. Raghavendra, Power efficient MAC protocol for mul-tihop radio networks, in: Proc. of IEEE International Personal, Indoorand Mobile Radio Communications Conference (1998) pp. 153–157.

[24] S. Singh, M. Woo and C.S. Raghavendra, Power-aware routing in mo-bile ad hoc networks, in: Proc. of the International Conference on Mo-bile Computing and Networking (1998) pp. 181–190.

[25] Y.C. Tseng, C.S. Hsu and T.Y. Hsieh, Power-saving protocols for IEEE802.11-based multi-hop ad hoc networks, in: Proc. of IEEE INFOCOM(2002).

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Jehn-Ruey Jiang received his Ph.D. degree in com-puter science in 1995 from National Tsing-Hua Uni-versity, Taiwan. He joined Chung-Yuan ChristianUniversity as an Associate Professor in 1995. He iscurrently an Associate Professor of the Departmentof Information Management, Hsuan-Chuang Univer-sity. He is a recipient of the Best Paper Award in In-ternational Conference on Parallel Processing, 2003.His research interests include distributed computing,mobile computing, distributed fault-tolerance, proto-

cols for mobile ad hoc networks and wireless sensor networks.

Yu-Chee Tseng received his B.S. and M.S. degreesin computer science from the National Taiwan Uni-versity and the National Tsing-Hua University in1985 and 1987, respectively. He worked for theD-LINK Inc. as an engineer in 1990. He obtainedhis Ph.D. in computer and information science fromthe Ohio State University in January of 1994. From1994 to 1996, he was an Associate Professor at theDepartment of Computer Science, Chung-Hua Uni-versity. He joined the Department of Computer Sci-

ence and Information Engineering, National Central University in 1996, andhas become a Full Professor since 1999. Since August 2000, he has becomea Full Professor at the Department of Computer Science and Information En-gineering, National Chiao-Tung University, Taiwan.

Dr. Tseng served as a Program Chair in the Wireless Networks and MobileComputing Workshop, 2000 and 2001, as a Vice Program Chair in the Inter-national Conference on Distributed Computing Systems (ICDCS), 2004, asan Associate Editor for The Computer Journal, as a Guest Editor for ACMWireless Networks special issue on “Advances in Mobile and Wireless Sys-tems”, as a Guest Editor for IEEE Transactions on Computers special on“Wireless Internet”, as a Guest Editor for Journal of Internet Technologyspecial issue on “Wireless Internet: Applications and Systems”, as a GuestEditor for Wireless Communications and Mobile Computing special issue on“Research in Ad Hoc Networking, Smart Sensing, and Pervasive Comput-ing”, as an Editor for Journal of Information Science and Engineering, asa Guest Editor for Telecommunication Systems special issue on “Wireless

Sensor Networks”, and as a Guest Editor for Journal of Information Scienceand Engineering special issue on “Mobile Computing”.

He is a two-time recipient the Outstanding Research Award, National Sci-ence Council, ROC, in 2001–2002 and 2003–2005, and a recipient of the BestPaper Award in International Conference on Parallel Processing, 2003. Hisresearch interests include mobile computing, wireless communication, net-work security, and parallel and distributed computing. Dr. Tseng is a SeniorMember of the IEEE.

Chih-Shun Hsu received his B.S. degree in com-puter education from National Taiwan Normal Uni-versity, Taiwan, in 1990, and the M.S. degree incomputer science from National Taiwan University,Taiwan, in 1992. He joined the faculty of the De-partment of Information Management, Nanya Insti-tute of Technology, Taiwan, as an instructor in 1996.He is currently a Ph.D. candidate of the Departmentof Computer Science and Information Engineering,National Central University, Taiwan. His current re-

search interests include wireless communications and mobile computing.

Ten-Hwang Lai received his Ph.D. in computer sci-ence from University of Minnesota in 1982. Sincethen he has been on the faculty of computer and in-formation science at the Ohio State University, wherehe is presently a professor. His research interestsinclude parallel and distributed computing, mobilecomputing, and wireless networking.

Lai was on the editorial board of IEEE Transac-tions on Parallel and Distributed Systems from 1993to 1996, and is currently an editor of ACM/Kluwer

Wireless Networks, Journal of Information Science and Engineering, and Wi-ley Encyclopedia of Computer Science and Engineering. He served as pro-gram chair and general chair of the 1998 and 2000 International Conferenceon Parallel Processing, respectively; and has been designated as program co-chair and general chair of the 2004 and 2005 IEEE International Conferenceon Distributed Computing Systems, respectively.


CROMA – An Enhanced Slotted MAC Protocol for MANETs

MARCEAU COUPECHOUXInstitut Eurecom and Alcatel Research & Innovation, France Route de Nozay, 91461 Marcoussis cedex, France

BRUNO BAYNATUniversity Pierre et Marie Curie, Paris, France

CHRISTIAN BONNETInstitut Eurecom, Sophia-Antipolis, France

VINOD KUMARAlcatel Research & Innovation, Marcoussis, France

Abstract. TDMA based MAC protocols can provide a very good utilization of the shared radio resources, especially at high input loads, insynchronized mobile ad hoc networks (MANETs). Global positioning systems like GPS or GALLILEO should provide a very good timingaccuracy for synchronization of nodes. This paper presents a new medium access protocol for mobile ad hoc networks, called CROMA.CROMA is collision-free and receiver-oriented. It operates in a slotted environment, in a dynamic and distributed way. In this protocol,receivers act as local base stations and can manage one or several communications on a single slot. Thus, sophisticated functions are allowedat higher layers. Moreover, the hidden terminal as well as the exposed terminal problems are handled by CROMA. A theoretical analysisand extensive simulations show that CROMA can reach very high throughputs.

Keywords: mobile ad hoc networks, MAC, conflict-free protocol, scheduling, dynamic slot allocation, TDMA

1. Introduction

In recent years a lot of effort has been spent in the design ofprotocols for mobile ad hoc networks. Such packet networksare mobile and multi-hop and operate without any fixed in-frastructure. This can be a low cost and easily deployabletechnology to provide high speed Internet access in a wirelessenvironment, to organize networks of sensors, or to comple-ment the coverage of future cellular networks.

In this paper, we pay special attention to the medium ac-cess control (MAC) sub-layer. It has a lot of impact on thesystem performance and its design is a very challenging is-sue. MAC should control access to the medium and share thechannel between source–destination pairs and/or flows of datain a dynamic and distributed way. Some desirable features ofthe access protocol are: to be able to reuse the resources asefficiently as possible, to avoid congestion and collisions, tobe fair, reliable, and energy efficient.

Many MAC protocols try to address these issues. In the lit-erature two categories of schemes have been proposed: (i) thecontention based schemes; (ii) the conflict-free schemes.

In the contention based protocols, the channel has tobe acquired by the nodes for each packet to be transmit-ted. Examples of contention based schemes are CSMA/CA,MACA [18], MACAW [5], FAMA [14], IEEE 802.11 [1].The latter seems to be very popular in most of the testbedsbecause the IEEE 802.11 family products are available offthe shelf. Although IEEE 802.11 is flexible, robust and sim-ple, a recent paper [29] claims that it may not do very well in

a multi-hop environment. According to [29], 802.11 has stillthe hidden terminal problem, does not handle the exposed ter-minal problem at all and its backoff strategy leads to severeunfairness. In this family of protocols, MACA-BI [26] wasthe first one to be receiver oriented, i.e., the transmission ofa packet is initiated by the receiver that sends a short controlpacket in order to reserve the channel and to invite the senderto transmit. As the receiver does not have the exact knowl-edge of packet queue at the sender, it must rely on a trafficprediction algorithm.

On the other hand, conflict-free protocols allow the reser-vation of the channel for a certain amount of time or data andtransmissions are conflict-free. TDMA deterministic schedul-ing may be preferred for networks with heavy load, carryingmixed traffic and realizing sophisticated functions at higherlayers. That is the reason why we propose in this paper a slotallocation protocol for mobile ad hoc networks.

Unfortunately, most of the scheduling problems are NP-complete. For example, Arikan [2] has shown that construct-ing an optimal schedule for the point-to-point schedulingproblem to optimize throughput is NP-complete. And thisis the same for the broadcast scheduling problem basedon throughput optimization, as proved by Ephremides andTruong [12]. Consequently, MAC designers have focused onsub-optimal, dynamic and decentralized solutions for the slotassignment problem.

A first class of scheduling protocols relies on the alloca-tion of priorities to nodes. A given slot is assigned preferablyto the node with the highest priority according to its offered

184 COUPECHOUX ET AL.

traffic. Slots can be allocated by using a control channel, e.g.,in [7]. Priorities of the neighbors are assumed to be known ateach node and are allocated in a pseudo-random way as in [3].Then different strategies can be applied for the allocation ofthe priorities in order to have a fair and efficient share of thechannel (see, e.g., [23]). However, some of these protocolssuffer from a high overhead due to the control channel. Oth-ers do not address the problem of the distributed and dynamicassignment of priorities.

On the other hand, time-spread protocols seem to be veryattractive because they are topology-independent (see, e.g.,[6] or [17]). However, the frame length makes them lessscalable and this class of protocols also faces the problem ofdistributed and dynamic code assignment.

At last, the necessity to address the problem of mobil-ity, topology changes, and scalability, gives rise to a familyof protocols where the reservation of the slots is done via arandom access, most of the time a handshaking, combinedwith a carrier sensing mechanism. FPRP [30] proposes afive-phase handshaking supported by a pseudo-Bayesian al-gorithm to enable a faster convergence of the reservationprocedure. CATA [27] uses four mini-slots in each time-slotto enable unicast and multicast transmissions. The protocolproposed in this paper comes within this family of protocols.It tries to make use of the advantages of the most popularcontention based protocols to a slotted environment in orderto increase their efficiency. In particular, the aim of CROMAis to achieve a high slot utilization, i.e., a high capacity, athigh input load thanks to an original reservation and pollingscheme.

The paper is organized as follows. In section 2, we givea precise description of our proposed MAC protocol. We ex-amine the correctness of this protocol in section 3. Section 4gives an analytical study of the protocol in a fully connectednetwork. At last, section 6 is the conclusion of the paper.

2. Protocol description

The Collision-free Receiver-Oriented MAC (CROMA) is amedium access protocol for mobile ad hoc networks thatschedules transmissions in a slotted environment. It is adynamic and distributed protocol that operates on a single-frequency channel with omni-directional antennas. CROMAhas been shortly presented in [9] and [8]. The present pa-per gives a full description of the protocol, integrates newadvanced features, and provides an enhanced performanceanalysis.

In CROMA, time is divided into frames, each of them di-vided into a fixed number L of time-slots. Each slot can betemporarily and locally attributed to the receiver of a com-munication link depending on topology changes and trafficpatterns. When a receiver is occupying a slot, it is allowedto poll several senders among its neighbors. The number ofcurrent communications for each slot is, however, limited bythe protocol to a pre-defined value K .

The polling packet sent by the receiver is used to reservethe channel and to invite a sender to send a data packet. In that

Figure 1. Frame structure of CROMA.

sense, CROMA is a receiver-oriented protocol since a slot inthe frame is associated to a single receiver.

CROMA does not rely on a traffic prediction algorithmat the receiver. Indeed, a requesting node has to reserve re-sources at its intended receiver during a random access phase.This reservation is needed only at the beginning of a packettrain (or message). When a receiver has no longer traffic topoll, communications are released and the slot is free for an-other receiver.

2.1. Frame structure

CROMA divides time into frames that are, in turn, dividedinto L equal time-slots. All mobile nodes are assumed to beperfectly synchronized.

Synchronization is a very critical issue for CROMA as forall distributed TDMA systems. A possible solution, now atlow cost, consists in making use of the GPS (Global Posi-tioning System) that provides a global synchronization forall nodes. Also the European satellite navigation system,GALILEO, will provide a very good timing accuracy [13].In this case, guard intervals have to be foreseen. Anotherway of research is local synchronization, where nodes tryto synchronize themselves by exchanging beacons with theirneighborhood [10,11]. The algorithms proposed in the lit-erature can be adapted in order to be used with CROMA.However, as in [30] and [27], this paper focuses on theprotocol description and considers that synchronization is arealistic assumption.

Throughout this paper, the following terminology has beenchosen. A requesting node is a node that has data packets tosend but has not yet succeeded in the reservation phase, itsintended receiver is the destination node of these data packets.A sender is a node that succeeded in the reservation phase andthat transmits data packets when it is polled by the receiver.A receiver is a node that polls senders on a given slot. Atlast, we will clearly distinguish the sender/receiver pair of acommunication as defined earlier from the source/destinationpair of a packet, that can be different for control packets.

Each time-slot is divided into three parts: two mini-slots,called REQ-mini-slot (request) and RTR-mini-slot (ready toreceive) for the signaling, and a data transmission phase,called DATA-mini-slot (see figure 1).

The REQ-mini-slot is used by requesting nodes during therandom access phase for sending a REQ to its intended re-ceiver. The RTR-mini-slot is used by their intended receiversto acknowledge requests as well as previous data transmis-

CROMA – AN ENHANCED SLOTTED MAC PROTOCOL FOR MANETS 185

Figure 2. Example of two parallel connections on a slot with CROMA.

sions, and to poll one of the senders that previously manageda successful reservation. During the DATA-mini-slot, thesender that has been polled in the RTR-mini-slot transmits adata packet. These data packets are of fixed length. Indeed, itis assumed that a higher layer is responsible for fragmentationand reassembly.

2.2. CROMA from an example

Before going into more details in the protocol description,let us illustrate the key feature of CROMA that is to allowmultiple reservations on the same slot. The receiver indeedmaintains a list of senders that managed a successful reserva-tion and will poll them in the successive frames. This featureis illustrated in figure 2, which shows two successive reserva-tions on the same slot i. In frame j , the REQ/RTR dialoguestarts the connection between nodes A and B: A sends a REQpacket with its address. B sends back a RTR, that contains afield to acknowledge the reservation (ackreq), and a field topoll node A (pol). The RTR is also received by node C that isnow aware of a communication on slot i with B as receiver.During the data phase, A, that has just been polled by B, isallowed to transmit a packet to B with its address A and asequence number (sn) 0. We say that B has got the floor onslot i. In frame j + 1, C establishes a connection with B.With the RTR, node B acknowledges the reservation with thefield ackreq, acknowledges the packet transmitted by node Ain frame j , and polls node C. In frame j + 2, B now polls A.With the RTR, it also acknowledges the data packet of C withsequence number 0. In frame j + 3, node B polls node C andacknowledges the data packet of A with sequence number 1.

2.3. The choice of a receiver-oriented protocol

The choice of a receiver-oriented protocol is justified by thefollowing arguments:

Figure 3. Packet formats of CROMA.

(i) this is a “natural” choice since only the zone that has tobe secured with respect to collisions is the zone aroundthe receiver, and thus, the spatial reuse of the radio re-sources is favored;

(ii) this choice allows the multiplexing of several communi-cations on a single slot. That implies finer flow controland QoS negotiation. If a slot is associated to a sender, itcannot easily multiplex communications with differentreceivers since they may not be available because of ahidden terminal;

(iii) if a slot is associated to a receiver, a current communi-cation on a given slot does not prevent a random accesson this slot. More bandwidth for the contention for thechannel implies less collisions and interference. If a slotis associated to a sender, it has to send at each frame acontrol packet (RTS) to give the address of its intendedreceiver. Moreover, the receiver has to respond with an-other control packet (CTS) in order to avoid the hiddenterminal problem. In CROMA, once the reservation hasbeen done, the REQ is not used any more for the dura-tion of the communication, and the REQ-mini-slot canbe used for new reservations.

2.4. Packet formats

This section describes the different packet formats and theMAC header of the data packets. It gives also the definitionof all the MAC fields. Their signification will become clearerin the protocol description (sections 2.5–2.7).

2.4.1. Common partsIn figure 3, the control packet formats and the MAC header ofthe data packets are shown. In all packets, generic informa-tion not described in this paper, like the protocol version, aregiven in the field fc that stands for frame control. The field fcs(frame check sequence) contains a CRC (cyclic redundancycode) calculated on all the fields of the MAC header and on


the frame body. The field source.ad gives the Ethernet addressof the source of the packet.

Note that all packets, including data packets, have a fixedsize, and each mini-slot is just long enough to allow the trans-mission of the associated packet. For example, the time totransmit a REQ including additional bits from the physicallayer, the transmit-to-receive turn around time, as well as asmall time interval to take into account the propagation de-lays equal the time of the REQ-mini-slot. Note also that it ispreferred that the size of the control packets are short com-pared to the length of the data packets (e.g., 512 bytes).

2.4.2. REQ control packetIn a REQ, the field dest.ad gives the ethernet address of thedestination of the packet (the intended receiver). The fieldqs is used by a requesting node to indicate to the intended re-ceiver the requested quality of service for the communication.This field may be used by higher layers to negotiate the QoS.It will be used in future versions of the protocol.

2.4.3. RTR control packetA RTR has three different functions, as illustrated in sec-tion 2.2 and in figure 2: respond to a REQ, poll the differentsenders on the current slot and acknowledge data packets.

In the RTR, the fields req.ad and r are used to reply to therequests sent on the same slot (during the REQ-mini-slot). Ifa request is correctly received and accepted, it is acknowl-edged by putting the address of the requesting node in thefield req.ad and the value ACK in the field r . If a request hasbeen correctly received, but the communication cannot be es-tablished, the field r is set to NACK. This situation is possibleif the requested QoS is not allowed or if the number of currentcommunications has reached its maximum, K . If the receiverdetects a collision of REQs, r is set to COL. If the receiver didnot received any request, or if the request cannot be decodedbecause of the channel conditions, r is set to NOTRECVD.The values NACK, COL, and NOTRECVD are useful infor-mation for the requesting nodes to reschedule their requests.

The field polled.ad is used by a receiver to poll a senderthat previously managed to establish a connection on this slot.If a sender reads its address in the field polled.ad, it is allowedto send a data packet during the DATA-mini-slot of the sameslot, just after receiving the RTR.

The acknowledgement of data packets is done thanks tothe field sn that stands for sequence number. Each node main-tains a counter that is incremented for each new data packet.Receivers keep the last received sequence number. If in time-slot i of frame j , a receiver has received a data packet withsequence number m, it sets the field sn to m in the RTR of theslot i of frame j + 1 and so, acknowledges the previous datapacket.

The byte n of a RTR gives information about the slot uti-lization. It is decomposed into seven bits that indicate thenumber k of current communications, and one bit t to informthat the receiver will not accept requests on this slot anymore.More details on the use of the bit t for fairness are given in

section 2.8. If k has reached the maximum K or if the bit t isset to 1, no more request can be done on this slot.

2.4.4. Data packetsIn data packets, the field dest.ad gives the address of the des-tination of the packet.

As previously explained, each sender maintains a counterthat is incremented for each new packet. This sequencenumber is put in the field sn and is used by the receiver toacknowledge the reception of the packet. Let us recall thatdata packets have a fixed size, that results of a higher layersegmentation or aggregation.

2.5. Reservation

Any communication between two nodes must be preceded bya preliminary reservation phase. In the reservation phase, re-questing nodes contend to get access to a receiver. This accessis done in a random way during the REQ-mini-slots and con-sists of five sub-phases: listening of an entire frame, choiceof a time-slot, transmission of the REQ on the chosen slot,listening of the RTR, and retry of a new reservation phase incase of failure (with or without random backoff). These fivesub-phases are now detailed.

2.5.1. Frame listeningThe first phase of the reservation consists in listening to theRTR-mini-slots during an entire frame, and maintaining foreach slot in the frame the state of the slot. This listeningprocess starts at the beginning of the reservation phase andlasts until the reservation has succeeded.

A slot can be in several states:

FREE: no activity has been sensed during the RTR-mini-slot,i.e., no receiver has got the floor on this slot. A request willbe possible on this type of slot.

OCC-NA: i.e., occupied and not available. This is the caseif a RTR has a source.ad different from the address of theintended receiver or if the requesting node has detected acollision during the RTR-mini-slot, or if it did not man-aged to decode the field source.ad in the RTR, or if therequesting node is itself a receiver on this slot. This is alsothe case if the field k of byte n has reached the maximumnumber of communications on a slot or if the bit t of byten is equal to 1. Note that a RTR collision detected on a slotdoes not necessary mean that the slot is free in a multi-hopsituation. A request will not be possible on this slot.

OCC-A-COL-k: i.e., occupied, available, collision, and k

communications. In this case, the source.ad of the RTRis the address of the intended receiver, a collision hasbeen detected by the receiver during the REQ-mini-slot(r = COL in the RTR), and there are currently k < K

communications on the slot. A request will be possible onthis slot.

OCC-A-NCOL-k: i.e., occupied, available, no collision, andk communications. In this case, the source.ad of the RTRis the address of the intended receiver, no collision has


Table 1Decision of a requesting node after listening to the RTR-mini-slot.

Reception req.ad r Decision

RTR decoded my_address ACK Enter the transmission phasemy_address NACK Retry on next frame

not my_address – Retry on next framebroadcast_address NOTRECVD Retry on next framebroadcast_address COL Start backoff algorithm

RTR not received or decoded – – Retry on next frame

been detected by the receiver during the REQ-mini-slot(r �= COL in the RTR), and there are currently k < K

communications on the slot. A request will be possible onthis slot.

It is important to emphasize that the slot states are up-dated continuously during the whole reservation phase. Inorder to reduce the energy consumption, slot states updatescan be, however, limited to a few frames before the reserva-tion process.

2.5.2. Choosing a time-slotThe choice of the time-slot depends on the chosen schedulingpolicy. This policy may have several objectives. For example,it may maximize the slot utilization, limit the amount of inter-ference in the network, establish connections that are robustto mobility. The impact of this choice is not detailed in thispaper. We present here a simple policy that favors free slotsfirst and therefore, aims at maximizing the slot utilization:

1. If there is at least one slot in state FREE,choose one randomly and exit, otherwise go to step 2.

2. If there is at least one slot in state OCC-A,select the slots having the lowest value of k. Among slots in this set:

(a) if there is at least one slot in state OCC-A-NCOL,choose one randomly and exit, otherwise go to step 2(b);

(b) otherwise, choose one slot in state OCC-A-COL randomly and exit;

Otherwise restart the reservation phase at the next frame.

2.5.3. Transmission of the request and RTR generationOn the chosen slot, the reservation is done by sending a REQduring the REQ-mini-slot. Two cases must now be consid-ered:

(i) The sender has chosen a free slot. If the intended re-ceiver can decode the REQ, it replies to the request bysending an RTR in the same slot and by using the fieldsreq.ad and r of this packet, as explained in section 2.4.3.Otherwise, the intended receiver does not reply. (Note,however, that the intended receiver may be aware that theslot is occupied, which can happen in a hidden terminalconfiguration. In this case, the receiver does not answerto the request. See section 3 for more details.)

(ii) The sender has chosen a slot that is already occupied bythe intended receiver. In this case, the intended receiver

replies with an RTR whether it can decode or not theREQ.

2.5.4. Listening of the RTR and decisionA requesting node that has sent a REQ during the first mini-slot of the chosen slot listens to the following RTR-mini-slot.Table 1 gives a summary of the decisions of the requestingnode after the RTR-mini-slot.

If the field req.ad has been set to its address and r to ACK,the requesting node enters the transmission phase. If r indi-cates a collision, the random backoff algorithm is started. Inall other cases, the requesting node is allowed to restart thereservation phase at the next frame. The random backoff al-gorithm is thus only used when a high load is detected for theintended receiver.

2.5.5. Backoff algorithmThe backoff algorithm starts when a requesting node has beeninformed that a collision occurred. An integer BO is randomlychosen between 1 and BACKOFFWND. This is a timer that isdecremented at the beginning of each frame and each timethe requesting node senses a slot in state OCC-A or FREE.As soon as BO reaches 0, a slot is chosen on the forthcomingframe according to the scheduling policy for a new request.With this algorithm, the load on the available slots is takeninto account.

The parameter BACKOFFWND is increased by a multi-plicative factor (1.5) at each successive retransmission anddecreased by one at each success. However, there are a lowerand an upper bound for it, called BOmin and BOmax, e.g., 2and 32.

2.6. Transmission

A sender whose request has been successful on a given slotstarts its transmission phase. During a transmission phase, re-ceivers of which resource has been reserved in the reservationphase, do a polling among their associated senders. Whena sender recognizes its address in the field polled.ad of theRTR, it sends in the same slot a data packet during the DATA-mini-slot.

Each sender maintains a counter of its transmissions thatis incremented at each new packet. This sequence number iscopied in the field sn of the packet header. With this method,the receiver is able to acknowledge the last correctly receiveddata packet. For that, a receiver copies in the field sn the se-quence number of the last received packet. At the sender side,


Figure 4. Polling during the transmission phase.

a sent data packet is stored until the receipt of the acknow-ledgement. If the next RTR is not received or if this RTR doesnot acknowledge the stored packet, a retransmission is nec-essary. After M retransmissions the stored packet is thrownaway. This loss can be treated by an upper layer.

Figure 4 shows an example of a transmission phase witha receiver and three senders. It only shows slots i of suc-cessive frames. On the upper part of the figure, the RTRs ofthe receiver are represented with the fields polled.ad and sn.A cyclic polling is pictured for the scheduling of the sendersand data packets are shown with their field sn.

It is clear that each receiver acts on a given slot as a localbase-station with respect to its associated senders. Thus, thepolling mechanism allows a high flexibility for the schedul-ing of different flows by higher layers and is a base for theimplementation of QoS algorithms. Moreover, several paral-lel communications are possible on a given time-slot.

2.7. Release

An established communication can be interrupted in threecases.

(i) The sender informs the receiver that it sends the lastpacket of the communication by setting the field sn ofthe packet’s header to the value EOT (end of transmis-sion). If the last packet is correctly received, the receiverdoes not re-schedule the sender any more. However, itacknowledges the last packet with its next RTR, and this,even if it has no sender to poll.

(ii) If a receiver has polled a sender and does not receiveany packet from the sender, a counter set to W isdecremented. When this counter reaches 0, the commu-nication is released, and the receiver does not poll thesender any more. If after a poll, a packet is received, thecounter is set again to W . After each polling, a senderstarts a timer. If it does not receive any polling from thereceiver when the timer expires, the connection is con-sidered to be broken.

(iii) During a communication, a sender may receive severalRTRs, i.e., there is a collision of RTRs. In this case,the sender considers that the current communication onthis slot is released. Indeed, sending a data packet couldimply a collision during the DATA-mini-slot. More pre-cisions about this specific aspect are given in section 3.

2.8. Fairness issue

CROMA includes a mechanism to ensure a local fairnessamong data flows. On a given time-slot, fairness among in-coming flows is assured by the receiver of the slot by meansof the RTRs. By using different polling strategy, a receivercan easily give a fair allocation of the slot to incoming flows.

However, if the number of slots in the frame is smallcompared to the number of potential receivers, situations ofunfairness can arise and flows can be completely starved. Thebit t included in the RTRs is used in order to avoid such situ-ations.

A receiver having the floor on a given slot counts the num-ber of consecutive full frames. A frame is full from the pointof view of a receiver, if it senses activity at each slot of theframe. In this case, it detects a potential blocking situationfor pair of nodes that cannot communicate because there areno free slots any more. If the number of monitored full framesreaches MAX_FULLFRAMES, the receiver sets the bit t to 1indicating that it will not accept new requests and that the cur-rent communications have to be released.

A sender detecting a bit t set to 1, sets the field sn of itsnext packet header to EOT and stops sending packets to thereceiver. This release is done even if the sender have stillpackets to transmit. A requesting node detecting a bit t set to1 in a RTR update the slot state to OCC-NA.

This strategy aims at avoiding blocking situations that canlead to unfairness. Indeed, these cases are detected by thereceivers that have to free their slot if the situation lasts.

3. Correctness

In this section, we will show that CROMA is correct, i.e., thatit is collision-free in both fixed and mobile environment. Thecapture effect is not considered here, so this section showsthat CROMA is collision-free in the common case providedthat a sender releases its communication as soon as it detectsa collision of RTRs.

Let us first consider a fixed and multi-hop topology. Wenow prove that two data packets cannot collide.

We suppose that a collision of two data packets occurs ata receiver R1. These packets have been sent by two differ-ent senders, namely S1 and S2. During the RTR-mini-slot, R1specified the MAC address of the sender, say S1, that was al-lowed to send its data in the current slot. As the MAC addressis unique, a single colliding data packet is destined to R1.Therefore, we know that the data packet of S2 was destined toanother receiver, R2.


Figure 5. Interference between two communications sharing the same slot.

Now, as R1 has received a data packet from S2 and linksare bi-directional, S2 has received the RTR of R1. More-over, S2 has also received a RTR from R2, since it sent a datapacket destined to R2. Thus, S2 has detected a collision ofRTRs in the current slot without interrupting its communica-tion with R2. This is impossible. As a conclusion, no datacollision can occur in a fixed topology.

Let us now consider the case of a dynamic topology. Twoconcurrent communications on a slot are shown on the top offigure 5, from node 1 to node 2 and from node 3 to node 4.These communications are sharing the same slot in frame j

and they are far away enough, so that they do not interfere. Incase of mobility and at the next frame j +1, node 3 can eitherstay out of range of nodes 1 and 2, enter the communicationrange of 1, 2, or both 1 and 2. Same alternatives can occur fornode 4. Thus, after mobility, a total of 16 relative new posi-tions are possible. Because of the symmetry of the problem,only 10 cases are shown in figure 5.

The left-hand side of figure 5 shows situations, wherea single communication is interrupted because the senderdetected a collision of RTRs on the considered slot. For ex-ample, in case (b), node 4 moved in the transmission range ofnodes 1 and 2. In frame j + 1, nodes 2 and 4 send simulta-neously an RTR. Node 3 receives correctly the polling of 4,whereas node 1 senses a collision during the RTR-mini-slot.Node 1 decides to interrupt the communication with node 2and does not send any data packet on this slot. If node 1 hasstill packet in its buffer, it has to enter a new reservation phase.

The central part of figure 5 shows exposed-terminaltopologies, where both communications can still share thesame slot. In case (e), node 4 moved in the transmission rangeof node 2. In frame j + 1, node 1 (resp. 3) decodes the RTRof node 2 (resp. 4) because it is out of the transmission rangeof node 4 (resp. 2). Both nodes 1 and 3 can send data packetduring the DATA-mini-slot.

The right–hand side of figure 5 shows topologies, wherecommunications are released because both senders detecteda collision of RTRs. Case (j) shows a configuration wherethe network of nodes is fully connected after mobility. Here,RTRs of nodes 2 and 4 collide at nodes 1 and 3. On detectingthe collision, they decide to interrupt their communication.

So, in the common case, in both fixed and mobile environ-ment, CROMA is collision-free. As in all protocols that relyon the exchange of short control packets, the capture effectmay, however, affect this conclusion.

4. Analytical study

In this section we calculate the approximate throughput, i.e.,the slot utilization of the protocol CROMA in a fully con-nected network. Following [27], we claim that this topologyis the worst case in terms of interference, contention, and spa-tial reuse because CROMA guarantees a collision-free trans-mission of data after reservation in a multi-hop environment.

4.1. Model for the slot utilization analysis

First of all, we describe our analytical model for the slottedMAC protocol CROMA. From this model will be derived theslot utilization of CROMA as a function of the probabilityp to send a REQ for a given source–destination pair. Let’senumerate the assumptions of our model.

1. We consider a fully-connected network of N synchro-nized nodes.

2. All packets are of constant length and are transmittedover an assumed noiseless channel.

3. There are L slots per frame.

4. The maximum number of connections on a slot is K , i.e.,when a receiver is already polling K different senders ona slot, no new REQ is allowed.


5. A receiver can only be associated with a single slot. Thishypothesis can be in practice relaxed, but for the sakeof tractability of the model, we limit the analysis to thiscase.

6. A node can be a sender on several slots of the frame.While being in communication on a slot, a node can senda REQ on another slot of the frame to start another con-nection.

7. The traffic between any two nodes s and d is a ON/OFFtraffic.

8. The ON periods are modeled by bursts of packets follow-ing a geometrical distribution. The length of a messagefollows a geometrical law with parameter q . Thus, theaverage message length (AML) is 1/(1 − q).

9. The OFF periods are modeled by series of slots withouttransmission following a geometrical distribution. If asource s does not communicate with a destination d , thereis a probability p that s wants to communicate with d atthe next frame.

10. A non-persistent policy is assumed for retransmissionsafter a failure. This hypothesis explains that we can con-sider a fixed probability p to start a communication.

The system is described by the number of parallel connectionson the slots at the end of the frame, (a0, a1, . . . , aL−1), where:

• ai is the number of current connections on slot i;

• 0 � ai � MIN(K,N − 1) (see assumptions 1 and 4);

• S = ∑L−1i=0 1{ai>0} � MIN(N,L), (see assumptions 3

and 5).

For the sake of simplicity, the states describe neither thereceiver associated to each slot, nor the list of associatedsenders. The vector (a0, a1, . . . , aL−1) is a discrete-time sto-chastic process, whose state space is also discrete. Moreover,this process is independent of its history because the geomet-ric law is memoryless. Consequently, this process is a discretetime Markov chain (DTMC). Since the state space is aperi-odic and finite, the chain is always ergodic.

From a frame to another, we can have the following tran-sitions on slot i:

• ai → ai + 1 (ai < K): a reservation has been successfulon slot i AND no communication has come to the end;

• ai → ai : (there is a successful reservation AND this isthe end of a communication) OR (there is no successfulreservation AND no message is ending);

• ai → ai − 1 (ai > 0): there is no successful reservationAND this is the end of a communication.

A transition probability between the two states (a0, a1, . . . ,

aL−1) and (b0, b1, . . . , bL−1) is assumed to be the product ofthe transition probabilities associated to each slot:

P((a0, a1, . . . , aL−1) → (b0, b1, . . . , bL−1)

)

=L−1∏

i=0

P(ai → bi). (1)

Figure 6. Discrete time Markov chain representing the state of the slot, forK � N .

Results will show that this assumption is a good approxima-tion.

4.2. One slot analysis

In this section L = 1. In this simple case, we can derive aclosed-form formula for the slot utilization.

The system is described by the number of parallel con-nections on the considered slot at the end of the frame (theDTMC is shown in figure 6). Let’s now compute the transi-tion probabilities ri,j of this Markov chain. Remember thatthe probability for a source–destination pair to enter a ON pe-riod is p. Thus, the probability that a node sends a request ona free slot is the probability that this node has a request for atleast one of the destinations:

p′ = 1 − (1 − p)N−1. (2)

Thus, on a free slot, a successful reservation occurs iff onlyone single node among N is sending a request during theREQ-mini-slot. Consequently the probability to have a suc-cessful reservation on a free slot is

θ(0) =(

N

1

)p′(1 − p′)N−1

. (3)

On an occupied slot with n connections, a receiver hasgot the floor on the slot and successively polls n senders thatmanaged to reserve resources. Here, a successful reservationoccurs iff only a single node among the N − (n + 1) nodesnot currently in connection is sending a request. Therefore,the probability to have a successful reservation on an occu-pied slot is

θ(n) =(

N − (n + 1)

1

)p(1 − p)N−(n+1)−1. (4)

In state 0 � n < K , there is a transition to state n + 1iff a successful request is received and this is not the end ofthe current communication. The transition state rn,n+1 is thusgiven by

rn,n+1 = θ(n)q. (5)

In state 0 < n < K , there is a transition to state n −1 iff there is no successful request and this is the end of acommunication, so

rn,n−1 = (1 − θ(n)

)(1 − q). (6)

From these two equations, we obtain directly rn,n for 0 < n

< K:

rn,n = 1 − rn,n+1 − rn,n−1. (7)


Figure 7. Slot utilization vs. input load, L = 1, N = 5, K = 3.

In state 0, the slot is free and so r0,1 = θ(0) and r0,0 =1 − r0,1. In state K , rK,K = 1 − rK,K−1. The transitionmatrix is given by

P = {ri,j }0�i,j�K. (8)

The steady state probabilities are obtained by solving thesteady state equations �π = �πP , that enable to express allthe probabilities in function of π0:

πn = π0

1 − q

[q

1 − q

]n−1 n−1∏

k=0

θ(k)

1 − θ(k + 1), (9)

for all n ∈ {1, . . . ,K}. The system is totally described withthe following equation:

∑Kn=0 πn = 1. At last, the slot uti-

lization of the protocol is given by U = 1 − π0:

U = 1 − 1

1 + ∑Kn=1

11−q

[ q1−q

]n−1 ∏n−1k=0

θ(k)1−θ(k+1)

. (10)

Figure 7 shows the slot utilization of CROMA, U , as a func-tion of the probability p for K = 3, N = 5 and differentaverage message length (AML = 2, 10 and 100 packets).Dotted curves have been obtained by simulations. These sim-ulations reproduce the assumptions of our model. We can seeon the one hand that the approximations of the analysis havea small impact on the performance evaluation. On the otherhand, it is clear that CROMA can achieve a very high slotutilization provided that the average message length is high.

From the DTMC, the average number of connections,Nc on the slot can also be derived:

Nc =K∑

n=0

nπn. (11)

Figure 8 shows the average number of connections for dif-ferent AML values. This mean number is clearly related tothe delay of transmission of a burst because the higher thenumber of connections on a slot is, the smaller is the resourceallocated to a single connection. Thus, a trade-off has to bemade between slot utilization and delay.

Figure 8. Average number of connections vs. input load, L = 1, N = 5,K = 3.

4.3. Multi-slot analysis

In this section, we extend the previous result to the generalcase with L slots. We first compute the transition probabil-ities, while distinguishing an occupied slot, a free slot and afull slot. For the sake of readability, we only consider the caseK � N .

Let’s consider a slot i occupied by the receiver d (this isthe case, where 0 < ai < K).

The number of nodes that are likely to send a REQ to d arenodes that are currently not in communication with d , theirnumber is N − 1 − ai . The probability for such a node s

to send a REQ on slot i is p (see assumption 9). Thus, theprobability of a successful reservation is:

θi =(

N − 1 − ai

1

)p(1 − p)(N−1−ai)−1. (12)

Note that if ai = N − 1, all nodes have a connection withthe considered receiver, so that there is no REQ on this slot,and θi = 0. Now the probability that a message is ending is(see assumption 8) 1 − q . We can now derive the transitionprobabilities for slot i:

P(ai → ai + 1) = θiq, (13)

P(ai → ai) = θi(1 − q) + q(1 − θi), (14)

P(ai → ai − 1) = (1 − θi)(1 − q). (15)

Let’s now consider a free slot i (ai = 0). There areS = ∑L−1

i=0 1{ai>0} occupied slots in the frame, i.e., S re-ceivers, since a receiver is associated to a single slot (seeassumption 5).

On the considered free slot i, N senders are likely to send aREQ for N−S possible receivers. Indeed, a node is allowed tosend traffic to several receivers in parallel on different slots,so all nodes are likely to start a new communication on i.Moreover, requests on i can be addressed to any of the N − S

nodes that are not receivers on another slot because i is notattributed.


Let’s consider a node s. The probability that s has n REQfor the N − S possible receivers is

p1(n) =(

N − S

n

)pn(1 − p)N−S−n (16)

if s also belongs to the S receivers, and

p2(n) =(

N − S − 1n

)pn(1 − p)N−S−n−1, (17)

otherwise. Thus, the probability that s has n requests is:

p(n) = p1(n)S

N+ p2(n)

N − S

N. (18)

Now, the probability that s sends a REQ on the free slot i is:

β =N−S∑

n=1

Pr[s sends a REQ on i | s sends n REQ]p(n)

=N−S∑

n=1

min

(n

L − S, 1

)p(n). (19)

At last, there are N possible senders like s, so the transi-tions probabilities for i are:

P(0 → 1) =(

N

1

)β(1 − β)N−1, (20)

P(0 → 0) = 1 − P(0 → 1). (21)

Let’s at last consider a full slot (ai = K). The transitionprobabilities are obvious:

P(K → K) = θi(1 − q) + q(1 − θi), (22)

P(K → K − 1) = 1 − P(K → K). (23)

The steady state equations �π = �πP are solved using anynumerical method, e.g., the iterative method of Gauss–Seidel(see [4] or [25]).

Figure 9 shows the slot utilization of CROMA as a func-tion of p for different average message lengths. Analysis andsimulations (dotted lines) are compared and the figure showsa good adequation of the two methods. As for L = 1, we cansee that CROMA can achieve very high slot utilization pro-vided that the AML is high. Note that values of p near 1 arenot realistic in a real implementation because of the backoffalgorithm. Simulations show that the point of operation ofa highly loaded CROMA network with backoff is always forp < 0.5. Figure 10 shows the influence of K on the systemperformance. There is a clear gain of channel utilization asK increases. However, this is obtained at the cost of higherdelays. This is shown in figure 11, where the average num-ber of connections per slot is plotted. A higher number ofconnections per slot implies a higher delay for the burst trans-missions.

Figure 9. Slot utilization vs. input load, L = 3, N = 5, K = 3.

Figure 10. Slot utilization vs. input load, influence of K, L = 3, N = 5,AML = 10.

Figure 11. Average number of connections vs. input load, influence of K,L = 3, N = 5, AML = 10.

5. Performance analysis in a multi-hop environment

In this section, we provide simulation results and the perfor-mance of CROMA and of the standard IEEE 802.11 (DCFmode) are compared.


5.1. Methodology

Studying MAC protocols in a multi-hop environment leadsto the problem of choosing an appropriate node topology.Literature on ad hoc networks has solved the problem byconsidering on the one hand typical networks, like the stringnetwork, or the grid network, and on the other hand randomlygenerated networks. In this paper, we adopted part of the twoapproaches by running CROMA over a classical and chal-lenging network and over a random network.

We will now describe the metrics used to evaluate the per-formance of the MAC protocols.

End-to-end delay. This is the average time spent by a packetfrom the traffic generator of a source to the reception mo-dule of the destination.

End-to-end delay jitter. This is the standard deviation of theend-to-end packet delay.

Aggregate throughput. This is the average number of bitssuccessfully received by all nodes in the network persecond. The input load is the average number of bits trans-mitted by all nodes per second.

Fairness index. This is the widely used index, f , definedin [16]. If a system allocates resources to n contendingentities, such that the ith entity receives an allocation xi ,then:

f (x) = (∑n

i=1 xi)2

n∑n

i=1 x2i

. (24)

If all entities get the same amount, i.e., xi’s are all equal,then the fairness index is 1 and the system is 100% fair.The choice of the metric depends upon the application. Inour case, we will consider that the entities are the flows ofdata between source–destination pairs (i, j) and the metricis their throughput, Ti,j .

5.2. Performance in a challenging environment

5.2.1. Throughput and delay analysisIn order to evaluate the performance of CROMA in ad hocnetworks, we considered a very simple multi-hop situationthat has been used in the literature for the evaluation of MACprotocols, e.g., in [15]. Nodes are assumed to be static,the traffic is ON/OFF with exponential distributions, and thepacket size is set to 512 bytes. Moreover, the channel is sup-posed to be perfect with a physical data rate of 2 Mbps. Thetransmission area of a node is a disk of radius R. Outside ofthe transmission area no communication is possible. Simula-tions have been done using the Network Simulator v2 (ns2,see [21]). The simulation parameter values are presented intable 2. Note that the mean OFF time is fixed and that themean ON time will vary in simulations.

In this configuration, eight nodes form a regular topol-ogy, flows of data are shown in figure 12. Four end-to-endcommunications are running in parallel: 0–1–2–3, 0–5–2–7,7–6–5–4, and 3–6–1–4, so that several nodes have to receiveand/or to relay several flows of data. A solid line without

Table 2Main parameter values for simulations.

Parameter Value

DATA Packet size 512 bytesBOmin 2BOmax 64K 3W 3M 7MAX_FULLFRAMES 30Inter-mini-slot time 10 µsPHY overhead 24 bytesPHY Data Rate 2 MbpsON distribution ExponentialOFF distribution ExponentialPeak Rate 256 KbpsMean OFF time 0.5 sSimulation time 200 sNumber of simulations per point 10

Figure 12. A multihop topology, the “squares topology”.

arrow between two nodes means that they are in the commu-nication range of each other, i.e., the transmissions from oneof them can be successfully decoded by the other one. A solidline with arrow means that at least one flow of data is usingthis link.

This configuration is interesting for several reasons:

(i) it exhibits a lot of hidden terminal situations. For exam-ple, nodes 6 and 2 are hidden from node 0, nodes 7 and3 are hidden from node 5;

(ii) spatial reuse is possible and there are situations of ex-posed terminal. For example, nodes 1 and 2 are exposed.Several flows can share the same slot, e.g., 1–4 and 2–7,or 4–0 and 7–3;

(iii) nodes and flows experience different contention situa-tions, nodes 0, 3, 4, and 7 have three neighbors, whilenodes 1, 2, 5, and 6 have five neighbors.

Figures 13 and 14 show the end-to-end packet delay andjitter as function of the input load for IEEE 802.11 andCROMA. The different curves for CROMA assume differentnumber of slots per frame.

In the case of low input load, IEEE 802.11 outperformsCROMA because the low level of contention implies a smallnumber of collisions and small backoff windows. At this levelof load, the network cannot fully take advantage of the reser-vation scheme because trains of packets are small.

In the case of higher input load, IEEE 802.11 nodes expe-rience more contention, and thus more collisions and widerbackoff windows: access delay increases drastically. On the


Figure 13. End-to-end delay vs. input load, squares topology.

Figure 14. End-to-end delay jitter vs. input load, squares topology.

other side CROMA takes advantage of packet bursts to re-duce the number of requests per transmitted packet. If aflow has made a successful reservation, long trains of pack-ets can be transmitted without contention. Delays and jittersof CROMA L = 8 remains, however, always above IEEE802.11 performance. It is clear that CROMA L = 8 is notwell dimensioned for the topology. Actually, the number ofslots is too high and the resource is not fully exploited, as it isshown in figure 15. To overcome this problem, a higher layercan split a link layer connection into two separate CROMAconnections. The slot utilization of CROMA L = 8 doesnot exceed 0.75. This is much less than CROMA L = 6 thatreaches 0.97. CROMA L = 3 and 4 fully exploit spatial reuseand exceed 1.1.

The reservation scheme, the synchronization, and the abil-ity of CROMA to handle the exposed terminal problem allowthe network to achieve high throughputs. Figure 16 showsaggregate throughput as a function of the input load. IEEE802.11 saturates at a throughput of 300 Kbps. In compari-son, CROMA L = 8 achieves a maximum throughput 350Kbps, although we have seen that it is obviously badly dimen-sioned for the topology. CROMA L = 6 reaches a maximumthroughput of 425 Kbps. For less slots per frame, a pro-blem of stability of the throughput arises. Although CROMAL = 3 and 4 achieve resp. 475 and 510 Kbps, the through-

Figure 15. Slot utilization vs. input load, squares topology.

Figure 16. Throughput vs. input load, squares topology.

put decreases for input loads higher than 525 Kbps. Indeed,the small number of slots implies a slight instability with theconsidered topology. However, curves show a slow decreaseleading to acceptable values even at high input load.

5.2.2. Fairness analysisWithout any fairness strategy and without the use of the bit t ,blocking situations can lead to severe unfairness. This is par-ticularly the case when the input load is high and the numberof slots per frame is small for the considered topology/trafficpattern. For example, in the topology of figure 12 with L = 4,if node 1 hears the RTRs of node 2 on slot 0, node 5 on slot 2,node 6 on slot 3, and sends RTRs on slot 1, 1 cannot send anyREQ since the frame is full. In case of low input load, thissituation is transient and has a low impact on the long termfairness. In case of high input load, however, the connection3–4 is completely starved leading to severe unfairness.

Figures 17 and 18 shows the benefit of use of the bit t

with the aforementioned fairness strategy. The fairness indexof CROMA L = 3 and L = 4 are compared to the index ofIEEE 802.11. For the IEEE standard and for CROMA without


Figure 17. Fairness index vs. input load, L = 3, squares topology.

Figure 18. Fairness index vs. input load, L = 4, squares topology.

the use of the t , the index is close to 1 for low to moderateinput load. After a threshold, the increase of input load leadsto a drop of the index. This threshold is 350 Kbps for IEEE802.11, and approximately 700 Kbps for CROMA. With theuse of the bit t , the fairness index of CROMA remains alwaysabove 0.95 for both L = 3 and L = 4.

5.3. Performance in a random network

In the previous section, we compared IEEE 802.11 andCROMA over a simple and pre-defined multi-hop topology.In this section, we consider a random connex network. 30nodes are drawn at random in a 1000 m×1000 m square area,each node having a transmission range of 250 m. This net-work is shown in figure 19. 10 connections are establishedbetween 10 random pairs of nodes. The traffic is assumed tobe exponential ON/OFF with the same parameters as in theprevious section. Figure 20 shows the aggregate throughputof the network as a function of the input load. While IEEE802.11 and CROMA L = 8 saturate at a load of approxi-mately 500 Kbps, CROMA L = 6, L = 4, and L = 3 reachresp. 600, 700, and 750 Kbps.

Figure 21 shows the mean end-to-end delay of the datapackets as a function of the input load. It is clear that the bet-ter performance of CROMA in term of throughput is obtained

Figure 19. Random topology with 30 nodes in a 1000 m × 1000 m area.

Figure 20. Throughput vs. input load, random topology.

Figure 21. End-to-end delay vs. input load, random topology.

at the expense of higher packet delays and jitters at low inputload (see figure 22). In this case, IEEE 802.11 outperformsCROMA. However, CROMA allows to extend the area of ac-ceptable delay and jitter by one third. For example, CROMAL = 6 still exhibits delays under 600 ms at an input load of700 Kbps. Note also that at low input load, the frame lengthof CROMA has little influence on the end-to-end delay.

In term of fairness, CROMA still outperforms IEEE802.11 in a random topology as shown in figure 23. Note thatit is very difficult to get statistically satisfying results over


Figure 22. End-to-end delay jitter vs. input load, random topology.

Figure 23. Fairness index vs. input load, random topology.

random topologies because of the simulation time. However,ten different random connex networks (not shown here) havebeen simulated and provide similar results.

6. Conclusion

In this paper, a new MAC protocol, called CROMA has beenproposed for mobile ad hoc networks. CROMA operates in aslotted environment, it is collision-free and receiver-oriented.The reservation of resources is made through a random accessphase on each slot of the frame. The transmission is donethanks to a polling by the receivers. Thus, receivers of a con-nection act as local base-stations and sophisticated functionsat higher layers can be easily implemented.

The correctness of CROMA has been proven. Even with adynamic topology, CROMA handles both the hidden and theexposed terminal problems.

Theoretical performance analysis and extensive simula-tions show that CROMA can reach very high throughputs ina fully connected network provided that the average messagelength is large. Moreover, CROMA outperforms IEEE 802.11at high input loads thanks to a better channel utilization.

References

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[19] Y.H. Kwon and D.C. Lee, An uplink packet relay protocol for CDMAcellular-like systems, in: Proc. of MILCOM’02, Vol. 2 (October 2002)pp. 940–945.

[20] H. Luo, S. Lu and V. Bharghavan, A new model for packet schedulingin multihop wireless networks, in: Proc. of ACM/IEEE MOBICOM’00(August 2000) pp. 76–86.

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[24] R. Ramaswami and K.K. Parhi, Distributed scheduling of broadcasts ina radio network, in: Proc. of IEEE INFOCOM’89, Vol. 2 (April 1989)pp. 497–504.


[25] W.J. Stewart, An Introduction to the Numerical Solution of MarkovChains (Princeton University Press, NJ, 1994).

[26] F. Talucci, M. Gerla and L. Fratta, MACA-BI (MACA by invitation) –a receiver oriented access protocol for wireless multihop networks, in:Proc. of IEEE PIMRC’97, Vol. 2 (September 1997) pp. 435–439.

[27] Z. Tang and J.J. Garcia-Luna-Aceves, A protocol for topology-dependent transmission scheduling in wireless networks, in: Proc. ofIEEE WCNC’99, Vol. 3 (September 1999) pp. 1333–1337.

[28] N.H. Vaidya, P. Bahl and S. Gupta, Distributed fair scheduling in awireless LAN, in: Proc. of ACM/IEEE MOBICOM’00 (August 2000)pp. 167–178.

[29] S. Xu and T. Saadawi, Does the IEEE 802.11 MAC protocol work wellin multihop wireless ad hoc networks?, IEEE Comm. Magazine 39(6)(2001) 130–137.

[30] C. Zhu and M.S. Corson, A Five-Phase Reservation Protocol (FPRP)for mobile ad hoc networks, in: Proc. of IEEE INFOCOM’98, Vol. 1(March 1998) pp. 322–331.

Marceau Coupechoux received M.Sc. in mathe-matics from the University Pierre et Marie Curie(Paris) in 1998 and a double Eng. Degree from theEcole Nationale Supérieure des Télécommunications(ENST), Paris, in 1999, jointly with the Universityof Stuttgart in 2000. He joined Alcatel Research &Innovation in 2000 and is Ph.D. student since 2001at the Institut Eurecom, Sophia-Antipolis. He is theauthor or co-author of 10 conference papers and 5patents.


Bruno Baynat received the M.Sc. degree from theInstitut National Polytechnique de Grenoble in 1988and the Ph.D. degree from the University Pierre etMarie Curie in 1991. Presently, he is Maître de Con-férence (Associate Professor) at the University Pierreet Marie Curie. His research interests are presently inthe development of models for the performance eval-uation of communication systems, with applicationsto wired networks (Multicast, QoS) and wireless net-works (Ad-Hoc, GPRS/EDGE/UMTS, Wi-Fi).


Christian Bonnet joined Institut EURECOM as anAssociate Professor in 1992. Since 1998 he hasbeen at the head of the Mobile Communications De-partment of EURECOM. He teaches distributed andreal-time systems, mobile communication systems,wireless LANs and protocols for mobility manage-ment. His main areas of research are MobilityManagement protocols, wireless access to IP Net-works and data communications in mobile networksincluding Mobile Ad Hoc networks. He is currently

participating in research projects related to UMTS in the field of QoS andIpv6.E-mail: [email protected]

Vinod Kumar received M.Sc., and Ph.D. fromEcole Nationale Supérieure des Télécommunications(ENST), Paris, France, in 1977 and 1980, respec-tively. He is Director in Alcatel Research & Innova-tion for projects related to wireless communications.He has more than 20 years of experience in digitalmobile communications, signal processing and per-formance evaluation of higher protocol layers, andhe has more than 20 patents, and more than 30 pub-lications in technical journals, books and conference

proceedings in above mentioned areas. He is Associate Professor in Univer-sity of Marne La Vallée, and Visiting Professor in several other institutes inFrance. He is Officiating Secretary of the Wireless World Research Forum(WWRF).E-mail: [email protected]


Dynamic Bandwidth Management in Single-Hop Ad Hoc WirelessNetworks ∗

SAMARTH H. SHAH, KAI CHEN and KLARA NAHRSTEDTDepartment of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

Abstract. Distributed weighted fair scheduling schemes for Quality of Service (QoS) support in wireless local area networks have notyet become standard. Therefore, we propose an Admission Control and Dynamic Bandwidth Management scheme that provides fairnessand a soft rate guarantee in the absence of distributed MAC-layer weighted fair scheduling. This scheme is especially suitable for smart-rooms where peer-to-peer multimedia transmissions need to adapt their transmission rates co-operatively. We present a mapping scheme totranslate the bandwidth requirements of an application into its channel time requirements. The center piece of our scheme is a BandwidthManager, which allots each flow a share of the channel, depending on the flow’s requirements relative to the requirements of other flowsin the network. Admitted flows control their transmission rates so they only occupy the channel for the fraction of time allotted to them.Thus co-operation between flows is achieved and the channel time is fair shared. As the available channel capacity changes and the trafficcharacteristics of various flows change, the Bandwidth Manager dynamically re-allocates the channel access time to the individual flows.Our simulation experiments show that, at a very low cost and with high probability, every admitted flow in the network will receive at leastits minimum requested share of the network bandwidth. We also present extensive testbed experiments with our scheme using a real-timeaudio streaming application running between Linux laptops equipped with standard IEEE 802.11 network cards.

Keywords: wireless single-hop ad hoc network, distributed weighted fair scheduling, bandwidth manager, max–min fairness

1. Introduction and motivation

In recent times, much effort has gone into solving the prob-lem of transmitting multimedia data over wireless networks.Three mutually orthogonal factors make this problem chal-lenging: (a) stringent QoS requirements of multimedia ap-plications, (b) bursty nature of some multimedia traffic, and(c) unreliable and dynamic nature of the wireless network.Network-specific QoS requirements of multimedia applica-tions include minimum throughput, maximum delay andmaximum jitter.

In a wireless network, the minimum throughput require-ment is more difficult to achieve than in a wireline net-work, because (a) this requires distributed co-operation be-tween nodes sharing a wireless channel, and (b) the flows inthe wireless network are exposed to various physical chan-nel errors. In smart-rooms and “hot-spot” networks, wirelessaccess-enabled nodes in a small area share limited channelbandwidth. Since the area is small, the wireless hosts per-vade through the entire network and are all within each other’stransmission range. There are a large number of hosts andhence connections. So, channeling all data through a sin-gle intermediate node, such as a base-station, is inefficient.Communication is pervasive, i.e., there are many source–destination pairs distributed throughout the network. Thesources must not all rely on a single entity, the base-station, to

∗ This work was supported by the DoD Multi-disciplinary University Re-search Initiative (MURI) program administered by the Office of Naval Re-search under Grant NAVY CU 37515-6281, and the NSF EIA 99-72884grant. Any opinions, findings and conclusions are those of the authors anddo not necessarily reflect the views of the above agencies.

relay their data to their respective destinations. They shouldbe able to directly communicate with their destinations. If abase-station is used as an intermediary, direct one-hop trans-missions are needlessly made two-hop. (The base-stationmust only serve as an access point to the wired Internet, not asa relay for peer-to-peer transmissions between mobile nodeswithin the wireless network.) Furthermore, in military anddisaster rescue environments, a group of people carrying mo-bile handheld devices should be able to communicate witheach other, with no time for planning and building a supportinfrastructure such as a base-station. The single-hop ad hocwireless network, without a base-station, thus accurately rep-resents the network used in smart-rooms, hot-spot networks,emergency environments, and in-home networking.

IEEE 802.11 has recently become the de facto MediumAccess Control (MAC) standard in connecting mobile hostsin an ad hoc network environment. It relies on the Distrib-uted Co-ordination Function (DCF) to resolve channel accesscontention in a distributed way. However, the IEEE 802.11DCF does not currently have any provision to guarantee QoS,such as minimum throughput, to flows accessing the channel.Without any co-ordination, if the sum of transmission ratesof all the hosts (or flows) is greater than the channel capac-ity, heavy channel contention will occur and thus QoS cannotbe guaranteed for any flow. Much research has been donein the area of distributed weighted fair scheduling (DWFS)[3,13,18,19,24] for IEEE 802.11 networks operating in theDCF mode. In DWFS, each flow is assumed to have a weightwhich defines its importance relative to other flows. A sched-uler combined with the MAC-layer IEEE 802.11 protocolthen schedules the flows’ packets on the channel such that

200 SHAH ET AL.

the throughput they receive is proportional to their weights.However, DWFS is not yet a standard part of the IEEE 802.11MAC protocol.

In the absence of distributed MAC-layer weighted fairscheduling in the current IEEE 802.11 standard, we proposea scheme at the higher layers of the OSI protocol stack to co-ordinate individual flows’ channel access in the single-hopad hoc network scenario, in order to promote co-operationbetween flows and provide minimum throughput guaranteefor each of them. To this end, we first determine the flows’weights based on their relative channel access requirements.The flow weights, in turn, determine the transmission rateof each flow. The flows’ transmission rates are controlled atthe application or middleware layers, without any MAC-layerscheduling support. Therefore, our scheme can be used overthe standard IEEE 802.11 protocol and is easily deployableusing today’s off-the-shelf 802.11 products. In case DWFSbecomes available at the MAC-layer in the future, our schemeis still required in order to provide the MAC-layer schedulerwith the flow weights, but enforcing the flow weights will beleft to the MAC-layer scheduler.

The exact share of network bandwidth allotted to a flow de-pends on its requirements relative to the requirements of otherflows. Each flow maps its minimum and maximum bandwidthrequirements to its minimum and maximum channel time pro-portion (CTP) requirements, respectively. We propose the useof a centralized Bandwidth Manager (BM), which obtainsfrom each flow its CTP requirements, at the start of its ses-sion. It uses this information to gauge what proportion of unitchannel time (CTP) each flow should be allotted. The CTPallotted by the BM to each flow (i.e., its “flow weight”) liessomewhere between the flow’s minimum and maximum re-quirements. The term channel time proportion is defined asthe fraction of unit time for which a flow can have the channelto itself for its transmissions. Since our network model allowsonly one node to transmit on the channel at a time, there is adirect correspondence between the channel time a flow usesand the share of the network bandwidth it receives. The BMmay also refuse to admit a flow, i.e., allot 0% channel time.This can happen if the flow’s minimum CTP requirement isso large that the network cannot support it, without violatingsome other flow’s minimum CTP requirement.

The problem with the admission control solution describedabove, however, is that it is a one-time procedure performedbefore the flow starts. It does not take into account thechanges in the wireless network over the duration of the flow’soperation. Not only can the perceived channel capacity varyover time due to varying contention [6] as flows arrive anddepart, but the channel capacity as perceived by different net-work nodes at the same time can also be different. The latterphenomenon is due to location-dependent fading errors andlocation-dependent interference from external objects.

When a new flow arrives and demands a share of the chan-nel, the respective CTPs allotted to already existing flows mayhave to be reduced in order to accommodate it. This revo-cation of channel time should not, however, result in theseexisting flows ceasing to meet their minimum CTP require-

ment. Similarly, when a flow ends, its CTP must be suitablyredistributed among the still existing flows so they can hopeto achieve a better QoS.

The BM must therefore not just perform one-time admis-sion control and teardown, but also perform dynamic band-width management. The BM must re-negotiate with each flowits CTP as its channel characteristics change, and as the num-ber of active flows in the network varies. The detection ofchange in channel characteristics, and adaptation of the flowto this change, happen continuously through the course of thesession. Bandwidth re-negotiation must also occur before aflow changes its packet transmission rate, as in the case ofbursty VBR traffic.

The rest of the paper is structured as follows. The nextsection describes the overall network topology, the architec-ture of the bandwidth management system and the bandwidthmanagement protocol. Section 3 presents our experimentalresults. Section 4 discusses some related work in the field.Finally, section 5 concludes the paper.

2. Bandwidth management system – design andimplementation

In the previous section, we motivated the need for admissioncontrol coupled with dynamic bandwidth management in asingle-hop ad hoc wireless network. In this section, we de-scribe the characteristics of the network we are concernedwith, the architecture of the bandwidth management systemand the communication protocol.

2.1. Network model

We design and implement our bandwidth management schemefor a wireless network consisting of heterogeneous computersand devices connected together over the IEEE 802.11 MAClayer. The network in our prototype testbed implementationconsists of handheld PCs and laptop computers with their802.11 interfaces configured in peer-to-peer ad hoc mode. Weassume that each node in the network is within the transmis-sion range of every other node. Hence, only one node cantransmit at a time over the channel. Since every node is withinthe transmission radius of every other node, routing is single-hop.

Unlike in [5], where a base-station determines the sched-ule of transmission for the entire network and all communi-cation is via the base-station, in our network, transmission isdistributed and peer-to-peer. The IEEE 802.11 MAC proto-col’s DCF, which is the one relevant to our network model,does not have a provision for a fixed transmission schedule.A node can send when it senses that the channel is not busy.A binary exponential backoff mechanism resolves collisionsthat might occur as a result of nodes transmitting at randomtimes. Moreover, any node in the network can transmit toany other node directly without using the base-station as anintermediary hop. Figure 1 illustrates our network model ascompared to the base-station model. The distributed, peer-to-peer and ad hoc nature of our wireless network model makes

DYNAMIC BANDWIDTH MANAGEMENT 201

Figure 1. Comparison of network models: (a) base-station model, (b) single-hop ad hoc network model.

the bandwidth management problem significantly harder tosolve than in the case of a base-station co-ordinated wirelessnetwork where the base-station has full control of the con-tending flows.

The wireless network has one system selected to host theBandwidth Manager (BM) program. In our prototype imple-mentation, we choose one of the more resource-rich nodes inthe network, i.e., one of the laptops, as the host system forthe BM. We assume that the BM program resides on a well-known port in a system whose IP address is well-known inthe wireless network. A service discovery mechanism suchas the ones described in [8,12] can be used to obtain the IPaddress and port number of the BM service. The BM has toregister with the service discovery system upon startup. If theBM suddenly becomes unavailable, due to a crash or due tomobility, an election algorithm can be run to elect a new oneafter a time-out.

Note that the base-station network is merely a special caseof the single-hop ad hoc network, but with no peer-to-peercommunication between mobile nodes. (All communica-tion, as mentioned before, is between the base-station andthe mobile nodes.) Most current wireless LANs, which adoptthe base-station network model, also use IEEE 802.11 DCF.Hence the contention characteristics are identical to those ina single-hop ad hoc wireless network. Our solution, whichis basically designed for the single-hop ad hoc network, thusalso works for the base-station network. Uplink and downlinktraffic between a particular mobile node and the base-stationcan simply be considered as two separate single-hop flows,and their respective channel time requirements can be allottedaccordingly by the BM. The BM in the base-station networkcan be situated at the base-station itself. In this paper, forbrevity, we focus only on the single-hop ad hoc peer-to-peernetwork model.

We assume a network has a set of flows F . Each flowg ∈ F is uniquely identified by its source IP address, sourceport number, destination IP address and destination port num-ber. We call this unique identifier the flow-id of the flow.A new flow f registers with the BM before beginning itstransmission. The application initiating flow f has a mini-mum bandwidth requirement Bmin(f ) and a maximum band-

Figure 2. Bandwidth management system architecture.

width requirement Bmax(f ). The flow f also has an esti-mate of the total network bandwidth Bp(f ). At the time ofregistration, it specifies its minimum and maximum CTP re-quirements, pmin(f ) and pmax(f ), to the BM. Section 2.3 dis-cusses how pmin(f ) and pmax(f ) are obtained from Bmin(f )

and Bmax(f ), respectively. In response, the BM adds flow f

to set F and allots it a certain channel time pa(f ), when theflow is admitted. Flow f then uses this allotted CTP pa(f )

to calculate its transmission rate. It transmits using this trans-mission rate until either it stops or until a new pa(f ) value isallotted to it. A new pa(f ) could be allotted to it when thereis a change in the channel characteristics or in the networktraffic characteristics.

We assume that the flows in the wireless network are well-behaved and co-operative, i.e., they will refrain from exceed-ing their allotted channel share (eating into other flows’ share)and will release any channel share allotted to them when theystop. If the flows are not well-behaved and co-operative, thena policing mechanism (see section 2.7) can be used to detectthe “rogue” flows and eliminate them from the system.

2.2. Bandwidth management system architecture

The architecture of the bandwidth management system con-sists of three major components as shown in figure 2: (a) theRate Adaptor (RA) at the application or middleware layer,(b) the per-node Total Bandwidth Estimator (TBE) at theMAC-layer and (c) the Bandwidth Manager (BM), which isunique in the entire single-hop wireless network. Our systemtakes advantage of cross-layer interaction between the appli-cation/middleware and link layers.

Rate Adaptor (RA). In our design, we assume the absenceof DWFS at the MAC layer. Hence, a flow’s bandwidth con-sumption in accordance with its allotted CTP is regulatedonly by the Rate Adaptor (RA). The RA converts a flow’sbandwidth requirements into CTP requirements, communi-cates this to the BM, and obtains an allotted CTP for this flow

202 SHAH ET AL.

from the BM. It then controls the transmission rate of eachflow depending on its allotted CTP. For the sake of simplicity,in our UDP simulation experiments and testbed experiments,the RA is built into the UDP application itself, to adapt its datageneration rate. Ideally, however, to avoid changing the ap-plication, we recommend that the RA be implemented sepa-rately as a module and be linked to the application at run-time.It would thus function as middleware, just below the applica-tion layer, and shape the applications’ traffic. Various queue-based rate controllers are available for this purpose [16]. Ourinterest is in the design of the overall bandwidth managementarchitecture, rather than the implementation of individual ratecontrol mechanisms. For our TCP simulation experiments,we simulate queue-based rate control by having an RA per-node at the network interface queue, rather than within theapplication.

Note that, in case DWFS is present at the MAC layer, shap-ing the traffic and enforcing flow rates can be left to it. TheRA’s function is deprecated to merely communicating withthe BM and determining the flow rate.

Total Bandwidth Estimator (TBE). The per-node TotalBandwidth Estimator is co-located with the IEEE 802.11 pro-tocol at the MAC layer. It estimates the total network band-width Bp(f ) for each flow f sourced at the node it resideson.1 Bp(f ) is what flow f perceives to be the total band-width of the network at a particular time. In other words, ata particular instant in time, Bp(f ) is equal to the theoreticalmaximum capacity of the channel (1, 2, 5.5 or 11 Mbps forIEEE 802.11) minus the bandwidth lost due to channel errors,caused by fading, interference and contention experienced byflow f ’s packets, at that instant. The physical channel errorsand contention at a particular instant in time is estimated fromthe errors and contention experienced in recent history. De-tails of the estimation method of Bp(f ) are in section 2.4.Note that the TBE is per-node whereas it performs total band-width estimation per-flow sourced at the node it resides on.

The TBE continuously measures the total perceived band-width for each flow. It periodically passes this up to the RAof the flow at the higher layers. The RA of a flow f usesit in the translation of flow f ’s bandwidth requirements toits CTP requirements. When the total bandwidth Bp(f ) per-ceived by flow f changes, the channel time requirements cal-culated using Bp(f ) also change. The TBE informs the RAof the new Bp(f ). The RA may now need to re-negotiate onbehalf of flow f with the BM, using flow f ’s new CTP re-quirements that are calculated with the new Bp(f ) estimate.Since CTP allotted to flow f is directly related to its share oftotal network bandwidth, if a flow perceives the total networkbandwidth as having decreased, its share of the bandwidth

1 In a single-hop peer-to-peer wireless network, we perform bandwidth man-agement per-flow, since each flow can have a different destination. In abase-station environment, we can perform bandwidth management per-node since every node only communicates with the base-station. In thebase-station scenario, each node, rather than application, specifies its band-width requirements to its RA, and bandwidth estimation is done only forlinks between mobile nodes and the base-station.

will also decrease. This may cause it to fall substantiallybelow its minimum bandwidth requirements. Hence the re-negotiation. We do not wish to re-negotiate for small changesin Bp(f ), however, in order to keep re-negotiation overheadsmall. The RA’s not reacting to small changes in Bp(f ) maythus cause small violations of the minimum bandwidth re-quirements. (But not minimum CTP requirements.) The mo-ment a large violation occurs, the RA immediately reacts andre-negotiates. The parameter that defines “small” and “large”is tunable. It trades off the hardness of the bandwidth guaran-tee with re-negotiation overhead.

Example. Assume a flow f in a 2 Mbps wireless network hasminimum bandwidth requirement 300 Kbps and perceives to-tal network bandwidth of 1.5 Mbps. (That is, the flow f per-ceives this to be the total capacity of the 2 Mbps channel.)Assume further that the CTP allotted to it is 20%, thus en-suring it just meets its minimum bandwidth requirement. Ifthe total network bandwidth, as perceived by f , decreases to1.2 Mbps due to an increase in physical channel errors or con-tention, then the 20% channel time is no longer sufficient forthe flow to meet its minimum bandwidth requirement. Its RAmust then re-negotiate for at least a 25% of the channel time.Similarly, if a flow perceives the total network bandwidth tohave increased, it must release any excess share of the channelit has been allotted, so that some other flow can use it.

Bandwidth Manager (BM). The Bandwidth Manager per-forms admission control at the time of flow establishmentand bandwidth redistribution at the time of flow teardown.Admission control involves revocation of some channel timefrom existing flows and re-allocation of this portion to the newflow. The BM also performs re-negotiation either when someflow detects a change in its perceived bandwidth or when itstraffic characteristics change.2

The BM admits a flow only if it can allot at least its mini-mum CTP requirement. Otherwise, the flow is rejected. Theremaining channel time as yet unallotted after all the admit-ted flows’ minimum channel time requirements are satisfied,is allotted on a max–min fair basis. We therefore deem ourchannel time allocation scheme at the BM max–min fair withminimum guarantee. Each flow receives whatever CTP isallotted to it by the max–min fair algorithm, in addition toits minimum CTP request, which is automatically guaranteedwhen it is admitted. A detailed description of the max–minfairness algorithm can be found in section 2.5 of the paper.

2.3. Bandwidth management protocol

This section describes the protocol used in the interactions be-tween the various components of the bandwidth managementarchitecture and the details of the BM’s operation. The BM is

2 The centralized BM does not take on the onus of channel bandwidth esti-mation, and leaves this to the individual per-node TBEs, because the avail-able channel capacity is different for different peer-to-peer flows, due tolocation-dependent physical errors.


Figure 3. Bandwidth management protocol.

Table 1Explanation of notation used in Bandwidth Management protocol.

Notation Meaning

F Set of flows admitted by the BMg ∈ F All individual flows previously admitted by the BMf New flow requesting admissionBmin(f ) Minimum bandwidth requirement of flow f

Bmax(f ) Maximum bandwidth requirement of flow f

Bp(f ) Total network bandwidth as perceived by flow f

pmin(f ) Minimum channel time proportion required by flow f

pmax(f ) Maximum channel time proportion required by flow f

prem 1 − ∑g∈F pmin(g): channel time remaining after

pmin(g),∀g ∈ F is metpnewmax(f ) pmax(f ) − pmin(f ): maximum channel time proportion

requirement for f that is input to max–min algorithmbecause pmin(f ) is already allotted

pmm(f ) Channel time proportion allotted to flow f by max–minalgorithm. This is in addition to pmin(f ) which was alreadyallotted before max–min algorithm began

pa(f ) Total channel time proportion allotted to flow f , i.e.,pmin(f ) + pmm(f )

invoked at the time of flow establishment, flow teardown, sig-nificant change in a flow’s perception of total bandwidth, orsignificant change in a flow’s traffic pattern. Figure 3 showsthe actions that occur when these events happen. Table 1 is anexplanation of the notation used in the protocol description.

Flow establishment. At the time of initiating a flow f ,the application specifies its required minimum bandwidthBmin(f ) and maximum bandwidth Bmax(f ), both in bits persecond, to its RA. The dRSVP [21] scheme also uses max-imum and minimum bandwidth requirements as the specifi-cation of utility. These values have to be each divided bythe flow f ’s perceived total network bandwidth Bp(f ) to ob-tain its requested minimum and maximum CTPs, pmin(f ) andpmax(f ), respectively. The total network bandwidth Bp(f )

perceived by a flow f is estimated by the TBE at the localnode. A best-effort flow will have Bmin(f ) = 0. Figure 4shows the shape of the utility curve of the application.

Figure 4. Utility curve of users.

Note that both the CTP consumed by the flow f ’s datapackets in the forward direction as well as CTP consumed bythe acknowledgements in the reverse direction, if any, mustbe included in f ’s CTP requirement. Still, it is sufficient todo bandwidth estimation at only one of the end-points of thelink. This is because both types of packets traverse the samewireless link, and hence face the same level of contentionand physical errors. The TBE simply quantifies the effect ofthese phenomena. We perform bandwidth estimation, usingthe TBE, at the source. Of course, the data and acknowl-edgements may be of different sizes and packets of differentsizes are affected differently by the same level of physical er-ror. Hence Bp(f ) is different for different packets of the sameflow. The TBE returns a single bandwidth estimate Bp(f ), forthe link flow f traverses, normalized to a standard packet size.(See section 2.4.) It must be appropriately scaled for differentflow packet sizes, using the reverse of the normalization pro-cedure, at the time of flow establishment and re-negotiation.For VBR–UDP flows, either the mean packet size can be usedor the VBR flow can be split into CBR components, as de-scribed later in this section. For TCP flows, separate Bp(f )

values can be derived for data and acknowledgement packetsfrom the single normalized value returned by the TBE.

It must be kept in mind that the TBE of flow f measuresthe perceived bandwidth Bp(f ) using MAC layer frames.These MAC layer frames include protocol headers from theintermediate layers of the protocol stack between the appli-cation and the link layers. The Bp(f ) scaling operation musttake into account the fact that the lower layers of the proto-col stack will add their respective headers to each packet, andthus consume some of the channel capacity. The size of thelower-layer headers must be added to the application packetsize in the scaling operation.

The RA of a node registers a new flow with the node’sTBE. Initially, the TBE has no estimate of the total networkbandwidth as perceived by this newly beginning flow. This isbecause it has to use the flow’s packets themselves for obtain-ing an estimate of the total network bandwidth, based on thephysical channel errors and contention these packets experi-ence. But the flow has not sent out any packets yet and is stillin the process of establishment. So, when initially computingthe flow’s requested minimum and maximum CTPs, the RAhas to use a hardcoded initial total bandwidth estimate.3 Oncethe flow begins, a more accurate total bandwidth estimate willbe available from the TBE. The requested minimum and max-imum CTPs can then be modified using this new, more ac-

3 In our prototype testbed implementation, we use a 2 Mbps network and weset this hardcoded value to 1.5 Mbps.

204 SHAH ET AL.

curate estimate, and re-negotiation done with these modifiedvalues.

Alternatively, in the case of a connection-oriented flow, thefirst few flow-establishing packets can be used in the totalbandwidth estimation instead of the hardcoded estimate. Forexample, the physical channel errors and contention faced byTCP’s three-way handshake messages can be used in the ini-tial measurement. If the application involves some other con-trol messages (e.g., client asking server if file exists or not),then these can be used. A current estimate being used by otherflows between the same end-points can also be used initially.A fourth option is to have the BM maintain a list of currenttotal bandwidth estimates for all flows. Then, a new flow canquery the BM for an initial estimate. The BM simply returnsthe average of the list of total bandwidth estimates.

Let the initial total bandwidth estimate, how ever itmay be obtained, for a new flow f be Bp(f ). The CTPpmin(f ), required to satisfy the new flow f ’s minimum band-width requirement Bmin(f ), is pmin(f ) = Bmin(f )/Bp(f ).pmin(f ) = 0 for best-effort flows. Similarly, the CTPpmax(f ), required to satisfy flow f ’s maximum bandwidthrequirement, is pmax(f ) = Bmax(f )/Bp(f ). The RA of thenew flow f sends the BM a request message containingthe flow-id of f , pmin(f ), pmax(f ) and a timestamp for or-dering.

The BM checks whether, for all flows g in the set F ofpreviously registered flows, 1 − ∑

g∈F pmin(g) � pmin(f ).If this is true, the new flow f is admitted (F = F ∪ {f }), elseit is rejected and a reply message offering it zero CTP isreturned to its Rate Adaptor. Note that a best-effort flow withpmin(f ) = 0 is always admitted. A rejected flow may attemptagain later to gain access to the channel. Flows are admittedstrictly in the order they arrive, to alleviate starvation of pre-viously rejected real-time flows. The problem of starvation ofa best-effort flow after admission is dealt with in section 2.6.

Once the new flow f is admitted, the BM must redistributechannel time within the new set of existing flows F . Since theoriginal admission test was passed by flow f , accommodat-ing it will not cause the CTP allotted to any flow g ∈ F tofall below its minimum CTP request. Hence, the BM initiallysets allotted CTP pa(g) = pmin(g), ∀g ∈ F . The remain-ing channel time, prem = 1 − ∑

g∈F pmin(g), is distributedamong the flows g ∈ F in max–min fair fashion. Our chan-nel time allocation policy is thus called max–min fair withminimum guarantee. The maximum CTP requirement foreach flow g ∈ F in the max–min fair computation is set topnewmax(g) = pmax(g) − pmin(g). This is because pmin(g)

has already been allotted to it and it only needs pnewmax(g)

more to fulfill its maximum CTP requirement. Thus, know-ing prem and pnewmax(g) ∀g ∈ F , the max–min algorithm cannow proceed. Details of the max–min fairness algorithm canbe found in section 2.5.

Suppose that out of the remaining channel time prem, theamount allotted to any flow g ∈ F by the max–min algorithmis denoted by pmm(g). Now, 0 � pmm(g) � pnewmax(g) and∑

g∈Fpmm(g) = prem. Then, the total CTP allotted to eachflow g ∈ F is pa(g) = pmin(g)+pmm(g). Note that for best-

effort flows, since pmin(g) = 0, pa(g) = pmm(g). In otherwords, channel time is allotted to best-effort flows only afterall the higher priority real-time flows are all allotted at leasttheir minimum share.

After the new flow f is admitted, the BM registers an entrypertaining to it in its flow table. This entry consists of: (a) thenew flow f ’s flow-id, (b) the socket descriptor of the socketused by the BM for communication with f ’s RA, (c) pmin(f ),(d) pmax(f ) and (e) pa(f ). The socket descriptor is stored inthe table so that if any re-negotiation needs to be done laterwith flow f ’s RA (for example, when newer flows arrive infuture or existing flows depart), this socket can be used. Inaddition, a timestamp indicating the freshness of the latestrequest message is also maintained for each flow. Thistimestamp is used for two purposes: (a) timing out stale reser-vations, and (b) proper ordering of multiple outstanding re-negotiation requests from the same flow. Since reservationscan time-out, the entries in the flow table are soft-state en-tries. If, for some reason, a flow’s reservation has timed-outbut the flow is still transmitting, this can be detected using apolicing mechanism. (See section 2.7.)

Finally, for every flow g ∈ F , the allotted CTP pa(g) isthen sent to flow g’s RA using a reply message. (Note thatthe name of the message is a misnomer in the case of all flowsg ∈ F except the new flow f because, in their case, the re-ply is gratuitous, not a response to any message they sent.)It may be the case that all flows g ∈ F do not need to be senta reply message. No reply message needs to be sent to aflow in F whose allotted CTP has not changed due to the ar-rival of the new flow f . Although we implement the replymessage as multiple unicast messages to individual RAs forreliability, it can also be implemented for efficiency as a sub-net broadcast message, containing flow-id and pa(g), ∀g ∈ F .A flow f is rejected using a unicast reply with pa(f ) = 0.Other existing flows’ allotted CTPs are not affected.

The RA of every flow that receives a replymessage, gra-tuitous or otherwise, from the BM sets its transmission raterespectively to pa(g) · Bp(g) bits per second (bps). The newflow f can now begin operation whereas the older flows sim-ply resume operation with their respective new rates.

The timestamp in the reply to flow g indicates the lastrequest received from g by the BM. The value of Bp(g)

used to compute pmin(g) and pmax(g) for this requestmustthen be used in the transmission rate formula above, since itis based on this value of Bp(g) that pa(g) was calculated bythe BM. As a new Bp(g) is returned by the TBE periodically,a new rate is also used periodically. If the Bp(g) change islarge since the last period, re-negotiation must occur, as ex-plained below.

Flow teardown. When a flow f terminates, its RA sendsa teardown message to the BM. The BM removes flow f

from the set of existing flows F i.e., F = F − {f }. It thenredistributes flow f ’s allotted CTP pa(f ) among the otherflows using the max–min fair algorithm with minimum guar-antees. The RA of each flow g ∈ F (the new set F ) istold of its newly allotted CTP by the BM. The socket de-


scriptors in the flow table are used to send gratuitous re-ply messages for this purpose. The entry for the terminatingflow f in the BM’s flow table is expunged. A teardown-acknowledgement message is sent to f ’s RA.

Change in a flow’s perception of total network bandwidth.The RA of every flow periodically obtains from the TBE theflow’s current perceived total bandwidth. The TBE updatesthe RA with the mean of the perceived total network band-width measured for each packet successfully transmitted bythe flow in recent history. The inter-update period could bein terms of number of packets transmitted or in terms of time.We recommend using a hybrid scheme for determining updateperiod: it should be based on time when the transmission rateof the flow is low and based on number of packets transmittedwhen it is high. In our experiments, we use high transmissionrates in order to determine the performance of our scheme un-der high network loads. Therefore, we use a perceived band-width update interval based on number of packets. We usea default interval of 100 transmitted packets in our experi-ments, but we also measure how various other intervals affectthe performance of the system.

In case a newly obtained perceived bandwidth valueNEWBp(f ) differs significantly from Bp(f ), the RA mustre-negotiate its flow’s CTP with the BM, as indicated in theexample in the previous section. It must also set the valueof perceived bandwidth Bp(f ) to the newly obtained valueNEWBp(f ). Note that the RA only sets Bp(f ) to NEWBp(f )

and re-negotiates with the BM using this new value whenthere is a significant change, not with every update. A newrate using the previously allotted CTP is, however, calculatedwith every update. In our experiments, we assume a deviationδ = 15% of NEWBp(f ) from Bp(f ) as significant enough towarrant re-negotiation. We also measure how other perceivedbandwidth deviation tolerance (δ) percentages affect systemperformance.

If re-negotiation has to be done, the RA of flow f sendsa request message to the BM with flow-id, pmin(f ) andpmax(f ). The values of pmin(f ) and pmax(f ) sent in therequest message are re-calculated using the new value ofBp(f ). The rest of the re-negotiation procedure is almostidentical to the one used for flow establishment, both at theBM as well as at the RA. (See figure 3.) The only differenceis that the BM does not have to add a new entry in its flowtable for f ; it only updates the already existing one.

Note that a flow f ’s re-negotiation request can be rejectedby the BM, i.e., it can receive pa(f ) = 0, in response tothe requested CTP. This means that the flow has been cut-off in mid-operation. Unfortunately, the nature of the wire-less network is inherently unreliable and as network resourcesdecrease, some flows will necessarily have to be cut-off inmid-operation so that others can be supported. Our schemeguarantees that each flow will obtain at least its minimum re-quested CTP for almost 100% of its active duration. If thesystem cannot guarantee the flow at least this level of QoS, itwill drop it altogether. In other words, a flow will either re-ceive (for nearly 100% of its active duration) at least its min-

imum requested CTP pmin(f ), or it will receive no channeltime at all. The guarantee in terms of bandwidth is that theallotted bandwidth never falls more than a factor of δ belowthe minimum requested bandwidth Bmin(f ), since if Bp(f )

changes by a factor of δ, re-negotiation occurs.Currently, we do not use any priority scheme to cut-off

particular flows. If perceived bandwidth decreases for allflows, the first flow initiating re-negotiation is cut-off. Al-ternate strategies to pick flows to cut-off in mid-operation arediscussed briefly in section 3.1.1.

Change in a flow’s traffic characteristics. When a VBR–UDP flow f (e.g., MPEG video stream) needs to send a burstof traffic at a rate different from its normal rate, it must informits RA. The RA will re-negotiate for a larger CTP for flow f

depending on the bandwidth of the burst. The re-negotiationprocedure is the same as in the case of change in perceivedbandwidth. At the end of the burst duration, the RA willagain re-negotiate to release the excess CTP. This solution isequivalent to splitting up a VBR stream in the time domaininto multiple CBR streams. There exists previous literaturein the context of ATM networks [11] in which VBR streamsare split into multiple CBR streams in the time domain. Sincethis scheme only involves re-organizing the traffic rather thanthe network, it can be directly applied from ATM networks towireless networks.

Figure 5 is an MPEG-4 trace of an hour-long, 25 framesper second, medium-quality, clip of the movie “Silence of theLambs”. The trace was taken from [10] and the referencestherein. On the x-axis is a running count of the frame number.On the y-axis is the frame size averaged over non-overlappingblocks of 50 frames. One possible way to split up this VBRflow into multiple CBR components is shown in figure 5 asthe contour of the plot. The CBR bandwidth component thusobtained is then used as the minimum bandwidth requirementBmin(f ) in negotiating with the BM.

Frequent bursts could result in an explosion in re-negotia-tion overhead. We deal with the problem of frequent burstsin one of two ways: (a) setting Bmin(f ), at the time of burst-induced re-negotiation, large enough to engulf multiple burstsand (b) having large buffering at the receiver to deal with theburst.

Figure 5. MPEG-4 trace of “Silence of the Lambs” clip with correspondingCBR components.

206 SHAH ET AL.

Figure 6. IEEE 802.11 unicast packet transmission sequence.

2.4. Total bandwidth estimation procedure

To determine pmin(f ) and pmax(f ), the RA of a flow f needsto have an estimate of the total bandwidth over the wirelesslink being used by the flow. To this end, we introduce a band-width measurement mechanism based on IEEE 802.11 DCFMAC layer, and demonstrate its robustness.

IEEE 802.11 relies on the DCF method to coordinate thetransmission of packets based on CSMA/CA without any cen-tral control unit. The packet transmission sequence is illus-trated in figure 6. Before transmitting a packet, a node sensesthe channel to make sure that the channel is idle; otherwise itbacks off by a random interval and senses the channel again.If the channel is idle, it transmits a RTS (Request-to-Send)packet to signal its intention to send a packet.4 On receiv-ing the RTS packet, the destination node replies with a CTS(Clear-to-Send) packet to give the sender a go-ahead signal,and to silence the destination node’s neighboring nodes. Afterreceiving the CTS packet, the sender sends the DATA packet,and it is then acknowledged by an ACK packet from the re-ceiver.

Similar to [14], we measure the throughput of transmittinga packet as TP = S/(tr − ts), where S is the size of the packet,ts is the time-stamp that the packet is ready at the MAC layer,and tr is the time-stamp that an ACK has been received. Notethat the time interval tr − ts includes the channel busy andcontention time. We keep separate throughput estimates todifferent neighboring nodes because the channel conditionsmay be very different. We only keep an estimate for activelinks, since we do not have any packets to measure tr − ts overinactive ones.

This MAC layer measurement mechanism captures the ef-fect of contention on a flow’s perceived channel bandwidth.If contention is high, tr − ts will increase and the throughputTP will decrease. This mechanism also captures the effect ofphysical errors because if the RTS or DATA packets are af-fected by channel errors, they have to be re-transmitted, uptothe re-transmission limit. This increases tr − ts and corre-spondingly decreases the flow’s perceived bandwidth. Sinceour MAC layer measurement of perceived bandwidth takesinto account the effects of both contention and physical errorsdue to fading and interference on a flow, we can have the flowreact suitably to these factors by monitoring the change in per-ceived bandwidth. It should be noted that the perceived band-width is measured only using successful MAC layer transmis-sions.

4 For very small packets, the sender may skip the RTS packet and directlysend out the DATA packet.

Figure 7. Raw throughput and normalized throughput at MAC layer.

It is clear that the measured throughput of a packet dependson the size of the packet. Larger packet has higher measuredthroughput because it sends more data once it grabs the chan-nel. To make the throughput measurement independent ofthe packet size, we normalize the throughput of a packet toa pre-defined packet size. Before being used by a flow of aparticular packet size, it must be scaled to that packet size. Infigure 6, Td = S/BWch is the actual time for the channel totransmit the data packet, where BWch is the channel’s bit-rate.Here we assume channel’s bit-rate is a pre-defined value. Thetransmission times of two packets should differ only in theirtimes to transmit the DATA packets. Therefore, we have:

(tr1 − ts1) − S1

BWch= (tr2 − ts2) − S2

BWch(1)

= S2

TP2− S2

BWch, (2)

where S1 is the actual data packet size, and S2 is a pre-definedstandard packet size. By equation (2), we can calculate thenormalized throughput TP2 for the standard size packet. Toverify the validity of this equation, we simulated a groupof mobile nodes within a single-hop ad hoc network usingthe ns-2 network simulator [23]. We sent CBR traffic fromone node to another, and varied the packet size from small(64 bytes) to large (640 bytes) during the course of the simu-lation. The measured raw throughput is normalized against astandard size (picked as 512 bytes). Figure 7 shows the resultof the measured raw throughput and its corresponding nor-malized throughput. Obviously, the raw throughput dependson the packet size; larger packet size leads to higher measuredthroughput. The normalized throughput, on the other hand,does not depend on the data packet size. Hence, we use thenormalized throughput to represent the bandwidth of a wire-less link, to filter out the noise introduced by the measuredraw throughput from packets of different sizes.

Another important issue is the robustness of the MAClayer bandwidth measurement. We measure the bandwidth ofa link in discrete time intervals by averaging the throughputsof the recent packets in the past time window, and use it to es-timate the bandwidth in the current time window. Obviously,this estimation may not be accurate because the channel con-dition may have changed. To evaluate the estimation error,we run a CBR flow over UDP with data rate 160 Kbps froma node to another in a 10 node one-hop environment. Back-ground traffic consists of 1 greedy TCP flow in the light chan-nel contention case, and 7 TCP flows in the heavy contention


case. Here we use TCP only to generate bursty cross-trafficto the UDP flow. We measure and normalize the throughputof the CBR flow every 2 seconds using the average of packetthroughputs in the past time window. Our results show thatunder light channel contention, over 97% of the estimates arewithin 20% of error; under heavy contention, still over 80% ofthe estimates are within 20% of error. We thus conclude thatusing average throughput of past packets to estimate currentbandwidth is feasible and robust.

It should be noted that the bandwidth estimation mecha-nism in no way alters the IEEE 802.11 protocol. Our band-width estimation mechanism, with the normalization exten-sion, was satisfactorily accurate for the scenarios in oursimulation and testbed experiments. However, the theorybehind the normalization may not be applicable for arbitrar-ily large packet sizes or arbitrarily high bit-error rates. Insuch cases, the TBE could keep an indexed table of sepa-rate estimates for different packet size ranges per active link,rather than maintaining a single normalized estimate per ac-tive link and scaling it to various packet sizes at the timeof flow establishment/re-negotiation. If the indexed tablemethod is used, the source and destination must both performtotal bandwidth estimation, for data and acknowledgements,respectively. The destination must periodically communicateits bandwidth estimate for acknowledgement packets with thesource using an in-band signaling mechanism. (The signal-ing itself consumes negligible bandwidth.) In the single nor-malized estimate method, the source alone does the estima-tion and appropriately scales the normalized estimate for bothdata and acknowledgement packet sizes. Thus, although theindexed table estimation method improves accuracy of the es-timate in certain special cases, it also incurs a small storagespace and in-band signaling overhead.

2.5. Max–min fairness

Fairness is an important issue in designing our BandwidthManager. In this paper, we adopt a max–min fairness algo-rithm with minimum guarantee in allotting channel time tothe flows. This section describes the max–min algorithm tocalculate how much channel time each flow gets beyond itsguaranteed minimum requested channel time, after the flowis admitted.

In max–min fairness [4], flows with small channel timerequests are granted their requests first; the remaining chan-nel capacity is then evenly divided among the more demand-ing flows. As described in section 2.3, pa(f ) is first set topmin(f ) for all the flows. The channel time that remains,prem, after satisfying the flows’ minimum requirements, isallotted to the flows in max–min fashion. The new maxi-mum requirement for each flow in the max–min algorithmis pnewmax(f ) = pmax(f ) − pmin(f ), because pmin(f ) hasalready been allotted to it and must be subtracted from theoriginal maximum requirement. We denote the channel timeallotted to flow f by the max–min algorithm as pmm(f ). Thisis in addition to pmin(f ) allotted before the max–min algo-rithm is even invoked.

Input. Channel time: p_rem; set of requests: p_newmax[f ]Output. Set of allocations: p_mm[f ]proc Max–min(p_rem, p_newmax[f ]) ≡

R := {}; //set of satisfied flowsN := size_of (p_newmax[f ]);p_mm[f ] := 0;while (true) do

total_satisfied = 0;foreach f ∈ R do

total_satisfied+ = p_mm[f ];odCA := (p_rem − total_satisfied)/(N − size_of (R));stop := true;foreach f /∈ R do

if (p_newmax[f ] < CA) thenR := R + {f };p_mm[f ] := p_newmax[f ];stop := false;fi

odif (stop) thenforeach f /∈ R dop_mm[f ] := CA;odbreak;fi

od

Figure 8. Max–min fair resource allocation algorithm.

The computation of the max–min allocation is as follows.Initially, the set of flows f , whose new maximum channeltime requirement pnewmax(f ) has been satisfied, is empty:R = ∅. Then, we compute the first-level allotment asCA0 = prem/N , where N is the total number of flows. Nowwe include all flows f with pnewmax(f ) < CA0 in set R, andallot each of them pmm(f ) = pnewmax(f ). Next, we com-pute CA1 = (prem − ∑

f∈R pnewmax(f ))/(N − ‖R‖). If forall flows g /∈ R, pnewmax(g) � CA1, then we allot each ofthem pmm(g) = CA1 and stop. Otherwise, we include thoseflows g with pnewmax(g) < CA1 in set R, allot each of thempmm(g) = pnewmax(g), and re-compute the next level CA2.When the algorithm terminates, the allocation pmm(f ) forall the flows is max–min fair. The pseudo-code for the al-gorithm is shown in figure 8. It is clear that the computationalcomplexity of this algorithm is O(N2). As mentioned earlier,after every flow f ’s pmm(f ) has been determined using themax–min algorithm, the BM sets pa(f ) = pmin(f )+pmm(f )

and returns this value to flow f ’s RA.

2.6. Alternate channel time allocation strategies

Although we use the max–min fairness with minimum guar-antee policy for bandwidth allocation in our implementation,a different fairness policy or even a biased, priority-basedscheme could also be used.

In our policy, as mentioned earlier, best-effort flows areonly given access to the channel after all the real-timeflows’ minimum requirements are satisfied. This could lead

208 SHAH ET AL.

to starvation of the best-effort flows, in the rare case that∑g∈F pmin(g) → 100%. One way to eliminate this prob-

lem would be to partition channel time into a large minimum-guarantee portion and a small max–min fair portion, similarto the bandwidth partitioning in [1]. The minimum require-ments of the real-time flows, i.e., all pmin(g) > 0, are al-lotted only from the minimum-guarantee portion. The max–min fair portion, along with any left over minimum-guaranteeportion, is used to allot the flows’ extra CTP pmm(g), usingjust a max–min scheme. Both real-time as well as best-effortflows, i.e., all flows with pnewmax(g) > 0, can vie for thisportion. The presence of a separate max–min fair portion en-sures that, however large the minimum requirements of thereal-time flows, some channel time is always available forbest-effort flows to vie for, so they are never starved. The dis-advantage of having a separate max–min fair portion is thatthe channel time available to satisfy minimum guarantees ofreal-time flows (the minimum-guarantee portion) is reduced,which could lead to more real-time flows being dropped.

Another alternate scheme involves pricing of channel timeand enforcing priorities based on flow budgets. The max–min fair policy with minimum guarantee lends itself to an ele-gant two-tier pricing scheme. The guaranteed minimum CTPpmin(g) is valued at a substantial price, whereas any chan-nel time pmm(g) in excess of this is relatively very cheap.Under this two-tier pricing scheme, users would be inclinedto request as little minimum guaranteed bandwidth as possi-ble, in order to save cost. High minimum requirements arethus “punished” while high maximum requirements carry nopenalty. The BM adjusts the price so as to trade-off blockingprobability of the flows with its revenue. If the price is toohigh, too few flows can afford it and hence blocking probabil-ity is high. If the price is low, blocking probability is low, butrevenue may suffer. Pricing for wireless networks has beenstudied previously [17,20,22,26], but our two-tier approach isespecially suitable for our bandwidth allocation policy.

2.7. Policing

In our bandwidth management scheme, policing refers to thetask of monitoring the users, to make sure that they conformto their allocated bandwidth. The bandwidth manager oper-ates in two modes: normal and policing. When operating inpolicing mode, the bandwidth manager listens promiscuouslyto the network traffic, and checks whether a flow, identifiedby the source and destination addresses and port numbers inits packet headers, is sending out packets faster than its al-lotted rate. Additionally, it can also catch those flows whohave not registered with the bandwidth manager. This can besome type of “denial of service” attack by a malicious users,or caused by some unmanaged applications.

Operating in policing mode is expensive. Therefore, thebandwidth manager should operate in this mode only whennecessary. To this end, the bandwidth manager relies on thesudden, sharp decrease of channel bandwidth as an indica-tion, in the re-negotiation process. If there is a sudden flock ofre-negotiation requests due to reduction in Bp(g), it is likely

that abnormally high channel contention has occurred. Subse-quently, the bandwidth manager switches into policing modeto monitor the activity of the network. It may be that the chan-nel contention is due to a sudden increase in physical errorsor it may be that it is due to a malicious or unmanaged flow.The policing scheme can identify which of the above is thecause. It could also happen that the unreliable subnet broad-cast reply message did not reach a particular RA, so a flowis continuing to transmit packets faster than its re-allotted rate.

3. Experimental results

We evaluate the performance of our Admission Control andDynamic Bandwidth Management system using both a proto-type testbed as well as simulations using the ns-2 simulator.We used our testbed when evaluating the performance of aflow in the presence of both physical channel errors caused byfading and interference effects as well as medium contentionfrom two other active stations, because there is no way to setup physical obstacles such as walls, ceilings and doors thatcause signal weakening in ns-2. We used ns-2 simulations toevaluate the performance of the system when there is heavymedium contention due to the presence of a large number ofactive stations.

3.1. Simulation experiments

For experiments with large numbers of nodes (�5 nodes)and flows, we used the ns-2 simulator. We comparedthe performance of an Admission Control and BandwidthManagement-enhanced IEEE 802.11 network (henceforthcalled “enhanced IEEE 802.11 scheme”) with an IEEE 802.11network without bandwidth management (henceforth called“base IEEE 802.11 scheme”). We used a 170 m × 170 m net-work area and the transmission range of each node was 250 m.Hence, the entire network area falls within every node’s trans-mission range. The maximum theoretical channel capacitywas 2 Mbps. We used the random waypoint mobility modelwith moderate node speeds in our simulations.

3.1.1. UDP throughput performanceOur first simulation scenario consisted of a 20-node networkwith 10 flows. Each flow had a minimum bandwidth require-ment of 100 Kbps and a maximum bandwidth requirementof 200 Kbps, which are typical of an audio streaming appli-cation. All the 10 flows used 512 byte packets. The sim-ulation ran for 600 seconds. The transmission rate used byour scheme at any instant was determined using the methoddescribed in section 2.3. The transmission rate used in thebase IEEE 802.11 scheme was a constant set to the maximumrequested rate of the CBR flow, as would be the case in anunmanaged application. The RA’s inter-update interval was100 packets and its perceived bandwidth variation-tolerancethreshold δ = 15%, by default.

Figure 9(a) is a plot of number of packets successfullytransmitted over every 1 second interval for each of the 10


(a) Base IEEE 802.11.

(b) Enhanced IEEE 802.11.

Figure 9. Comparative throughput performance of base and enhanced IEEE802.11 for 10-flow scenario.

flows using the base IEEE 802.11 scheme. Figure 9(b) is thesame plot using the enhanced IEEE 802.11 scheme. Note thatin our scheme two flows needed to be cut-off in mid-operationso that other flows’ minimum CTP requirements are not vio-lated. One of these is cut-off at time 149 seconds and the otherat time 264 seconds. These times indicate the respective firstoccasions when the flows in question requested a minimumCTP that could not be supported. When a new flow is admit-ted, contention increases for all the existing flows. In general,the flow that notices an “unacceptably” poor channel qual-ity and “complains” first is dropped. Alternate flow droppingstrategies can also be employed, such as dropping the flowlast admitted. Pricing could also pay a role here: the flowpaying the least can be dropped.

It is clearly evident from the plots that our protocol dra-matically improves throughput fairness. In the base IEEE802.11 scheme, flows often fall far below their minimumbandwidth requirement over the 1 second measurement in-terval, resulting in a chaotic plot. Using our scheme, flowsalmost never fall below their minimum bandwidth require-ment shown with the horizontal line at 24 packets per second.(100 Kbps/4096 bit packets is approximately 24 packets persecond.) Even when they do, it is only by a small amount.Our scheme thus ensures that the minimum bandwidth re-quirements of the flows are met with a far higher probabilitythan the base IEEE 802.11 scheme. Figure 10 is a 100-secondsnapshot from the combined plot of figures 9(a) and 9(b) thatshows the comparative behavior of a single flow (flow 1).

Figure 10. Comparative behavior of a single flow over base 802.11 versusenhanced 802.11.

(a) Without smoothing.

(b) With smoothing.

Figure 11. Perceived bandwidth and re-negotiations corresponding to its vari-ation.

Figure 11(a) shows the variation of perceived bandwidthfor one of the flows in the above experiment as measured byits TBE at the MAC layer. The superimposed stepwise curveshows the bandwidth last used for re-negotiation in the aboveexperiment. Recall that δ = 15%. We also experimented withsmoothed perceived bandwidth estimates, which reduced theoverhead of re-negotiation frequency. Figure 11(b) is a plotof a running average of the measured perceived bandwidthwith exponential decay, which is used for smoothing of theestimate. The smoothed estimate falls as contention increasesand rises when the two flows are dropped and contention de-creases. Other methods to reduce re-negotiation overhead aredescribed in the next subsection.

In section 1, we mentioned that improving fairness is es-sential for providing minimum throughput guarantees to wire-less multimedia applications. The key factor enabling our

210 SHAH ET AL.

scheme to provide minimum bandwidth requirement guar-antees with a high probability, is its improved fairness. Noflow takes up excess bandwidth during a particular intervalthereby depriving another flow of bandwidth and resulting ina large throughput discrepancy (i.e., poor fairness) betweenthe flows. Our scheme also reduces jitter in throughput ascompared to base IEEE 802.11. Throughput jitter is the dif-ference in throughput observed over two consecutive same-sized time intervals. It should be as low as possible for a CBRflow. We use 1 second time intervals. We thus designate fair-ness and throughput jitter as the key performance measuresthat characterize the performance of our system. The betterthese measures, the higher the probability of the flows meet-ing their minimum bandwidth requirements.

While our scheme focuses on ensuring that flows receivetheir minimum throughput, the delay and delay jitter arealso improved as a by-product of our bandwidth managementscheme. Since we co-operatively control the sending rate ofthe flows, we observe a negligible packet loss rate when usingour scheme. Due to the rate control, queue length is uniformlyshort, queuing delay is small, and congestion loss is avoided.Since contention is uniformly low, delay jitter is also im-proved. With base IEEE 802.11, however, since the transmis-sion rate is set to the maximum, a 33% packet loss rate resultsdue to congestion and the resultant queue overflow. Whenusing our scheme without perceived bandwidth smoothing,each flow re-negotiates its allotted CTP once every 14 sec-onds on average. In section 3.2.2, we determine that each ofthese re-negotiations can take upto 60 ms in the presence ofcontention. This does not affect the flow too much because itcontinues sending at a rate dictated by the previously allottedCTP and current value of Bp(f ) during this interval. It doeshowever represent a small amount of network traffic over-head. The mean throughput of an active flow for our schemein the above scenario is 8% lower than that of an active flow inbase IEEE 802.11. We believe that this lower mean through-put is a small price to pay for the vastly improved stabilityin throughput. The latter property is essential for multimediaapplications. In the next subsection, we will discuss the rea-sons for throughput deterioration and present mechanisms toreduce the flow-initiated re-negotiation overhead.

3.1.2. Overhead for UDP experimentsThere exists a trade-off between network traffic overhead andperformance in terms of fairness and jitter. We need to be ableto quantify the fairness and throughput jitter so that we canmeasure how much they are affected when we try to reduceoverhead.

In our simulations, we measure the number of packets ofeach flow transmitted over each 1 second interval in the 600second run. Let us denote the number of packets transmit-ted by flow f over second i as N

fi . Let the average over all

flows of number of packets transmitted in second i be denotedas Ni . Let the set of active flows, i.e., flows that have been es-tablished but not yet torn down or cut-off, during second i

be A. We only measure throughput per second for the dura-



Figure 12. Comparative throughput performance of base and enhanced IEEE802.11 for 3-flow scenario with identical bandwidth requirements.

tion in which all flows are active together. Assume that thenumber of seconds for which the measurement is done is n.

We define a fairness metric FM = ∑f∈A|Nf

i − Ni |/‖A‖.We also define a throughput jitter metric for a flow f , JMf =∑n−1

i=1 |Nfi − N

f

i+1|/(n − 1). The overall jitter metric JM isthe mean of the JMf ’s, i.e., JM = ∑

f∈AJMf /‖A‖.For the experiments in this subsection, we use a differ-

ent network scenario in which there are 6 nodes in the ns-2-simulated wireless network and 3 flows. The flows each re-quire a minimum throughput of 200 Kbps (approximately 48packets/sec.) and a maximum throughput of 600 Kbps. Weran this simulation scenario for a duration of 300 seconds. Allother simulation parameters exactly remain the same from theprevious subsection. We used the period when all three flowsare active for all our measurements.

Figures 12(a) and 12(b) show the number of packets trans-mitted over every 1 second for base IEEE 802.11 and en-hanced 802.11, respectively. Once again, it is evident fromthe plots that our scheme performs better in terms of both fair-ness and throughput jitter. However, we apply our metric todetermine exactly how much our scheme improves these per-formance measures. We obtained a value of FM = 6.72 pack-ets for base IEEE 802.11 versus FM = 4.06 packets for ourscheme. (Lower FM means better fairness.) We also obtainedJM = 8.80 packets for base IEEE 802.11 vs. a JM = 4.93packets for our scheme. (Lower JM means lower through-put jitter.) We conclude that for this particular scenario, ourscheme results in a 60–80% improvement in performance.Each flow in our scheme requests a re-negotiation of CTPonce every 7 seconds, without perceived bandwidth smooth-


Table 2Effect of Bp(f ) inter-update period on performance and overhead.

Inter-update period FM JM Overhead(pkts.) (pkts.) (pkts.) (requests/flow/sec.)

50 3.62 4.37 0.5100 4.06 4.66 0.143150 4.15 4.93 0.059200 4.18 5.10 0.019

ing. This is lower than the 14 seconds for the scenario in theprevious section because the transmission rate is higher andhence the 100-packet inter-update interval is reached faster.

As in the case of the scenario in the previous subsection,there is a 28% packet drop rate in the case of base IEEE802.11, but negligible drop rate using our scheme. Also asin the previous scenario, the mean throughput of base IEEE802.11 is 15% higher during the period under measurement(all 3 flows are active) than our scheme. This is because ofthree reasons: (a) the flows are pumping data into the net-work as fast as possible in order to get as much throughput asthey can in the base IEEE 802.11 scheme while we are usingrate control, (b) our TBE is configured to return a conserva-tive estimate for Bp(f ), and (c) in the Dynamic BandwidthManagement scheme, the re-negotiation messages betweenthe various RAs and the BM consume some network band-width.

The conservative Bp(f ) estimate was used to minimizepacket drop rate. The cost of using such a conservative esti-mate is that our enhanced IEEE 802.11 scheme under-utilizesthe network. Mean throughput is less than it would be underfull network utilization. However, the TBE’s estimate can besuitably tuned so that throughput of our scheme approachesthat of the base IEEE 802.11 scheme and network utilizationincreases. On the other hand, this will also increase the packetdrop rate of our scheme and thereby degrade performance aspackets are dropped randomly from flows. So, there exists atrade-off between throughput and packet drop rate.

In addition to the perceived bandwidth smoothing de-scribed in the previous section, we now discuss two othermethods to minimize re-negotiation overhead and hence thenetwork bandwidth re-negotiation consumes. One method isto increase the inter-update period between successive per-ceived bandwidth updates from the TBE to the RA. Recallthat we use 100 packets as the default inter-update intervalin our experiments. Table 2 shows how overhead and perfor-mance vary with different inter-update intervals. As the inter-update interval increases, some changes in perceived band-width go undetected and cannot be responded to. Hence, thefairness and throughput jitter worsen while the overhead im-proves. The overhead is measured as the frequency of re-negotiation requests per flow. The threshold tolerance to per-ceived bandwidth changes was set at the default of δ = 15%for this experiment.

The other method to reduce re-negotiation overhead isto increase the tolerance to changes in perceived bandwidthBp(f ). Recall that we define significant change as a δ = 15%change in perceived bandwidth. If we define significant

Table 3Effect of various Bp(f ) variation tolerance levels δ on performance and over-

head.

Tolerance level FM JM Overhead(%) (pkts.) (pkts.) (requests/flow/sec.)

10 3.22 4.36 0.33315 4.06 4.66 0.14320 4.89 5.19 0.05625 5.77 5.37 0.026

change as, say, a δ = 25% change, then we can reducere-negotiation overhead because the RA now waits longerand tolerates more Bp(f ) fluctuation before initiating re-negotiation. Again, this worsens the performance of the sys-tem because fidelity to bandwidth variations is reduced. Ta-ble 3 shows how overhead and performance vary with differ-ent levels of tolerance to Bp(f ) variation. The inter-updateinterval was set to 100 packets for this experiment. Tables 2and 3 both show that for a small price in terms of perfor-mance, we can obtain large gains in overhead reduction.

3.1.3. Additional UDP performance resultsIn this section, we present results for two additional scenar-ios: (a) when the flows have different minimum bandwidthrequirements and (b) when the arrival time of the flows isstaggered. We use the 6-node, 3-flow scenario used in theprevious section, with the default perceived bandwidth toler-ance of δ = 15% and the default inter-update interval of 100packets.

Figure 13 shows the comparative base IEEE 802.11 andenhanced IEEE 802.11 throughput performance when the 3flows each have different minimum bandwidth requirements.The minimum requirements of the 3 flows are 100 Kbps,200 Kbps and 400 Kbps, respectively. The maximum band-width requirement, 600 Kbps, is the same for all 3 flows.The plots show that while no guarantee can be made withbase IEEE 802.11, we can make coarse guarantees with ourscheme.

While in all our previous scenarios, all participating flowsstarted at around the same time, figure 14 shows the through-put performance of the enhanced IEEE 802.11 scheme whenthe start times are staggered. All simulation parameters areidentical to those in section 3.1.2, except the staggered starttimes and the length of the simulation run, which is set to200 seconds. The bandwidth requirements are identical forall 3 flows, as in section 3.1.2. This plot is similar to figure 1from [5] and figure 11 from [2], which were for a base-stationnetwork with centralized scheduling. We have produced asimilar effect for a single-hop ad hoc network that uses theIEEE 802.11 protocol’s DCF.

3.1.4. TCP experimentsSo far our simulation experiments have focused on multime-dia applications and UDP flows. In this section we investi-gate the behavior of TCP flows and their interactions with theBM scheme. To this end, we simulate three TCP flows, eachrunning between different nodes, in a single-hop ad hoc net-work managed by a BM, i.e., using enhanced IEEE 802.11.

212 SHAH ET AL.



Figure 13. Comparative throughput performance of base and enhanced IEEE802.11 for 3-flow scenario with different minimum bandwidth requirements.

Figure 14. Enhanced IEEE 802.11 performance for 3-flow scenario withstaggered start times.

TCP traffic is best-effort and elastic, so pmin(f ) is set to zeroand pmax(f ) to 100%. As mentioned in section 2.3, differentBp(f ) values derived from the same normalized bandwidthestimate are used for data and acknowledgements, due to theirdifferent packet sizes, when obtaining their respective CTPrequirements. The size of the network interface queue is 50packets, and the maximum congestion window size for a TCPflow is 128 packets. The experiment lasts 200 seconds. Whilefor the UDP experiments, rate-control using the RA is donein the UDP application, in the TCP experiments, queue-basedrate control is done per-node at the network interface queue.The interface queue only releases packets at the rate allottedby the BM.



Figure 15. TCP congestion window behavior when interface queue size issmaller than congestion window limit.

Figure 15(b) shows the congestion window sizes of thethree TCP flows, in the enhanced IEEE 802.11 case. Theyeach expose the same behavior: the window size increaseseach time to 50 packets, cuts back and the cycle repeats. Thisbehavior is due to TCP’s additive-increase multiplicative-decrease (AIMD) congestion control algorithm, where thecongestion window size will decrease only when a packet lossevent is encountered. Packet loss occurs only when the queueoverflows, because of co-ordinated channel access ensured bythe RA. Queue overflow occurs only when congestion win-dow exceeds the maximum queue size. A TCP flow will keepincreasing its congestion window size up to the queuing limit.In fact, this “probing” of congestion window size is TCP’sway of aligning itself to the available bandwidth of the net-work. Without knowing the BM’s allocated rate for this node,a TCP flow has to fill the router queue before it cuts backits congestion window size, which incurs unnecessary longqueuing delay for the packets. However, this behavior doesnot forfeit its allocated bandwidth, as TCP always keeps thequeue non-empty.

As comparison, we run the same TCP experiments overa single-hop ad hoc network without the bandwidth manage-ment, i.e., using base IEEE 802.11. Figure 15(a) shows thatthe congestion window sizes of the three flows follow thesame “saw-tooth” pattern as in figure 15(b). But the maxi-mum window size that each flow can reach may not be ex-actly the same, because of the unfairness in the channel ac-cess, and hence in time to first packet loss, for each queue.


Table 4Performance and throughput loss comparison using TCP with interface queue

size smaller than congestion window limit.

Scheme FM JM Pkts. T’put(pkts.) (pkts.) dropped (total acks recvd.)

Base IEEE 802.11 6.70 9.40 565 45065Enhanced IEEE 802.11 2.12 2.39 33 35698

Figure 16. TCP congestion window behavior when interface queue size islarger than congestion window limit.

Unmanaged release of packets from the queue results in un-equal congestion window growth and causes unfairness. Asa result, the fairness metric (FM) and jitter metric (JM) ofthe flows deteriorates, and the number of dropped packets aresignificantly larger than that in the BM managed scheme, asshown in table 4. The total number of dropped packets isgreater in the base IEEE 802.11 case because an entire win-dow of packets may be dropped at a time before TCP resetsits congestion window size, whereas in the enhanced IEEE802.11 case, a single packet loss results in window reset. Theoverall throughput of the TCP flows in the enhanced IEEE802.11 case, however, is smaller than that in the base IEEE802.11 scenario. This is similar to the result for UDP flowsas shown in section 3.1.2. We also experimented with lessconservative Bp(f ) estimates, which resulted in a decreasein throughput disparity between the base and enhanced IEEE802.11 cases, at the cost of some performance deterioration.Thus the Bp(f ) values can be used to trade-off performance(as measured by the FM and JM) and throughput loss, as withthe UDP experiments.

Another scenario of running TCP over BM is setting eachnode’s interface queuing limit to be larger (150 packets) thanthe congestion window limit (128 packets) of a TCP flow. Werun the experiments for this scenario for 150 seconds. TCP’scongestion window size can never reach the maximum inter-face queue size, and hence there is no packet loss as result ofqueue overflow. In this case, we can expect TCP’s congestionwindow size to stay at its maximum limit without fluctuat-ing, because there is no packet loss at the MAC layer either.Figure 16 shows this behavior. Note that the slow conver-gence speed of TCP’s congestion window size does not im-pact its throughput efficiency, as the interface queue is keptnon-empty at all times. However, in order to minimize queu-ing delay, it is advisable to set TCP’s congestion windowlimit to a small value when running over a bandwidth man-

Table 5Performance and throughput loss comparison using TCP with interface queue

size larger than congestion window limit.

Scheme FM JM Pkts. T’put(pkts.) (pkts.) dropped (total acks recvd.)

Base IEEE 802.11 6.53 8.50 0 33804Enhanced IEEE 802.11 2.51 2.72 0 26577

Figure 17. Single-hop ad hoc network testbed.

aged network. Table 5 compares the fairness performance andthroughput loss for the base and enhanced IEEE 802.11 sce-narios for the case where congestion window limit is less thanthe interface queue size. From the plot in figure 16, it is ob-vious that the throughput disparity, as a percentage, betweenthe base and enhanced IEEE 802.11 cases in this scenario,decreases with time.

3.2. Testbed experiments

We used our testbed experiments to evaluate the throughputperformance and the request-reply delay overhead in the pres-ence of both physical channel errors as well as contentionfrom a limited number of active stations. Our testbed (see fig-ure 17) consisted of 3 IBM ThinkPad laptops, each equippedwith an ORiNOCO PCMCIA 802.11b wireless card config-ured in peer-to-peer ad hoc mode.

We used a rate-adaptive CBR audio streaming applicationover UDP in our testbed experiments. The audio streamingapplication could operate at 5 different QoS levels between32 Kbps and 256 Kbps depending on the available channelcapacity perceived by the TBE. At the maximum QoS (256Kbps), all audio samples were transmitted while at lower lev-els fewer samples were sent, and the audio was reconstructedthrough interpolation at the receiver. The purpose of the test-bed experiments was to study the feasibility of our scheme ina testbed with a realistic single-hop ad hoc network environ-ment. The RA in the application and the TBE communicatedvia the /proc interface.

214 SHAH ET AL.

Figure 18. Indoor testbed experiment plot.

3.2.1. Throughput performanceWe conducted two throughput experiments, one indoors andone outdoors. In each case, we started some unmanaged pingsessions, as shown in figure 17, to bring about contention.The ping ICMP packet transmission on the channel also ar-tificially reduced its bandwidth so that the bandwidth per-ceived by the audio streaming application actually fluctuatedbetween 32 and 256 Kbps depending on the physical errors.In the absence of the pings, the reduction in perceived band-width brought about by the physical errors alone was not suf-ficient to cause the audio streaming application to adapt itsquality. The physical errors, at their worst, reduced the per-ceived bandwidth by a few hundreds of Kbps. Given a 2 Mbpschannel and an application with a peak rate of 256 Kbps,these errors thus had no effect on the application. Its qual-ity level did not fluctuate. To bring about adaptation on thepart of the application, the physical errors had to vary theavailable channel capacity for the application between 32 and256 Kbps. Hence, we used the pings to contend with the ap-plication for the channel and thus artificially reduce the avail-able channel capacity it perceives to the necessary range. Thepings brought down the available channel capacity to around500 Kbps so that fading and interference errors could then re-duce it further below the 256 Kbps threshold needed for theapplication to adapt.

Figure 18 shows the throughput performance for the in-doors scenario. On the x-axis is time in 45 second units. They-axis shows the adaptation of the audio streaming applica-tion, between 32 and 256 Kbps, to the change in availablechannel capacity. The channel bit-rate was fixed at 2 Mbpsat the network cards. The perceived bandwidth variation-tolerance was set at δ = 15% and the inter-update intervalwas 100 packets. The BM was located on the same machineas the sender, 12.0.0.11.

The flurry of re-negotiations with the BM on the left-hand side of the plot corresponds to our moving the sender(12.0.0.11) down to a secluded portion of the basement of thebuilding while the receiver (12.0.0.12) and the third laptop(12.0.0.10) remained in the lab on the second floor. Whilein the basement, the sender moved around, down narrow cor-ridors, over staircases and through fire doors. As the levelof fading and interference changed drastically, the perceivedchannel capacity also changed drastically and hence the flurryof channel time re-negotiations. The contending pings alsowere affected by the physical errors and produced variablecontention, thus inducing even greater instability in the appli-cation QoS.

We then brought the sender back to the second floor, theperceived bandwidth returned to around 500 Kbps, and thequality of the audio returned to its maximum. We then placed


Figure 19. Outdoor testbed experiment plot.

the sender and receiver next to each other so that physical er-rors were rare. The 3 dips in the graph on the right-hand sidecorrespond to experiments with no physical errors, but 3 dif-ferent levels of contention due to 3 different ping rates. All3 of these ping rates were greater than those used for the firstpart of this experiment. In the first part, the pings reduced theavailable channel capacity to around 500 Kbps and the phys-ical errors dragged it further down. In this part there wereno physical errors, but the larger ping rates themselves tookthe available channel capacity below 256 Kbps, causing re-negotiation from the application. The purpose of this experi-ment with no physical errors was to demonstrate the effect ofthe contending ping sessions: they produce a reduction in theperceived available channel capacity of the managed audiostreaming application, in a controlled fashion, and the reduc-tion is a constant one.

Next, we performed another set of experiments outdoors.The channel bit-rate was set at 5.5 Mbps for this experiment.As before, we had pings produce contention to artificially re-duce available channel capacity for the audio streaming flow.Other parameters such as the value of δ and the inter-updateinterval were the same as in the indoor experiment. In theoutdoor scenario, we used only two of the laptops. The BMwas once again co-located with the sender, 12.0.0.11. At

the start of the experiment, the sender 12.0.0.11 and the re-ceiver 12.0.0.12 were next to each other on the sidewalk ofa street. Then, keeping the receiver 12.0.0.12 stationary onthe sidewalk, the sender 12.0.0.11 was moved away by a per-son walking at a normal pace down the street on the sidewalk.When the sender was around 150 meters away, the availablechannel capacity perceived by the audio flow began fluctu-ating due to signal fading effects. This resulted in a flurry ofre-negotiations shown in figure 19. The sender then wanderedfor a while around the point 150 meters away before return-ing to the starting position. As the sender moved closer to thereceiver, at one point, the available channel capacity returnedto its ping-induced constant level and the application returnedto its highest quality level.

We repeated our experiments using ARS (auto rate selec-tion) feature of the wireless card, instead of using constantrates 2 Mbps and 5.5 Mbps mentioned above. Our resultswere very similar when using ARS as compared to whenusing fixed rates. We also experimented with the BM atthe destination node, with no change in performance. Therequest-reply delay overhead for re-negotiation requests doesnot affect performance much because the application paral-lelly continues transmitting at its previously allotted CTP un-til the re-negotiation reply arrives, a few milliseconds later.

216 SHAH ET AL.

3.2.2. Request-reply delayThe request-reply delay is the time delay between the send-ing of a request message and the receipt of a reply mes-sage. All our control messages had a 32-byte payload. Thisexchange of messages occurs both during flow establishmentas well as when perceived bandwidth changes significantly.We set the bandwidth of the network to be 5.5 Mbps, as inthe case of the outdoor experiment. We used all 3 laptops forthe request-reply delay experiments, with 12.0.0.11 being thesender, 12.0.0.12 being the receiver and the BM being locatedon 12.0.0.10. We found that, if there is no contention, eachrequest-reply round-trip took 23 ms on average. In the pres-ence of ping-induced contention, each request-reply round-trip took 61 ms on average. Flow establishment occurs onlyonce per flow, obviously, and if the perceived bandwidth doesnot change much, then the 20–60 ms request-reply delay is asmall one.

4. Related work

In this section we discuss two areas of related work: (a) cen-tralized channel allocation, and (b) distributed fair schedulingin single-hop and multi-hop wireless networks.

In wireless network environment, past research has fo-cused on flow scheduling at the access-point to achieve cer-tain fairness criteria between flows competing for the wirelesschannel [5,7,15]. Bianchi et al. [5] proposed the “utility fair”criteria in bandwidth allocation, where each user’s bandwidthis allocated in such a way that their individual utility is equal-ized. It assumes that the central manager at the base stationhas exact knowledge of the asymptotic utility curves of all theapplications, which might be difficult to obtain. The flows inour scheme can specify a simple linear utility curve using justtwo points. In our scheme, the BM guarantees a minimumbandwidth for each flow, and allots the rest of the channelcapacity in a max–min fashion to each flow up to its maxi-mum request. We believe our approach is simple yet effectivein a smart-room where random users walk up to the room andshare the wireless channel. Another difference is that we use adistributed peer-to-peer transmission (details in section 2.1),rather than an access-point model, in allocating the channelresources.

Another wireless network channel allocation scheme is theeffort-limited fair scheduling by Eckhardt and Steenkiste [9].It adjusts the “air time” of a flow to meet its minimum band-width requirement in response to channel error rates, only upto a certain factor (called the “power factor”), to avoid starv-ing other best-effort flows. The usage of air time to measurethe bandwidth requirement of a flow is similar to the “chan-nel time” in our scheme. However, it is unclear how the powerfactor can be chosen for different flows because this will givepreferential treatment to certain users. In our scheme, whena flow’s minimum requirement cannot be satisfied, the flowis simply rejected. This creates incentive for the users to seta minimum channel time requirement as small as possible toreduce the possibility of being denied access to the channel.

In [25], the authors propose an admission control schemefor a peer-to-peer, single-hop, ad hoc wireless network modelsimilar to the one we have used. Their scheme requires the useof special probe packets to obtain the service curve, which isan estimate of network load. Using the service curve, one-time admission control is performed. In contrast, our schemeestimates network load using the data packets of the connec-tion itself. Moreover, we perform dynamic bandwidth re-negotiation over the course of the connection, in addition toadmission control at flow startup.

Another area of related work is the distributed weightedfair scheduling (DWFS) schemes in single-hop and multi-hop wireless networks [3,13,18,19,24]. As mentioned before,our bandwidth management scheme is required to assist theDWFS scheme when it is available. At the same time, asshown in our experiments, our scheme also works well with-out any underlying DWFS schemes. This is a very importantfeature because today’s IEEE 802.11 network interface cardonly implements the standard DCF MAC protocol withoutany DWFS extensions. Therefore, our bandwidth manage-ment scheme, of which we already have a working prototype,is highly deployable in today’s smart-rooms.

5. Conclusion

In this paper, we presented an Admission Control schemeto determine what fraction of channel time each flow ina single-hop ad hoc wireless network receives. To thisend, we mapped the bandwidth requirement at the applica-tion/middleware layer to a channel time proportion (CTP)requirement at the MAC layer. We presented an applica-tion/middleware layer rate control mechanism to ensure thatflows conform to their respective CTPs. Since one-time ad-mission control is not sufficient to handle the changes in net-work and flow characteristics, we also presented a DynamicBandwidth Management system that adapts the flows’ respec-tive CTPs during the course of their operation. The adaptationcan be a response to change in the network environment orchange in a particular flow’s traffic characteristics. The sim-plicity and robustness of our system enables the incorporationof elegant pricing and security features into it. We have devel-oped a prototype implementation of the system and we haveused this implementation in a testbed, in addition to extensivesimulations, to demonstrate the feasibility and utility of ourscheme.

References

[1] G. Ahn, A. Campbell, A. Veres and L. Sun, Swan: Service differen-tiation in stateless wireless ad hoc networks, in: Proceedings of IEEEInfoCom, New York (June 2002).

[2] O. Angin, A. Campbell, M. Kounavis and R. Liao, The mobiwaretoolkit: Programmable support for adaptive mobile networking, IEEEPersonal Communications, Special Issue on Adapting to Network andClient Variability 5(4) (1998) 32–44.

[3] B. Bensaou, Y. Wang and C. Ko, Fair medium access in 802.11 basedwireless ad hoc networks, in: Proceedings of IEEE MobiHoc, Boston,MA (August 2000).


[4] D. Bertsekas and R. Gallager, Data Networks, 2nd ed., chapter 6(Prentice-Hall, 1992).

[5] G. Bianchi, A. Campbell and R. Liao, On utility-fair adaptive servicesin wireless networks, in: Proceedings of IEEE/IFIP IWQoS, Napa, CA(May 1998).

[6] F. Cali, M. Conti and E. Gregori, Dynamic tuning of IEEE 802.11 pro-tocol to achieve a theoretical throughput limit, IEEE/ACM Transactionson Networking 8(6) (2000) 785–799.

[7] T. Chen, P. Krzyzanowski, M. Lyu, C. Sreenan and J. Trotter, A sum-mary of Qos support in SWAN, in: IEEE/IFIP IWQoS 1998, Napa, CA(May 1998).

[8] S. Czerwinski, B. Zhao, T. Hodes, A. Joseph and R. Katz, An archi-tecture for a secure service discovery service, in: Proceedings of ACMMobiCom, Seattle, WA (August 1999).

[9] D. Eckhardt and P. Steenkiste, Effort-limited fair (ELF) scheduling forwireless networks, in: Proceedings of IEEE InfoCom, Tel Aviv, Israel(March 2000).

[10] F. Fitzek and M. Reisslen, Mpeg-4 and h.263 video tracesfor network performance evaluation, IEEE Network Magazine15(6) (2001) 40–54, http://www-tkn.ee.tu-berlin.de/research/trace/stat.html

[11] M. Grossglauser, S. Keshav and D. Tse, Rcbr: A simple and efficientservice for multiple time-scale traffic, in: Proceedings of ACM Sig-Comm, Cambridge, MA (August 1995).

[12] E. Guttman, C. Perkins, J. Veizades and M. Day, Service location pro-tocol, version 2, RFC 2608 (June 1999).

[13] V. Kanodia, C. Li, A. Sabharwal, B. Sadeghi and E. Knightly, Distrib-uted multi-hop scheduling and medium access with delay and through-put constraints, in: Proceedings of ACM MobiCom, Rome, Italy (July2001).

[14] M. Kazantzidis, M. Gerla and S. Lee, Permissible throughput networkfeedback for adaptive multimedia in AODV Manets, in: Proceedings ofIEEE ICC, Helsinki, Finland (June 2001).

[15] J. Kim and M. Krunz, Bandwidth allocation in wireless networks withguaranteed packet-loss performance, IEEE/ACM Transactions on Net-working 8(3) (2000) 337–349.

[16] A. Kuznetsov, Linux traffic control (tc), http://www.sparre.dk/pub/linux/tc

[17] R. Liao, R. Wouhaybi and A. Campbell, Incentive engineering in wire-less LAN based access networks, in: Proceedings of IEEE ICNP, Paris,France (November 2002).

[18] H. Luo, S. Lu and V. Bharghavan, A new model for packet schedul-ing in multihop wireless networks, in: Proceedings of IEEE MobiCom,Boston, MA (August 2000).

[19] H. Luo, P. Medvedev, J. Cheng and S. Lu, A self-coordinating approachto distributed fair queuing in ad hoc wireless networks, in: Proceedingsof IEEE InfoCom, Anchorage, AK (April 2001).

[20] P. Marbach and R. Berry, Downlink resource allocation and pricing forwireless networks, in: Proceedings of IEEE InfoCom, New York (June2002).

[21] M. Mirhakkak, N. Schult and D. Thomson, Dynamic bandwidth man-agement and adaptive applications for a variable bandwidth wirelessenvironment, IEEE JSAC 19(10) (2001) 1984–1997.

[22] Y. Qiu and P. Marbach, Bandwidth allocation in ad-hoc networks:A price-based approach, in: Proceedings of IEEE InfoCom, San Fran-cisco, CA (March–April 2003).

[23] The network simulator ns-2, http://www.isi.edu/nsnam/ns/,updated October 2001.

[24] N. Vaidya, P. Bahl and S. Gupta, Distributed fair scheduling in a wire-less LAN, in: Proceedings of ACM MobiCom, Boston, MA (August2000).

[25] S. Valaee and B. Li, Distributed call admission control in wireless adhoc networks, in: Proceedings of IEEE VTC, Vancouver, Canada (Sep-tember 2002).

[26] Y. Xue, B. Li and K. Nahrstedt, Price-based resource allocation in wire-less ad hoc networks, in: Proceedings of IEEE IWQoS, Monterey, CA(June 2003).

Samarth Shah received his B.E. degree in com-puter science and engineering from the University ofMadras, India, in 1998. He is currently a Ph.D. can-didate in the Department of Computer Science at theUniversity of Illinois at Urbana-Champaign. His in-terests include quality of service (QoS) in wirelessnetworks.E-mail: [email protected]

Kai Chen received his B.Eng. degree in computerscience from Tsinghua University, Beijing, China, in1995, and M.S. degree in computer science from theUniversity of Delaware, Newark, Delaware, USA, in1998. Currently he is a Ph.D. candidate in the De-partment of Computer Science at the University ofIllinois at Urbana-Champaign. From 1998 to 2000,he worked as a research programmer at the NationalCenter for Supercomputing Applications (NCSA) atUrbana, IL, USA. His research interests include mo-

bile ad hoc networks, transport layer issues in mobile networks, quality ofservice, incentive engineering, and pervasive computing.E-mail: [email protected]

Klara Nahrstedt is an Associate Professor at theUniversity of Illinois at Urbana-Champaign, Com-puter Science Department. Her research interests aredirected towards multimedia middleware systems,Quality of Service (QoS), QoS routing, QoS-awareresource management in distributed multimedia sys-tems, and multimedia security. She is the coauthor ofthe widely used multimedia book Multimedia: Com-puting, Communications and Applications publishedby Prentice Hall, the recipient of the Early NSF Ca-

reer Award, the Junior Xerox Award, and the IEEE Communication Soci-ety Leonard Abraham Award for Research Achievements. She is the editor-in-chief of ACM/Springer Multimedia Systems Journal, and the Ralph andCatherine Fisher Associate Professor. Klara Nahrstedt received her B.A. inmathematics from Humboldt University, Berlin, in 1984, and M.Sc. degreein numerical analysis from the same university in 1985. She was a researchscientist in the Institute for Informatik in Berlin until 1990. In 1995 she re-ceived her Ph.D. from the University of Pennsylvania in the Department ofComputer and Information Science.E-mail: [email protected]


High Speed Networking Security:Design and Implementation of Two New DDP-Based Ciphers

N. SKLAVOSElectrical & Computer Engineering Department, University of Patras, Patras 26500, Greece

N.A. MOLDOVYANSpecialized Center of Program Systems, SPECTR, Kantemirovskaya Str. 10, St. Petersburg 197342, Russia

O. KOUFOPAVLOUElectrical & Computer Engineering Department, University of Patras, Patras 26500, Greece

Abstract. Using Data-Dependent (DD) Permutations (DDP) as main cryptographic primitive two new ciphers are presented: ten-roundCobra-H64, and twelve-round Cobra-H128. The designed ciphers operate efficiently with different plaintext lengths, 64 and 128-bit, forCobra-H64 and Cobra-H128, respectively. Both of them use very simple key scheduling that defines high performance, especially in the caseof frequent key refreshing. A novel feature of Cobra-H64 and Cobra-H128 is the use of the Switchable Operations which prevent the weakkeys. The offered high-level security strength does not sacrifice the implementation performance, of both ciphers. Architecture, designand hardware implementation of the two ciphers are presented. The synthesis results for both FPGA and ASIC implementations provethat Cobra-H64 and Cobra-H128 are very flexible and powerful new ciphers, especially for high-speed networks. The achieved hardwareperformance and the implementation area cost of Cobra-H64 and Cobra-H128 are compared with other ciphers, used in security layers ofwireless protocols (Bluetooth, WAP, OMA, UMTS and IEEE 802.11). From these comparisons it is proven that the two proposed are flexiblenew ciphers with better performance in most of the cases, suitable for wireless communications networks of present and future.

Keywords: networking security, data-dependent permutations, Cobra-H64, Cobra-H128, encryption

1. Introduction

Security is a primary requirement of any wired and wirelesscommunication. Encryption algorithms are meant to providesecure communications applications. However, if the sys-tem is not designed property, it may fail. New encryptionalgorithms have to perform efficiently in a variety of cur-rent and future applications, doing different encryption tasks.All hardware implementations have to be efficient, with theminimum allocated number of logic gates. This means sim-plicity in cipher’s architectures with enough “clever” datatransformation components. The implementation of a com-munication protocol, demands low power devices and fastcomputation components which imply that the number andcomplexity of the encryption operations should be kept assimply as possible. A basic transformation in the operation oftoday’s ciphers is needed, including transformations in bothdata and key blocks size. The ciphers of the near future haveto be key agile. Many applications need a small amount oftext to be encrypted with keys that are frequently changed.Many well-known applications, like IPsec, use this way of ci-pher’s operation. Although the most widely used mode of op-eration is encryption with the same key for all the amount oftransport data, the previous mode is also very useful for futureapplications. Ciphers that requiring subkeys precomputationhave a lower key agility due to the precomputation time, andthey also require extra RAM to hold the precomputed sub-

keys. This RAM requirement does not exist in the implemen-tations of encryption algorithms, which compute their keysduring the encryption/decryption operation. Cellular phonestechnology demands specific characteristics of the cryptog-raphy science. Ciphers have to be compatible with wirelessdevices restricted standards in hardware resources.

Data-Dependent (DD) Permutations (DDP) performedwith so called Controlled Permutation (CP) boxes [6,14,16]appears to be very efficient cryptographic primitive for fasthardware encryption. Security estimation of the DDP-basedciphers CIKS-1 [11] and SPECTR-H64 [10] against linearcryptanalysis has shown that DDP are efficient, provided theyare combined with other non-linear operations. The DDP-based ciphers are proposed for hardware implementation withlow cost.

In this paper we present two new DDP-based ciphersCobra-H64 and Cobra-H128 and the results of their hard-ware implementations. The design of the presented cipherstakes into account some recommendations arising from thelinear and differential analysis of other DDP-based ciphers[5,10,11]. Both proposed ciphers Cobra-H64 and Cobra-H128 have been implemented in ASIC and FPGA hardwaremodules. Two different VLSI architectures are examined foreach one of the proposed ciphers. The synthesis results of allhardware integrations are presented in detail.

The paper is organized in the following way. In section 2we consider construction of the controlled operational boxes

220 SKLAVOS ET AL.

performing DDP. We present the design criteria and we de-scribe the structure of the two new DDP-based block ciphers:ten-round Cobra-H64 with 64-bit data input and twelve-roundCobra-H128 with 128-bit data input. Section 3 describes indetails the encryption algorithm Cobra-H64. A feature of thisiterative cryptosystem is the use of two sellula-automaton-likenon-linear operations in its round transformation, as an addi-tional primitive. Section 4 describes the cipher Cobra-H128that has structure similar to the structure of Cobra-H64. Insection 5 we discuss the key scheduling and present resultson security estimation against differential analysis. Section 6presents the hardware implementations cost and performance(FPGA and ASIC). Comparisons of the proposed implemen-tations, of both Cobra-H64 and Cobra-H128, with other blockciphers are given. Finally conclusions and observations arediscussed in the last section.

2. Design of the controlled permutations

Controlled permutations can be easy performed with wellknown interconnection networks (IN) [1,3] which were pro-posed to construct key-dependent permutations [4,17]. How-ever such use of IN do not effectively thwarts differentialcryptanalysis [20]. Regarding cryptographic applications itis more attractive to use IN to perform DDP on data sub-blocks [15] and subkeys [12]. An operational box Pn/m, per-forming permutations on n-bit binary vectors depending onsome controlling m-bit vector V , is called Controlled Per-mutation box (CPB). In the case that the controlling vectordepends on a data subblock, the CP box performs DDP. Thefast CP boxes can be constructed using elementary switchingelements P2/1, figure 1(a), as elementary building blocks per-forming controlled transposition of two one-bit inputs x1 andx2. In general case, each P2/1-box is controlled with one bitν and forms two-bit output (y1, y2), where y1 = x1+ν andy2 = x2−ν .

Taking into account that it is very desirable to mini-mize the time delay, while performing CP-box permutations,the layered topology of IN can be considered as the main onesince it permits to design very fast CPB. Layered CPB areconstructed as superposition of S = 2m/n active layers, sep-arated with S − 1 fixed permutations π1, . . . , πS−1 that areimplemented in hardware as simple connections. Each activelayer, figure 1(b), in a CPB with n-bit input is represented bythe set of n/2 parallel elementary boxes P2/1. The generalstructure of the layered CPB is shown in figure 1(c). Its nota-tion is presented in figure 1(d). In all figures of this paper, thesolid lines indicate data movement, while the dotted lines cor-responding to CP-boxes indicate controlling bits. A CP-boxinverse of the box Pn/m is denoted as P−1

n/m.We assume that in a layered CP-box all elementary switch-

ing elements are consecutively numbered from left to rightfrom top to bottom and the j th bit of vector V controls thej th switching element P2/1. In accordance with the num-ber of layers the vector V can be represented as concate-nation of 2m/n vectors V1, V2, . . . , V2m/n ∈ {0, 1}n/2, i.e.,V = (V1, V2, . . . , V2m/n).

Figure 1. (a) P2/1-box, (b) structure of one active layer, (c) general structure

of the layered CP boxes and (d) P−1n/m-box.

Controlled permutations performed with the box Pn/m,can be characterized using an ordered set of the modifi-cations {�0,�1, . . . ,�2m−1}, where each modification �i ,i = 0, 1, . . . , 2m − 1, is a fixed permutation of some n-bitsets. Permutations �i are called CP-modifications.

Notation. Let {0, 1}n denote the set of all n-bit binary vec-tors X = (x1, . . . , xn). Let X denote also decimal value (orsimply value) of the vector X: X = ∑n

i=1 xi2i−1.Let X ⊕ Y denote the bit-wise XOR operation performed

on X,Y ∈ {0, 1}n and XY denote bit-wise AND operation.Let Y = X≪k denote the cyclic rotation of the word X

by k bits (0 � k < n), where Y = (y1, . . . , yn) is the output,∀i ∈ {1, . . . , n − k} we have yi = xi+k , and ∀i ∈ {n − k +1, . . . , n} we have yi = xi+k−n.

Let Xl = (x1, . . . , xn/2) and Xl = (xn/2+1, . . . , xn) de-note the least and the most significant bits of X ∈ {0, 1}n.

Definition. CP-boxes Pn/m and P−1n/m are mutual inverses, if

for all possible values of the vector V the corresponding CP-modifications �V and �−1

V are mutual inverses.

One active layer can be considered as some single-layerCP box Sn. It is evidently that P2/1 = P−1

2/1, therefore

Sn = S−1n . A layered CP box Pn/m can be represented as

superposition Pn/m = S(V1) ◦ π1 ◦ S(V2) ◦ π2 ◦ · · · ◦ πs−1 ◦S(V2m/n). The respective box P−1

n/m has the following struc-

ture P−1n/m = S(V2m/n) ◦ π−1

2m/n−1 ◦ S(V2m/n−1) ◦ π−12m/n−2 ◦ · · · ◦

π−11 ◦ S(V1). Thus, to construct inverse of the CP-box Pn/m it

is sufficient to number the boxes P2/1 from left to right frombottom to top and to replace πi by π−1

2m/n−i . We shall assume

that in the boxes P−1n/m switching elements P2/1 are consecu-

tively numbered from left to right from bottom to top. Notethat the vector Vj corresponding to the j th active layer in the

HIGH SPEED NETWORKING SECURITY 221

Figure 2. Structure of boxes: (a) P8/12 and (b) P−18/12.

Figure 3. Structure of the CP-boxes: (a) P32/96 and (b) P−132/96.

box Pn/m controls the (2m/n − j + 1)th active layer in P−1n/m

(see figure 2).The cipher Cobra-H64 (Cobra-H128) uses the boxes P32/96

and P−132/96 (P64/192 and P−1

64/192). Each one of them isconstructed using four (eight) parallel boxes P8/12 and four(eight) parallel boxes P−1

8/12 which are shown in figure 3 (fig-

ure 4). The P8/12 boxes are connected with the P−18/12 boxes in

accordance to the principal “each to each”.While designing the single key encryption algorithms

Cobra-H64 and Cobra-H128 our strategy was oriented to theextensive use of the controlled permutations that are very fastand with low cost for hardware implementation. Our designcriteria were the following:

1. The encryption algorithm should be an iterated 64-bit or128-bit block cipher.

2. The cipher should be fast, in the case of frequent key re-freshing. Therefore the encryption algorithm should beable to perform encryption and decryption with simple andfast change of the used subkeys sequence.

3. Round transformations of data subblocks should be char-acterized by high parallelism.

Figure 4. Structure of the CP-boxes: (a) P64/192 and (b) P−164/192.

4. Except DDP some additional non-linear operation shouldbe used in the round transformation.

The encryption/decryption schemes of Cobra-H64 andCobra-H128 are described by the following formulas: C =T(e=0)(M,K) and M = T(e=1)(C,K), where M is the plain-text, C is the ciphertext (M,C ∈ {0, 1}64 for Cobra-H64and M,C ∈ {0, 1}128 for Cobra-H128). K is the secret key,T is the transformation function, and e ∈ {0, 1} is a parame-ter defining encryption (e = 0) or decryption (e = 1) mode.The secret key is considered as concatenation of four subkeys:K = (K1,K2,K3,K4). For i = 1, 2, 3, 4 Ki ∈ {0, 1}32 forCobra-H64 and Ki ∈ {0, 1}64 for Cobra-H128. The ciphersuse no preprocessing to generate subkeys. The extended keyQ(e) is formed as simple sequence of subkeys Ki taken inrespective order.

The both ciphers use the same iterative structure which isshown in the following figure 5. Encryption begins with theInitial Transformation. Then r rounds of data transformationare based on procedure Crypt(e), followed by the Final Trans-formation. First, the data input X is divided to subblocks L

and R. Then Initial Transformation is executed which per-forms XOR-ing each between the data subblocks and two dif-ferent subkeys: L0 = L ⊕ O3 and R0 = R ⊕ O4.

The encryption procedure is performed in accordance withthe following pseudo-algorithm:

For j = 1 to r − 1 do:Execute transformation:

(Lj , Rj ) = Crypt(e)(Lj−1, Rj−1,Q(e)j );

Swap the data subblocks:Rj = T , Rj := Lj , Lj := T ;

End For loop;Execute transformation:

(Lr , Rr) = Crypt(e)(Lr−1, Rr−1,Q(e)r ).

Procedure Crypt(e) represents round encryption function,where Q

(e)j is the round key, used in the j th encryption round.

Encryption finishes with procedure of final transformation:L′ = Lr ⊕ O1 and R′ = Rr ⊕ O2. The ciphertext block isY = (L′, R′). The ciphers use different procedures Crypt(e)

222 SKLAVOS ET AL.

Figure 5. Structure of Cobra-H64 and Cobra-H128.

and different numbers of rounds r = 10 for Cobra-H64 andr = 12 for Cobra-H128.

3. The block cipher Cobra-H64

3.1. Formation of the round keys

Each of the round keys Q(e)j consists of four e-dependent

round subkeys A(1), A(2), A(3), A(4) ∈ {0, 1}32, i.e., Q(e)j =

(A(1), A(2), A(3), A(4))(e)j . Table 1 specifies the rounds sub-

keys and their correspondence to the secret key.For each one of the ten rounds each of these subkeys is

used while performing two operations G. Taking into ac-count that the three inputs of the operation G are different, therole of each subkey changes from one round to another one.While data decryption the subkeys are generated as simpleswapping subkeys Qj with single-layer box P(e)

128/1 which is

represented by two parallel boxes P(e)64/1 (e = 0 for encryption

and e = 1 for decryption). The box P(e)64/1 is some single-layer

CP box in which all elementary switching elements are con-trolled with the same bit e. The pairs (K1,K3) and (K2,K4)are inputs of the corresponding boxes P(e)

64/1 (figure 6). Four

32-bit outputs of two boxes P(e)64/1 are the e-dependent sub-

keys Oi (i = 1, 2, 3, 4). Thus, we have Oi = Ki , if e = 0,and O1 = K3, O2 = K4, O3 = K1, O4 = K2, if e = 1.

Correct change of the encryption mode for the decryptionone is also defined by the respective change of the fixed per-mutation π(e) in procedure Crypt(e) presented in figure 7.

Table 1Specification of the subkeys A(i) in Cobra-H64.

j = 1 2 3 4 5 6 7 8 9 10

A(1)j

= O1 O4 O3 O2 O1 O1 O2 O3 O4 O1

A(2)j = O2 O1 O4 O3 O4 O4 O3 O4 O1 O2

A(3)j = O3 O2 O1 O4 O3 O3 O4 O1 O2 O3

A(4)j = O4 O3 O2 O1 O2 O2 O1 O2 O3 O4

Figure 6. Swapping of subkeys.

Figure 7. Procedure Crypt(e) in Cobra-H64.

3.2. Data-dependent permutations

To perform DDP Cobra-H64 uses CP boxes P(V )32/96 and

(P−132/96)

(V ′) described in section 2 (figure 3). Controlling vec-tors for these CP boxes are formed using the same extensionbox E implemented with simple connections. The input ofthe E-box is the current value of the left data subblock L. Letthe vector V = (V1, V2, V3, V4, V5, V6) be the 96-bit outputof the E-box. The extension box provides the following rela-tions:

V1 = Ll, V2 = L≪6l , V3 = L≪12

l ,

V4 = Lh, V5 = L≪6h , V6 = L≪12

h .

Inverting a bit in L causes the inversion of three bits of V .Thus, each bit of L influences three boxes P2/1 in the box

P(V )32/96 and three P2/1-boxes in (P−1

32/96)(V ′). While designing

the box E we used the following criterion.


Figure 8. Structure of the switchable permutation.

Criterion. For arbitrary given vector L the permutation ofeach input bit of the CP box must be defined by six differentbits of L.

Due to realization of this criterion each bit of L influencesexactly six bits of R. Such distribution of the controlling bitsprovides that arbitrary input bit of the boxes P32/96 and P−1

32/96moves to each output position with the same probability pro-vided L is a uniformly distributed random variable.

3.3. Switchable fixed permutation π(e)

Switchable fixed permutation π(e) performs permutation π(0)

when enciphering, and π(1) when deciphering. Change of thepermutation is performed as simple swapping outputs of thefixed permutations π(0) and π(1) with single-layer box P(e)

64/1

(see figure 8). Permutations π(0) and π(1) contain two cy-cles. The first cycle corresponds to identical permutation ofthe most significant input bit x32. The second cycle is de-scribed by the following equations:

π(0)(x1, x2, . . . , x31) = (x1, x2, . . . , x31)≪5,

π(1)(x1, x2, . . . , x31) = (x1, x2, . . . , x31)≪26.

The role of the fixed permutation, with such structure, is toprovide each input bit of the CP-box P(V )

32/96 influences each

output bit of the CP-box (P−132/96)

(V ′). Indeed, let consider

the case that V = V ′. Each input bit of the P(V )32/96-box with

the same probability moves to each input digit of the opera-tion π(e). If it moves to the most significant digit, then it re-turns to its initial digit at the output of the CP box (P−1

32/96)(V ′).

If it moves to arbitrary other digit at the input of π(e), then itcan be moved in all output digits of the (P−1

32/96)(V ′)-box ex-

cept its initial position.Thus the permutation π(e) improves the resultant DDP cor-

responding to performing sequential operations P(V )32/96 and

(P−132/96)

(V ′). Indeed even in the case of V = V ′ the super-

position P(V )32/96 ◦π(e) ◦ (P−1

32/96)(V ′) forms an effective CP-box

permutation, and all modifications are permutations havingthe same cycle structure (all modifications contain one cyclewith length 1 and one cycle with length 31).

Actually, during encryption in general case we haveV �= V ′, since after execution of the operation P(V )

32/96 the

permutation I is performed on L. Investigating the role ofthe fixed permutation between two mutually inverse CP-boxoperations we have performed many statistic experiments.These experiments have shown that the use of such permu-tation, significantly improves the properties of the transfor-mation performed with two mutually inverse CP-boxes. Suchfixed permutation defines some internal mechanism of the op-timization of the distribution of the influence of the left datasubblock on the elementary switching elements of the bothCP-boxes.

3.4. Permutational involution I

Permutational involution I performed on the left data sub-block is used to strengthen the avalanche effect. Let yi′ andyj ′ be the output bits corresponding to the input bits xi andxj . To design involution I we have used the following twocriteria:

(i) ∀i, j : |j − i| � 3 should be |j ′ − i ′| � 4;

(ii) ∀i should be |i − i ′| � 6.

Integers 4 and 6 are selected in order to define each bit ofthe left subblock influences as many as possible bits after theoutputs of the both operations G be XORed with R. Thesecriteria are satisfied by the involution

I = (1, 17)(2, 21)(3, 25)(4, 29)(5, 18)(6, 22)(7, 26)(8, 30)

(9, 19)(10, 23)(11, 27)(12, 31)(13, 20)(14, 24)(15, 28)

(16, 32).

For example, Out(1) = In(17) and Out(17) = In(1). Theinvolution I provides that changing one bit of the left datasubblock causes inversion from 2 to 8 bits of the right datasubblock after the outputs of both operations G are XORedwith R.

3.5. Non-linear operation G

Operation GA′A′′(L) is described by the following expression:

W = L0 ⊕ A′0 ⊕ L2L3 ⊕ L1L2 ⊕ L1L3 ⊕ L2A

′′1 ⊕ A′

1L3

⊕ A′′0L1L2,

where binary vectors Lj , A′j , and A′′

j are expressed as fol-lows:

L0 = L = (l1, l2, . . . , l32),

L1 = (1, l1, l2, . . . , l31),

L2 = (1, 1, l1, l2, . . . , l30),

L3 = (1, 1, 1, l1, l2, . . . , l29),

A′0 = A′ = (

a′1, a

′2, . . . , a

′32

),

A′1 = (

1, a′1, a

′2, . . . , a

′31

),

A′′0 = A′′ = (

a′′1 , a′′

2 , . . . , a′′32

),

A′′1 = (

1, a′′1 , a′′

2 , . . . , a′′31

),

A′′2 = (

1, 1, a′′1 , a′′

2 , . . . , a′′30

).

224 SKLAVOS ET AL.

Figure 9. Round function Crypt(e) of Cobra-H128.

Table 2Specification of the subkeys A(i) in Cobra-H128.

j = 1 2 3 4 5 6 7 8 9 10 11 12

A(1)j

= O1 O4 O3 O2 O1 O3 O3 O1 O2 O3 O4 O1

A(2)j

= O2 O3 O4 O1 O2 O4 O4 O2 O1 O4 O3 O2

A(3)j

= O3 O2 O1 O4 O3 O1 O1 O3 O4 O1 O2 O3

A(4)j

= O4 O1 O2 O3 O4 O2 O2 O4 O3 O2 O1 O4

4. The block cipher Cobra-H128

Procedure Crypt(e) of Cobra-H128 is presented in figure 9.

4.1. Formation of the round keys

Each one of the twelve round keys Q(e)j consists of four

e-dependent round subkeys A(1), A(2), A(3), A(4) ∈ {0, 1}64,i.e. Q

(e)j = (A(1), A(2), A(3), A(4))

(e)j . Each of the sub-

keys A(1), A(2), A(3), and A(4) is specified in table 2 viakey elements O1,O2,O3,O4 as one of the subkeys Ki (i ∈{1, . . . , 4}) depending on j ∈ {1, . . . , 12} and e ∈ {0, 1}.Each one of the subkeys Ki is used while performing twooperations (one CP-box operation and one operation G), foreach one of the twelve rounds. Note that the role of each sub-key changes from one round to another one.

Change of the ciphering mode is performed using twomechanisms: (i) changing key scheduling and (ii) swap-ping vector L and output �(L) of the permutation � be-fore fulfilling two operations G. To provide correct changeof the transformation mode ∀j ∈ {1, . . . , 12} the condi-tion (A(1), A(2), A(3), A(4))

(1)j = (A(3), A(4), A(1), A(2))

(0)13−j

must be hold. Fast change of the key scheduling is performed

Figure 10. (a) Swapping subkeys and (b) vectors L and �(L).

with two single-layer CP-boxes P(e)128/1. Each of the CP-boxes

P(e)128/1 contains 64 parallel P(e)

2/1-boxes controlled with the

same bit e. The left (right) inputs of the P(e)2/1-boxes corre-

spond to the left (right) 64-bit input of the CP-box P(e)128/1.

The pairs of subkeys (K1,K3) and (K2,K4) are inputs of thecorresponding boxes P(e)

128/1 (figure 10(a)). Four 64-bit out-

puts of two boxes P(e)128/1 are denoted as Oi (i = 1, 2, 3, 4).

Thus, we have Oi = Ki , if e = 0, and O1 = K3, O2 = K4,O3 = K1, O4 = K2, if e = 1. The vectors L and �(L) areswapped with the third CP box P(e)

128/1 (figure 10(b)).

4.2. Data-dependent permutations

To perform DDP Cobra-H128 uses CP boxes P(V )64/192 and

(P−164/192)

(V ′) described in section 2 (figure 4). In Cobra-H128, the controlling vectors V and V ′ are formed using thesame procedure. This procedure includes two XOR transfor-mations and one fixed permutation π that defines which bit ofthe vector V controls which elementary CP box P2/1.

Let consider the formation of the vector V . The cor-respondence between bits of the controlling vector V =(V1, . . . , V6) and elementary switching boxes P2/1 of the

P(V )64/192-box is given in table 3, where rows indicate ac-

tive layers and numbers correspond to indices of the bitsof vectors L, L(1), and L(4). The rows correspondingto vectors V1 and V4 indicate bits of L. The rows cor-responding to vectors V2 and V5 indicate bits of L(1).The rows corresponding to vectors V3 and V6 indicate bitsof L(4). For example, accordingly to table 3 we have:V3 = (l

(4)13 , l

(4)14 , . . . , l

(4)32 , l

(4)1 , . . . , l

(4)12 , l

(4)10 , l

(4)11 , l

(4)9 ), V4 =

(l33, l34, l35, . . . , l64). The vector V ′ is formatted with simi-lar way.

The distribution of the bits of the controlling data subblockis a critical part in the design of the DDP-based ciphers. Whencomposing table 3, several iterations have been performed,followed by differential analysis and modification of the dis-tribution table. Combining subkeys with controlling data sub-block makes the DDP in Cobra-H128 to be key-dependent. Itis easy to see that table 3 satisfies the criterion of section 3.2.This provides that (i) each bit of L influences exactly sixbits of R and (ii) arbitrary input bit of the boxes P64/192 andP−1

64/192 moves to each output position with the same proba-bility, provided L is a uniformly distributed random variable.

In one round between the operations P(V )64/192 and

(P−164/192)

(V ′) the permutational involution I is performed. Itis described as follows: Y = (Y1, . . . , Y8) = I(X1, . . . , X8),


Table 3Distribution of the controlling bits in the box P(V )

64/192.

V1 31 32 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2V2 10 24 25 26 29 13 27 16 1 2 31 32 3 4 19 6 7 8 9 23 11 12 28 15 14 30 17 18 5 20 21 22V3 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 2 3 4 5 6 7 8 12 10 11 9V4 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64V5 55 56 57 58 59 60 61 62 63 64 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54V6 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 33 34 35 36 37 38 39 40 41 42 43 44

where 64-bit input (output) binary vector X(Y ) is representedas concatenation of eight bytes Xi(Yi), i ∈ {1, . . . , 8}, andY1 = X≪4

6 , Y2 = X≪45 , Y3 = X≪4

4 , Y4 = X≪43 ,

Y5 = X≪42 , Y6 = X≪4

1 , Y7 = X≪48 , Y8 = X≪4

7 .Use of this permutational involution provides that in one

round each input bit of the box P(V )64/192 influences all output

bits of the box (P−164/192)

(V ′) improving statistical propertiesof the round function. Permutation I introduces no time delaysince it is implemented in hardware as simple connections ofwiring.

4.3. Non-linear operation G

The role of the fixed permutation � in procedure Crypt(e)

is to make each bit of the left data subblock influences moredifferent digits of R while combining outputs of two opera-tions G with R. The permutation � contains four cycles ofthe length 16 and is described as follows:

(1, 50, 9, 42, 17, 34, 25, 26, 33, 18, 41, 10, 49, 2, 57, 58)

(3, 64, 43, 24, 19, 48, 59, 8, 35, 32, 11, 56, 51, 16, 27, 40)

(4, 7, 28, 47, 52, 23, 12, 63, 36, 39, 60, 15, 20, 55, 44, 31)

(5, 14, 13, 6, 21, 62, 29, 54, 37, 46, 45, 38, 53, 30, 61, 22).

The transformation W = GA′A′′(L) defining the operationGA′A′′ is described as follows:

W = L0 ⊕ A′0 ⊕ L1A

′′0 ⊕ L2L5 ⊕ L6A

′1 ⊕ A′

2A′′1

⊕ L3L4 ⊕ L1L4L6 ⊕ L2L6A′′1 ⊕ L1L2L4A

′′1,

where ∀j ∈ {0, 1, . . . , 6} we have Lj = L≪64−j and A0 =A, A1 = (1, a1, . . . , a63), A2 = (1, 1, a1, . . . , a62), (A = A′or A = A′′).

5. Discussion

5.1. Peculiarities of Cobra-H64 and Cobra-H128

In ciphers Cobra-H64 and Cobra-H128 extensive use of theCP-box operations is applied. They are used in three differ-ent ways: (i) as DDP that are one of two basic cryptographicprimitives, (ii) to swap subkeys when changing encryptionmode for decryption one, and (iii) to switch permutation π(e)

when changing ciphering mode. In addition to DDP, theseciphers use the non-linear operation G. The round transfor-mation of both ciphers is characterized by high parallelismthat provides pre-requisites of their high performance. Due tothe use of very simple key scheduling the ciphers Cobra-H64

and Cobra-H128 are fast in the case of key refreshing, sincethey are free of “external” key scheduling.

The ciphers Cobra-H64 and Cobra-H128, in comparisonwith the prototype SPECTR-H64, have the following fea-tures:

1. They use the initial secret key in each round.

2. The round transformation includes fixed permutational in-volution, performed on one of the data subblocks.

3. The round transformation includes one switchable opera-tion that prevents the weak keys with the structure K =(X,X,X,X).

The ciphers Cobra-H64 and Cobra-H128 have very sim-ilar structure of the procedure Crypt(e). The differencesare based on the use of (i) the operations having differentsize of input (32-bit for Cobra-H64 and 64-bit for Cobra-H128), (ii) different switchable operations: switchable per-mutation π(e) in Cobra-H64 and e-dependent swapping inCobra-H128, (iii) the use of the subkeys to calculate control-ling vectors in Cobra-H128, while in Cobra-H64 the control-ling vectors depend only on the current value of the left datasubblock.

5.2. Security estimations

We have considered different variants of the differential crypt-analysis. We have obtained that the fewer active bits in thedifference the higher the probability of the differential charac-teristic. This corresponds to the results of the analysis of otherDDP-based ciphers. Our best attack against Cobra-H64 andCobra-H128 corresponds to two-round difference with oneactive bit. This difference passes two rounds with probabilityp(2) = 1.16 · 2−19 for Cobra-H64 and p(2) = 1.13 · 2−29 forCobra-H128. Probability of the best two-round characteristicof SPECTR-H64 is p(2) = 1.15 · 2−13. Minimal number ofrounds required to thwart differential attacks is 8 for Cobra-H64 and 10 for Cobra-H128 and SPECTR-H64.

Our preliminary linear analysis of Cobra-H64 and Cobra-H128 has shown that they are secure against linear attacks fornumber of rounds r � 5. Accordingly to [10] SPECTR-H64is secure against linear attack for values r � 6. High degree ofthe algebraic normal form and the complexity of the Booleanfunction describing round transformation of the developed ci-phers prevent the interpolation and high order differential at-tacks. Our statistic experiments has shown that 4 round ofCobra-H64 and 5 round of Cobra-H128 are sufficient to sat-isfy test criteria proposed for the AES finalists [18].

226 SKLAVOS ET AL.

Because of the very simple key scheduling used in the pro-posed ciphers it appears to be important to study how a singlebit of key statistically influences ciphertext (key’s propagationproperty). For this purpose we have used the criteria of [18]considering the secret key as input vector and fixing differentplaintexts. Such statistic testing has shown that five roundsof Cobra-H64 and six rounds of Cobra-H128 are sufficient tosatisfy test criteria.

The used key scheduling is secure against basic related-key attacks. In spite of the simplicity of the key schedulethe keys K ′ = (X, Y,X, Y ) or K ′′ = (X,X,X,X), whereX,Y ∈ {0, 1}32 for Cobra-H64 or X,Y ∈ {0, 1}64 for Cobra-H128, are not weak, since encryption and decryption requirechanging the parameter e. It seems to be difficult to calculatea semi-weak key-pair for presented ciphers, if it is possibleat all. Thus, the role of the switchable permutation π(e) ispreventing the weak keys. For example, for SPECTR-H64which uses no switchable operations for all X the 256-bit keyK = (X,X,X,X,X,X,X,X) is weak.

6. Hardware implementation

6.1. ASIC and FPGA devices

Hardware implementations of both proposed ciphers are de-signed and coded in VHDL hardware description language.Both Cobra-H64 and Cobra-H128 were implemented usingtwo complete different implementation hardware modules:Application Specific Integrated Circuit (ASIC) and Field Pro-grammable Gate Array (FPGA). The performance character-istics of both ASICs and FPGAs are substantially differentcompared with a general-purpose microprocessor. ASICs andFPGAs have the advantage that can use all the resources for

pipelining data transformation, or parallel processing. Onthe other hand, the internal structure of the microprocessorsfunctional units limits the parallel processing and pipeliningtransformation. In addition, the instruction parallelism levelis a factor of great importance that must be taken under con-sideration for microprocessor performance. Furthermore, thehardware devices of these types can operate on arbitrary sizeof words, in contrast with processors, that operate only onfixed-sized words.

ASICs are in general far more expensive devices due totime consuming and high cost fabrication procedure, whichis done by expertise industry departments. FPGAs can bereached from anyone, since they are enough cheaper and canbe programmed or reconfigured by the designers/researchers.FPGAs have the major advantage that can perform a com-pletely different task/function after simple designers’ recon-figuration. ASICs performance is tight and can not be mod-ified after the chip’s fabrication. The offered reconfigurationhas a speed penalty in these devices. ASICs have higher speedperformance in comparison with FPGAs. This is due to thefact that the reconfiguration in FPGAs causes delays, intro-duced by the dedicated circuit’s parts needed to reconfigura-tion. In general, any implemented system of digital logic inan FPGA, is slower than ASIC implementation of the samesystem.

6.2. Implementation architectures for Cobra-H64 andCobra-H128

Both Cobra-H64 and Cobra-H128 are examined in hardwareimplementation by using two different architectures: FullRolling and Pipeline for both ASIC and FPGA devices. Theused Full Rolling architecture is shown in figure 11(a). It isa typical architecture for secret key block cipher implementa-

Figure 11. (a) Full Rolling and (b) Pipeline architectures.


tion. This architecture operates efficiently for both encryptionand decryption process. According to this architecture onlyone block of plaintext/ciphertext is transformed at a time. Thenecessary number of clock cycles to encrypt/decrypt a datablock is equal to the specified number of cipher rounds (10for Cobra-H64 and 12 for Cobra-H128). The key expansionunit produces the appropriate round keys which are stored andloaded in the used RAM blocks. One round of the encryptionalgorithm is performed by the Data Transformation RoundCore. This core is a flexible combinational logic circuit and itis supported by a n-bit register and n-bit multiplexer (64-bitfor Cobra-H64 and 128-bit for Cobra-H128). In the first clockcycle, the n-bit plaintext/ciphertext is forced into the DataTransformation Round Core. Then in each clock cycle, oneround of the cipher is performed and the transformed data arestored into the n-bit register. According to Full Rolling archi-tecture a 64-bit data block is completely transformed every 10clock cycles for Cobra-H64 (10 transformation rounds). Theoperation of Cobra-H128 (12 transformation rounds) needs12 clock cycles in order a 128-bit plaintext/ciphertext to begenerated

The second proposed architecture, figure 11(b), is aN-stage pipeline architecture. The main characteristics of thisare: (i) the pipelining used technique, and (ii) the usage of aRAM for the round keys storage and loading, which are pre-computed. Pipelining is not possible to be applied in manycryptographic applications. However, Cobra-H64 and Cobra-H128 block ciphers structures provide the availability to beimplemented with pipelining technique. The pipelining ar-chitecture offers the benefit of the high-speed performance.The implementation can be applied in applications with hardthroughput needs. This goal is achieved by using a numberof operating blocks with a final cost to the covered area. Theproposed architecture uses 10 basic round blocks for Cobra-H64 and 12 basic round blocks for Cobra-H128, which arecascaded by using equal number of pipeline registers. Basedon this design approach, 10 and 12 different n-bit data blockscan be processed at the same time, for Cobra-H64 and Cobra-H128, respectively. Pipeline proposed architecture producesa new plaintext/ciphertext block every clock cycle.

6.3. VLSI implementations synthesis results

The synthesis results are shown in table 4, for Cobra-H64and Cobra-H128, for both hardware implementations (ASICand FPGA), where: D Flip-Flops (DFFs), Configurable LogicBlocks (CLBs), Function Generators (FGs).

The above synthesis results for both implementations(ASIC and FPGA) prove that the Pipeline architectures ofCobra-H64 and Cobra-H128 have very high speed perfor-mance. Especially Cobra-H64 throughput is up to 5.5and 7.1 Gbps for FPGA and ASIC implementation, respec-tively. Cobra-H128 throughput reaches the values of 11 and12.1 Mbps for the same implementation devices. On the otherhand, Full Rolling architectures for both proposed ciphersallocate minimized area resources with good data rate. Ofcourse, for the Full Rolling architectures, of both Cobra-H64and Cobra-H128, the main goal is the minimized allocatedarea resources, with good achieved throughput. The opera-tion frequency for both ciphers is very high, for both proposedarchitectures and for all the examined hardware modules. Es-pecially, Cobra-H64 frequency ranges between 82 MHz to110 MHz. Cobra-H128 operates up to 90 MHz for FPGAdevices and up to 95 MHz for ASIC approaches. It is obvi-ous that according to the applications major demands, areaor performance, the designer can use the Full Rolling or thePipeline architecture respectively, for each one of the two pro-posed ciphers. The kind of application and the characteristicsof the application itself, will determine the use of the FPGAor the ASIC as the integration device, for the implementa-tion either Cobra-H64 or Cobra-H128. Both Cobra-H64 andCobra-H128 architecture simplicity in addition to the offeredhigh-level security strength make their hardware integrationvery useful in wireless communications networks.

In order to evaluate the very good performance, in hard-ware terms, of both Cobra-H64 and Cobra-H128 we com-pare the proposed ciphers implementations with the encryp-tion algorithms that are used in today’s wireless protocols.Especially in figures 12 and 13, the proposed ciphers perfor-mance is compared with the best hardware implementationsof ciphers used in wireless communications protocols. InIEEE 802.11 first versions (a–d), the Wired Equivalent Pri-vacy (WEP) is widely used to ensure privacy in the trans-

Table 4Implementation synthesis results.

Architecture\Hardware device FPGA technology ASIC technology(Xilinx) (0.33um)

Covered area F Rate Area F Rate

CLBs FGs DFFs(MHz) (Mbps) (sqmil) (MHz) (Mbps)

Cobra-H64 615 1229 204 82 525 2694 100 640(Full Rolling)Cobra-H64 3020 6040 640 85 5500 14640 110 7.1 Gbps(10-stage Pip.)

Cobra-H128 2364 4728 399 86 917 6364 90 1 Gbps(Full Rolling)Cobra-H128 22080 44160 1.5 Gbps 90 11500 48252 95 12.1 Gbps(12-Stage Pip.)

228 SKLAVOS ET AL.

Figure 12. Proposed ciphers implementations FPGA comparisons.

mission channel. Especially WEP is based on RC4. Thelatest working group (802.11i) adopts AES as the block ci-pher of this IEEE protocol. Bluetooth security is based onSAFER+ cipher. UMTS uses both AES and SAFER+ in order

to achieve security due to the external attacks over the trans-mitted data. Finally, the Wireless Transport Layer Security(WTLS) ensure encryption in both Wireless Application Pro-tocol (WAP) and Open Mobile Alliance (OMA). DES, IDEA


Figure 13. Proposed ciphers implementations ASIC comparisons.

Figure 14. FPGA Area/Performance comparison.

230 SKLAVOS ET AL.

and RC5 are the alternative ciphers that can be used for bulkencryption in WTLS.

For the FPGA implementation approach as it is shownin figure 12, Full Rolling (FR) architectures of both Cobra-H64 and Cobra-H128 has minimized area resources com-pared with FPGA approaches of AES [13], RC4 [7], SAFER+[9], IDEA [2], and DES [8] (for the last ciphers we present thebest results corresponding to different implementation archi-tectures). The pipeline architectures (P)s of Cobra-H64 andCobra-H128, as it was expected needs more resources thanthen (FR)s. The main advantage of pipeline architectures, asit is proven from figure 12, is the very high speed perfor-mance, compared with the achieved throughput of the otherconventional ciphers [2,7–9,13].

In addition, the proposed architectures are compared withother ASIC implementations [19,21–23] in figure 13. For theASIC approaches, a covered area comparison is not efficientbecause of the variety in units (mm2, sqmil, gates, transistors)of the reported area of the conventional architectures [19,21–23]. Both Cobra-H64 and Cobra-H128 higher performancecompared with AES [21], IDEA [23], RC5 [19], and DES[22], in all of the cases.

Furthermore, for the FPGA implementations, all theproposed Cobra ciphers architectures are compared in theArea/Performance model. This model can easily be calcu-lated, by the equivalent coved area resources (in CLBs) andthe throughput (Mbps), by using the formula: Area/Perfor-mance model = Area/Performance. The Area/Performancemodel comparison is illustrated in figure 14. It is obvious thatthe FRs architectures of both Cobra-H64 and Cobra-H128 aremuch better compared with RC4 [7], SAFER+ [9], IDEA [2],and DES [8], while they are too close to AES [13] value.Cobra-H64 (P) architecture as well as Cobra-H128 (P) Area-Delay model have higher value than FRs architectures. Thisis the paid penalty of the very good performance (figure 12)compared with the other conventional ciphers.

7. Conclusions

In this work, we propose two new fast ciphers: Cobra-H64and Cobra-H128. These ciphers are based on DDP transfor-mations. Security analysis has shown that the both ciphersare secure against known attacks. Due to high parallelism ofcomputations in one round and the use of the switchable op-erations one can use very simple key scheduling that makeshardware implementation cheaper and faster in the case offrequent change of keys. Both ciphers achieve high-speedperformance in FPGA devices and especially for ASIC ap-proaches. The implementation cost and the performance ofthe proposed ciphers are compared with the security layers ofthe most widely used wireless protocols: IEEE 802.11, Blue-tooth, WAP, OMA, and UMTS. These comparisons presentthe advantages of the proposed ciphers in terms of area re-sources, operating frequency, and throughput. These advan-tages prove the suitability of the proposed ciphers in wirelesscommunications, which set hard specifications in security im-plementations.

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sis of the ICE encryption algorithm, in: Proceedings of the 6th Interna-tional Workshop Fast Software Encryption – FSE’98, Lecture Notes inComputer Science, Vol. 1372 (Springer, 1998) pp. 270–283.

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[22] D.C. Wilcox, L.G. Pierson, P.J. Roberston, E.L. Witzke and K. Gass,A DES ASIC suitable for network encryption at 10 Gbps and bey-oned, in: Proceedings of CHES’99, Lecture Notes in Computer Sci-ence, Vol. 1717 (Springer, 1999) pp. 37–48.

[23] R. Zimmermann, A. Curiger, H. Bonnenberg, H. Kaeslin, N. Felberand W. Fichtner, A 177 Mb/s VLSI implementation of the internationaldata encryption algorithm, IEEE Journal of Solid State Circuits 29(3)(1994) 303–307.

Nicolas Sklavos is a Ph.D. Researcher with the Elec-trical and Computer Engineering Department, of theUniversity of Patras, Greece. His interests includecomputer security, new encryption algorithms de-sign, wireless communications, and reconfigurablecomputing. He holds an award for his Ph.D. researchon “VLSI Designs of Wireless Communications Se-curity Systems” from IFIP VLSI SOC 2003. He isa referee of International Journals and Conferences.He is a member of the IEEE, the Technical Chamber

of Greece, and the Greek Electrical Engineering Society. He has authored orcoauthored up to 45 scientific articles in the areas of his research.E-mail: [email protected]

Nikolay A. Moldovyan is an honoured inventorof Russian Federation (2002), a chief researcherwith the Specialized Center of Program Systems“SPECTR”, and a Professor with the Saint Pe-tersburg Electrical Engineering University. Hisresearch interests include computer security,cryptography, and currently developed concept of thevariable transformations as a new direction in ap-plied cryptography. He received his Diploma andPh.D. in Academy of Sciences of Moldova (1981).

He is a member of the IACR.E-mail: [email protected]

Odysseas Koufopavlou received the Diploma ofElectrical Engineering in 1983 and the Ph.D. degreein electrical engineering in 1990, both from Univer-sity of Patras, Greece. From 1990 to 1994 he was atthe IBM Thomas J. Watson Research Center, York-town Heights, NY, USA. He is currently an Asso-ciate Professor with the Department of Electrical andComputer Engineering, University of Patras. Hisresearch interests include VLSI, low power design,VLSI crypto systems, and high performance commu-

nication subsystems architecture and implementation. Dr. Koufopavlou haspublished more than 100 technical papers and received patents and inventionsin these areas.E-mail: [email protected]


Media Synchronization and QoS Packet Scheduling Algorithmsfor Wireless Systems ∗

AZZEDINE BOUKERCHE and HAROLD OWENS IISchool of Information Technology and Engineering (SITE), University of Ottawa, Canada

Abstract. Wireless multimedia synchronization is concerned with distributed multimedia packets such as video, audio, text and graphicsbeing played-out onto the mobile clients via a base station (BS) that services the mobile client with the multimedia packets. Our focus is onimproving the Quality of Service (QoS) of the mobile client’s on-time-arrival of distributed multimedia packets through network multimediasynchronization. We describe a media synchronization scheme for wireless networks, and we investigate the multimedia packet schedulingalgorithms at the base station to accomplish our goal. In this paper, we extend the media synchronization algorithm by investigating fourpacket scheduling algorithms: First-In-First-Out (FIFO), Highest-Priority-First (PQ), Weighted Fair-Queuing (WFQ) and Round-Robin(RR). We analyze the effect of the four packet scheduling algorithms in terms of multimedia packet delivery time and the delay betweenconcurrent multimedia data streams. We show that the play-out of multimedia units on the mobile clients by the base station plays animportant role in enhancing the mobile client’s quality of service in terms of intra-stream synchronization and inter-stream synchronization.Our results show that the Round-Robin (RR) packet scheduling algorithm is, by far, the best of the four packet scheduling algorithms in termsof mobile client buffer usage. We analyze the four packet scheduling algorithms and make a correlation between play-out of multimediapackets, by the base station, onto the mobile clients and wireless network multimedia synchronization. We clarify the meaning of bufferusage, buffer overflow, buffer underflow, message complexity and multimedia packet delay in terms of synchronization between distributedmultimedia servers, base stations and mobile clients.

Keywords: distributed algorithms, media synchronization, mobile multimedia, wireless communications, packet scheduling algorithm,quality of service (QoS)

1. Introduction

Wireless communication technological advancements havecreated a new paradigm known as mobile distributed multi-media systems. In a mobile distributed multimedia system,diverse packets can be simultaneously manipulated. Packetssuch as text, images, audio and video can be played out on amobile client via a base station. Some multimedia packets aretime dependent upon each other [25]. Because of the time de-pendency that exists between some packets, we can classifythese packets as discrete packets or continuous packets. Thediscrete packets include text and images, while the continuouspackets have both audio and video.

The basic abstraction, for a time constrained media ele-ment, is a timed stream of media components (video frameor audio sample). Typically, during play-out on the mobileclient, media components must be kept in temporal order.This ordering process is known as media synchronization.There are two kinds of timing aspects for constrained ele-ments: (1) intra-media continuity is subject to a real-time-constraint in handling media packets and (2) inter-media con-tinuity is subject to temporal correlation during playbackof media packets [25]. The synchronization problem, inwired communication systems, has been extensively stud-ied [4,22,29]. In a wired communication system, multime-

∗ This work was partially supported by Research Grants from NSERC, theCanada Research Program, Canada Foundation for Innovation, OntarioDistinguished Researcher Award (OIT/ODRA#201722).

dia synchronization is much easier when compared to a wire-less environment. This is mainly because of a plethora ofresources like memory, power and bandwidth. In contrast, awireless systems is a bit more complicated because of a dearthof resources (memory, power and bandwidth) that needs to bemanaged efficiently. Due to the abundance of resources inthe wired system and limited resource in the wireless system,communication between wired and wireless systems causesa major communication problem solved by performing syn-chronization between the two types of systems.

In a wireless communication system, a base station (BS)must be used in order to deliver the multimedia packets to themobile client [8,9]. The resources between the base stationand the mobile client are limited in terms of bandwidth, mem-ory and power. If the multimedia packets are received fromthe wired systems to the mobile client too quickly (flooding)or too slowly (starvation), the limited resources could be com-promised. The memory at the mobile client could be over-flowed causing lost multimedia packets to be retransmittedor underflowed causing un-smooth video and audio play-out.This potential compromise of the mobile client’s resources, iswhy we investigate wireless multimedia synchronizationandQoS packet scheduling.

This paper considers transmission of live audio stream andthe corresponding video stream from the distributed multi-media servers to the base stations which service the mobileclients. We investigate the effect of QoS packet schedulingalgorithms at the base station on the overall network syn-

234 BOUKERCHE AND OWENS

chronization. We investigate four packet scheduling algo-rithms: First-In-First-Out (FIFO), Priority-Queuing (PQ), andRound-Robin (RR) and Weighted Fair-Queuing (WFQ) queu-ing. In today’s wireless and wired communication systemsFIFO is used because of the simplicity of the algorithm [14].PQ is an priority-based scheduling algorithm useful for thetransfer of real-time traffic but usually produces unfairnessamong traffic classes [14]. RR scheduling creates fairness bygiven each packet a certain amount of play-out time. WFQscheduling, at different network layers, gives fair sharing ofbandwidth among various network traffic.

2. Related work

Media synchronization control is paramount for multimediaapplications such as video and audio stream over wired/wire-less networks. Media synchronization control is necessaryfor preserving the temporal relationships among plural mediastreams by compensating for network delay jitters [13]. Me-dia synchronization can be classified as intra-stream synchro-nization and inter-stream synchronization [26]. The intra-stream synchronization refers to the temporal constraintswithin a single stream such as the time intervals between twosuccessive packets of a video or audio stream. Inter-streamsynchronization pertains to synchronization among multiplestreams of concurrent packets such as voice and audio stream(lip sync). To ensure quality of service of multimedia servicesand to minimize end-to-end multimedia packet delay for themobile client, intra-stream synchronization and inter-streamsynchronization must be guaranteed as illustrated in figure 1.

Research has been done in the area of QoS packet schedul-ing. Most of the research and publications deal with networklevel performance like packet overhead, throughput and de-lay. For overall QoS of the mobile client, several areas suchas scheduling for synchronization, feedback techniques for

Figure 1. Intra-stream and inter-stream synchronization.

synchronization [24], network-based schemes [28], buffering-based schemes [4], and reactive control schemes have beenstudied over the past few years. Network synchronizationtechniques have been used to improve the quality of service,in terms of multimedia synchronization for smooth play-outof audio and video, at the mobile client. In our earlier work,we have discussed MoSync, an algorithm [8] is the only algo-rithm that deals with continuous media in high layers, partic-ularly audio and video at the application layer. The primarydifference between packet-level performance and continuous-media level performance is at the continuous-media level, thetemporal structure of the media is the focus [8].

To achieve the overall goal of having a unified network(wired/wireless), extensive research has been done on theeffect of packet scheduling synchronization in the network.Earlier network designs (wireless and wired) were meant fornon-delay affected applications such as text and graphics.The new paradigm for, both wired and wireless networks,are resource intensive. Applications like animated graphics,on-demand video and voice-over IP require bounded delaysand guarantee throughput, but tolerate some errors [19]. Thegrowth of smart phones, PDAs and laptop computers haveincrease the need for unification between wired and wire-less networks. However, like all technological advancement,new technology invents new problems, particularly synchro-nization with multimedia packets, being played out on themobile client. The combination of wired and wireless net-works differ greatly from the typical wired networks. First, allpackets must pass through a base station before reaching itsmobile terminal destination [8]. Second, a mobile client’s re-sources (battery, memory, screen, bandwidth, etc.) are verylimited. Due to resource limitations, conventional synchro-nization strategies for delivery of multimedia packets to themobile client, cannot be applied in a mixed environment.A single base station buffers multimedia packets for multiplemobile clients and must service each mobile client in its cellarea. Packet buffering can cause congestion at the base stationand slow the processing of multimedia packets at the base sta-tion. On the other hand, a burst of multimedia packets, fromthe base station to a limited resource (battery, memory, screen,bandwidth, etc.) mobile client, can cause buffer overflow atthe mobile client. If multimedia packets do not arrive on timeat the mobile client from the base station, buffer underflow atthe mobile client can occur. The base stations assume that thepackets will be available when multimedia packets requestsare made from the mobile client. The Internet only providesbest-effort and cannot guarantee on-time multimedia packetdelivery to the base station. The mobile client must informthe multimedia server and base station about its play-out con-ditions. The combination of wired and wireless network com-munication brings about a new synchronization for multime-dia applications methodology.

In earlier studies, the MoSync algorithm [8], upon whichwe expounded, implemented a feedback-based synchroniza-tion for multimedia applications. MoSync is a synchroniza-tion scheme for wireless clients and distributedmultimedia systems that uses a Quasi-sink to control synchro-

MEDIA SYNCHRONIZATION AND QOS PACKET SCHEDULING 235

nization. The proposed solution copes with network jitters,end-system jitters, clock drift and changing network condi-tions [4,20]. MoSync can be employed for both intra-streamduring playback video and resynchronization streams in awireless network. Since MoSync uses delay time to allowintra-synchronization, inter-synchronization is guaranteed bythe MoSync algorithm [8]. A set of experiments was doneearlier assuming FIFO play-out of the multimedia packets onthe mobile client from the base station. In this paper, we studythe hand-off problem and propose tow hand-off schemes toenhance the performance of the media synchronization forwireless systems, as well as investigate the multimedia packetplay-out onto the mobile client in RR, PQ, and WFQ from thebase station.

3. MoSync algorithm

The MoSync algorithm [8] assumes that the network can pro-vide sufficient resources to deliver multimedia packets to themobile client. The multimedia data consists of many mul-timedia packets that may be progressively transmitted overthe network. MoSync does not use a global synchronizationclock. In the MoSync algorithm, the servers time stamp eachmultimedia packet, with current local time to allow the basestation to calculate round trip delay, jitters and inter-arrivaltime. The multimedia packets carries the server’s number andpacket sequence number but these additions are negligible somultimedia synchronization is the primary focus of the thispaper.

The MoSync model consists of K scalable server nodesand L mobile clients. A base station can communicate withmultiple mobile clients because the communication betweenthe base station and mobile client is wireless. Figure 2 illus-trates the servers, base station and mobile client connectivity.The system can contain multiple servers (data, voice, graph-ics, video) depending upon the need of the mobile client. Allmobile clients receives the multimedia services from the sameservers. The communicate is done from the the servers to themobile client via the base station. The BS communicates withmany servers and mobile client at any given time. The BS inthe MoSync algorithm have special roles when servicing mo-bile client in their area. The base station has three roles: as amessenger it passes multimedia units to the mobile client, asa filter it sends request to the servers per request of the mobile

Figure 2.

client and last, as a Quasi-receiver, it only receives the firstpackets from the servers. When a mobile client request mul-timedia packets from a server the multimedia packets are sentvia the base station. The servers can control the supply rateof multimedia packets to mobile clients by using the mobileclient’s feedback messages. MoSync focuses on three areas tosolve the combination of wired and wireless communicationsynchronization problems:

1. No jitters and constant delay case (extending Biersack andGeyer [4]).

2. Intra-stream and inter-stream synchronization problem(using network load).

3. Resynchronization (using exponential smoothing forecast-ing).

MoSync uses three simple protocols. In case one, to avoidbuffering of early arrival multimedia packets sub-stream thestart-up protocol is used. This allows evaluation of the roundtrip delay for different sub-streams. In case two, end-systemjitters and network jitters is counteracted. Both end-systemand network jitters must be countered to avoid underflow oroverflow at the mobile client. For multimedia packets to play-out smoothly the servers data transmission rate must matchthe available service rate at the mobile client. The overallgoal when dealing with network and end-system jitters, is tokeep the average delay, buffer underflow and buffer overflowat an acceptable minimal level. In case three, to solve theaverage delay changes, clock drift and server dropouts expo-nential smoothing forecasting is used [8]. When a mobileclient receives a multimedia packet, the mobile client calcu-lates the expected arrive time of the next multimedia packetand sends the calculation to the base station. The base stationthen forwards the information to the servers. In both case twoand three the MoSync algorithm uses either the pessimistic oroptimistic synchronization protocol when requesting and re-ceiving multimedia packets from the servers. With MoSync’spessimistic protocol, the base station updates the servers oneach multimedia packet request made by the mobile client.With MoSync’s optimistic protocol, the server sends the mul-timedia packets directly to the mobile client without contin-uous updates being made by the base station once the firstmultimedia packet from the server arrives to the mobile client.The MoSync algorithm operates at the network’s applicationlayer.

3.1. Basic concepts of the MoSync algorithm

The MoSync algorithm uses three types of nodes: server,Quasi-receivers and receivers [8]. The algorithm containsK servers, N base stations and L mobile clients, where K

servers are of type servers, M base stations are of type Quasi-receiver and the L mobile clients are of type receivers. Themobile client synchronizes the multimedia packets, receivedfrom the multimedia server, by reporting buffer usage tothe base station and updating the base station with multime-dia packet arrival time differences. After a base station has


information about the arrival time of each packet, it calcu-lates the synchronization time for the next packet. A sched-uler, in the server, manages the on-time transfer of thesubframes, as a part of the frames to the mobile client. Themobile client requests multimedia service from the multi-media servers, through the base station. When a base sta-tion requests the first group of multimedia packets, it sendsthe synchronization point information to all servers [8]. Thebase station can receive messages from the mobile client andfrom neighboring base stations. The mobile client messagesare for requesting multimedia packets from servers and donemessages to let base stations know all packets have been re-ceived by the mobile client. The neighboring base stationmessages are used to inform the base station if its neighboris on or off, to set-up hand-offs to neighboring base stations.It is the mobile client’s job to calculate the latest arrival timeand the differences between every multimedia packet’s arrivaltime, the buffer usage and update its serving BS with the newdata.

3.2. MoSync algorithm and hand-off management schemes

A wireless network consist of a mobile client, a BS that ser-vices the mobile client and MSC that performs the hand-offbetween base stations. The BS is connected to a wired servernetwork that provides the mobile client with services, such asdata, graphics, audio and video, via the BS, through a wire-less link. Each BS has a coverage area known as a cell. Whena mobile client, receiving services from a BS, moves out ofthe coverage area of the servicing BS and into the cell of an-other BS, the mobile client’s services must be transfer to thenew BS to ensure continuous service. Hand-offs in a wire-less network are classified as Soft Hand-offs and Hard Hand-offs.

In a hard hand-off, the connection between the mobileclient and servicing BS, is broken before new services are es-tablished with the new cell’s BS. The mobile client can onlybe connected to one BS at any time, with a hard hand-off,because frequencies between base stations must be differentto avoid interference between mobile client signals. Becausea mobile client can only be connected to one BS at a time,during hand-off, delays between services are introduce intothe network. Hard hand-off is mostly used in FDMA andTDMA [1].

In a soft hand-off, the connection between the servicingBS and the mobile client is maintained while another con-nection between the mobile client and the new cell’s BS isestablished. The mobile client can be connected to one ormore base stations at any time, in a soft hand-off and con-current connections between mobile clients and base stationscan exist because the frequency band at the base stations dif-fers. Soft hand-off is mostly used in CDMA [1]. In a wirelessnetwork, soft hand-off is the preferred method of hand-off be-cause the mobile client’s services are not interrupted when ahand-off is performed. Because there is no interruption in ser-vice, using soft-hand off, audio and video is played-out on themobile client smoothly.

We have investigated several hand-off schemes for Mo-Sync [10]. In our experiments, we settle with the two-handof scheme, which consists of two parts: setup hand-off andend hand-off. In phase one, setup hand-off updates new ar-rival base stations and maintains synchronization for newlyarrived mobile clients. If a mobile client can communicatewith another BS, the mobile client will send a “new BS ar-rived” message to its primary BS. For the mobile client torequest service from the new BS, the new BS must have delaytime information about each multimedia server. If the newbase station does not have delay time information about themultimedia servers, then the new BS sends a “Request” mes-sage to all multimedia servers. When a mobile client receivesmultimedia packets, the mobile client calculates the latest ar-rival time and the differences between multimedia packets us-ing the MoSync algorithm. In phase two, the end hand-offdeals with the ordering of multimedia packets and the flowof the multimedia packets to the mobile client. Any BS canserve as the newly selected primary BS. After informing themobile client, BS and multimedia servers, the end hand-offphase selects the closest common node from the primary BSand the newly selected BS. The common node must be withinthe wireless network. If there is no common node, then theMSC of the current primary BS will be the common node un-til a common node can be found. The common node reroutesthe multimedia packets in time-stamped order using the newlyselected primary BS. Once the mobile client moves withinthe coverage area of the selected BS, the end hand-off phaseis terminated. The MoSync algorithm works with both hardhand-off and soft hand-off to synchronize multimedia packetdata flows [9].

4. Qos packet scheduling algorithms

In a wireless network, the mobile client requests multimediaservices through a base station. The base station forwardsthe multimedia packet request to the distributed multimediaserver. The multimedia server sends the multimedia packet tothe base station where the packets are buffered and sched-uled for play-out onto the mobile client that requested themultimedia service. With today’s wireless technology, themobile client can request video, audio, text and graphics ser-vices from distributed multimedia servers via the base station.The base station cannot play-out the multimedia packets ontothe mobile client until the latest packet has arrived. If an on-demand-video movie is requested by the mobile client, thebase station must receive both audio and video packets for aparticular frame from the distributed multimedia servers be-fore the frame can be played-out onto the mobile client. Themulti-data stream dependence between multimedia packets,such as video and audio, is known as inter-stream synchro-nization or lip-synchronization. Each audio and video mul-timedia packet must be kept in temporal order for smoothplay-out audio and video at mobile client. The time de-pendency of multimedia packets of the same data stream isknown as intra-stream synchronization. The mobile client


calculates the arrive of the multimedia packets and informsthe base station of the arrival time difference. The updatedmessages from the mobile client to the base station allowsthe base station to synchronize its request to the multime-dia server and synchronizes the multimedia packets play-outonto the mobile client. In the MoSync algorithm the base sta-tion schedules play-out of multimedia packets on the mobileclient in a FIFO order. The resources at the mobile clientare limited in terms of power, bandwidth and memory. Theobject of multimedia synchronization is to keep the bufferat the mobile client full so smooth play-out of multimediaservices can be presented. If the multimedia packets fromthe servers are scheduled too early, buffer overflow occursat the mobile client. If multimedia packets from the serversare scheduled after the deadline to maintain temporal order,buffer underflow occurs at the mobile client. Mobile clientbuffer underflow and overflow is a network problem that hasto be dealt with through synchronization of packet delivery,to the mobile client from the servers, via the base station.The scheduler in the base station effects network synchro-nization in terms of buffer usage, overflow, underflow, mes-sage complexity and multimedia packet delay at the mobileclient.

In this paper, we wish to investigate how packet schedul-ing algorithms may affect the media synchronzation in wire-less and mobile communication systems. We have investigateseveral packet scheduling schemes, In this paper, we settledown with the following QoS packet scheduling algorithms:FIFO, PQ, WFQ and RR.

4.1. First-In-First-Out (FIFO)

In today’s wired and wireless systems, First-In-First-Out(FIFO) is the most common scheduling algorithm. There isonly one queue and all packets are treated equally. Packetsare stored at a buffering location and processed on a first comefirst serve basis. FIFO does not recognize priority or classesof traffic. If the buffering area does not have space to storearriving packets, the buffer area discards the packets [14]. Interms of data flow, since large flows may arrive first and cap-ture large bandwidth, FIFO is a since of unfairness amongflows. On the average, a flow burst received is usually trans-mitted in a similar burst using FIFO scheduling. FIFO is thefastest queuing method. It is effective for large links that havelittle delay and minimal congestion.

4.2. Priority Queuing (PQ)

Some multimedia requests, like on-demand-video, requireslarge bandwidth and guaranteed processing order of multime-dia packets. In this special case, Priority Queuing (PQ) canbe used to guarantee the bandwidth, memory and process-ing time for a particular high priority data flow. PQ sets acertain number of priority classes according to some parame-ter. In our experiment, the packet with the lowest play-outtime receives higher priority. In a wired and wireless com-munication system the priority can be an IP address, proto-col or interface [14]. The packets are played out according

to the highest priority. PQ serves the highest priority pack-ets first and then moves to the lowest after all higher packetshave been processed. Priority-based algorithms reduce delaysfor high priority classes that are bandwidth sensitive, such asvideo, audio and interactive traffic. PQ reduce delays for highpriority classes with out reducing the overall throughput ofthe lower classes. PQ can lead to unfairness among classesif not monitored closely because higher priority classes thatarrives on a continuous basis will always be processed beforethe lower priority classes are processed. To reduce starvationamong lower priority classes, PQ monitors the traffic and willupdate the lower priority classes priority to enable process-ing of lower priority classes during continuous high priorityclasses traffic flows.

4.3. Round Robin (RR)

Lately, there has been an increase in multimedia traffic suchas audio and video. The heavy demand for multimedia trafficincreases QoS needs to enhance the multimedia communica-tion. Packet scheduling in the networks play an important partin the QoS of each client in the network. These multimediaservices are bursty and bandwidth hungry. To ensure fairnessbetween processing of packets in the network, Round Robin(RR) packet scheduling can be used. With RR packet schedul-ing, arriving packets are store in the same buffer on a First-In-First-Out (FIFO) strategy. The packets are removed from thebuffer in a FIFO basis. Each packet is guaranteed a certainquantum of processing time. If the packet finish playing-outduring the allowable processing time, the packet is removefrom the queue. If the packet does not finish processing dur-ing the allowable time, the packet is stamp with informationsuch as arrival time and burst time and is place back into thebuffer as a new arriving packet to be processed. The RoundRobin algorithm creates fairness among packets by guaran-teeing play-out time and by isolating each traffic flow frommisbehaving traffic. Bursty packet like audio and video thatarrives for a particular flow are spread out and transmitted oneat a time according to the RR packet scheduling algorithm.RR does not easily adjust to the network load due in part tothe quantum slice that is static.

4.4. Weighted fair queuing (WFQ)

Multiple traffic flows can be identical in priority and their net-work resource usage. With the multimedia traffic of today,assigning a priority or guaranteeing play-out time of multi-media packets is not enough to ensure high QoS for a client.The Weighted Fair Queuing (WFQ) algorithm combines thePQ and RR into a dynamic fair queuing that divides the avail-able bandwidth among traffic queues base on weight or pri-ority. In today’s networks, video packets are given a higherpriority than audio packets, audio packets are given a higherpriority than graphic packets and graphic packets are given ahigher priority than text during play-out of multimedia pack-ets. WFQ ensures that each packet is treated fairly accordingto its weight. Packets of the same priority is stored in the same


buffer. Multimedia packets of the same priority are played-out in a round robin order. Hence, satisfactory response timeis a result of WFQ fairness. For delay sensitive applicationssuch as video-on-demand, animated graphics and interactiveapplications, WFQ is the most appropriate scheduling algo-rithm. Furthermore, WFQ dynamically adjusts to traffic flow.Low volume and high volume traffic both get fair allocationof available bandwidth. WFQ may not perform well in a wire-less environment where the resources such as memory, powerand bandwidth are limited.

5. Simulation experiments

We have developed a discrete-event model to simulate a cellu-lar wireless multimedia system, on a wireless and wired net-work. Our assumption is that the communication betweenservers and base stations are wired and connections betweenmobile clients and base stations are wireless. In our model,there are 300 channels available for 60 cells, with the systemload equally distributed over all cells. Each mobile client hasa buffer size that’s at least 3 times that of the largest multi-media packets and at most six time as large as the multimediapackets received from the server. In this simulation, we em-ploy a two-phase hand-off scheme using soft-hand hand-offin CDMA [5].

We have performed several tests, using many jitters and re-synchronization. We evaluate MoSync’s performance in twoenvironments: uniform and nonuniform jitters using FIFO,PQ, WFQ and RR scheduling. Table 1 describes the parame-ters we use in our simulation experiments. In the uniformcase, for the four packet-scheduling algorithms, we evalu-ate each algorithm, assuming uniform multimedia unit de-lay, under jitters. All cells have the same multimedia packetsdemand and the requests from the mobile client have inter-arrival time λ and average service time µ. We distribute thedelay evenly among all multimedia requests, maximum andminimum delay time, assuming minimal delay time of 50 ms.Due to the distribution of the delays, the jitters will cause dif-ferent effects on the synchronization algorithm.

In the nonuniform case, we have assumed exponential dis-tribution of network delay jitters. This distribution was usedin earlier studies of the MoSync algorithm [8]. For our exper-iments, there are multimedia media delay ranges as follows:(1) 0–200 msec; (2) 0–400 msec; (3) 0–600 msec. The ranges

Table 1Simulation parameters.

Packet scheduling algorithms FIFO, PQ, WFQ, RRNumber of cells 60Number of multimedia servers 4Mobile client buffer size 30 times of a multimedia packetsPlay-out time/multimedia packets 100 msecMean service time/session µ

Arrival rate in normal cell λ

Round-Trip-Time (RTT) torequest/deliver a multimedia packets 50 msecJitters (uniform) 0–20, 20–40, 40–60, 60–80 msecJitters (non-uniform) 0–200, 0–400, 0–600 msec

are exponentially distributed, with mean values of 20, 40 and60, respectively. The upper bound delay is set in the nonuni-form distribution to prevent failure, caused by large delays,in the stimulation. The mean communication session is set to20 multimedia packets and the mean buffer size of a mobileclient is set to three times the size of the multimedia pack-ets [8]. Our results will that each range of delay jitters causethe network to behave differently.

5.1. Method of experiment

The BS plays-out the multimedia packets, sent by the mul-timedia servers, onto the mobile client that requested thepackets. To avoid mobile client buffer overflow and bufferunderflow and to ensure intra-stream and inter-stream syn-chronization of the multimedia packets, multimedia packetsrequest and delivery must be synchronized with the mobileclient that requests the multimedia packets.

The multimedia servers, which send the multimedia pack-ets request, have a built-in scheduler that delivers the pack-ets on time. The base station receives and buffers multime-dia packets from the multimedia servers and schedules thepackets for play-out onto the mobile client that requested themultimedia packets. In earlier studies, using the MoSyncalgorithm, the base station plays-out the multimedia pack-ets onto the mobile clients in a FIFO order. We will makea correlation between the scheduling of multimedia pack-ets by the BS, to be played-out onto the mobile client andthe effect the multimedia packets scheduling, by the BS, hason intra-stream synchronization, inter-stream synchronizationand mobile client’s buffer usage. We assume that the wirednetwork can provide efficient data flow from the multimediaservers, to the mobile client, per request of multimedia pack-ets.

5.2. Performance metrics

To assess the performance of each of the scheduling algo-rithms, we choose the same performance metrics as in earlierwork [8,9]. Our areas of focus are as follows:

• Message complexity: measures the overhead in terms ofthe number of messages needed to satisfy the user’s mul-timedia requests.

• Buffer usage: measures the synchronization behavior foreach mobile client.

• Overflow rate: measures the average number of multime-dia units that overflow a mobile client buffer.

• Underflow rate: measures the average deficiency in theclient’s buffer of the number of multimedia units currentlyneeded for smooth play-out.

• Multimedia unit arrival rate: measures the reactionof the network and the network condition by moni-toring multimedia packets arrival time to client (intra-synchronization).


• Delay between multiple data streams: measures the delaybetween the data streams that are serving the client con-currently (inter-synchronization).

6. Simulation results

In this section, we analyze the results from the simulation ex-periments we have obtained using FIFO, PQ, WFQ and RRpacket scheduling. We simulate MoSync with both intra-synchronization and inter-synchronization. We use a controlmechanism to control buffer availability and reduce the num-ber of messages between mobile client and BS. Because ar-rival time is highly dependent on the network queue delay,distributed clock drifts and violation of network bandwidthguarantee, it is easier to delay arriving multimedia packets,using this control mechanism, than to speed up the arrivaltime of the packets.

6.1. Buffer usage

Figures 3 and 4 illustrate the buffer usage for uniform andnonuniform cases. In figure 3, at 20 ms uniform delay jit-ters, FIFO, PQ, WFQ and RR scheduling algorithms performequally. To play-out smoothly, all of the four algorithms re-quire a buffer size sixteen times that of the multimedia pack-ets. The four scheduling algorithms perform equally becausethe network compensates for the small jitters by shifting thedelay to the upper layers, where the multimedia packets arebuffered and propagated with little delay between multime-dia packets. At 40 ms uniform delay jitters, all algorithmsallow smooth play-out at fifty requests, but FIFO requiresabout twice the buffer space to play-out smoothly. FIFO re-quires twice the buffer space because the rate of multime-dia packet arrival is almost proportional to the departure rate.At 60 ms uniform delay jitters, all packet scheduling algo-rithms synchronize equally requiring no more buffer space ormultimedia packets to play out smoothly. With 60 ms uni-form jitters, the mobile client’s and distributed multimediaserver’s receive and send rates are matched. MoSync effi-ciently buffers the data at the application layer and allows forsmooth play-out of multimedia packets by the BS onto theclient. At 60–80 msec delay jitters in figure 4, FIFO needstwice the number of multimedia packets to play out smoothlycompared to RR, PQ and WFQ. FIFO is really effected by thelarge delays in the network whereas PQ, WFQ and RR areaffected because synchronization is easier and traffic is morepredictable, so RR, WFQ and PQ the MoSync algorithm cansynchronize accordingly. Notice up to 20 multimedia pack-ets request with 60–80 ms delay jitters all algorithm behavesthe same because for small numbers of multimedia packet re-quest MoSync is able to buffer packets and play-out packetssmoothly onto the mobile client. After 20 multimedia pack-ets request FIFO is affected by the non-predictability of thearriving multimedia packets. With WFQ, PQ and RR algo-rithm we know what multimedia packets will be serviced nextby the BS. Figure 4 illustrates the nonuniform buffer usage.

Figure 3. Buffer usage for two-phase algorithm: uniform case.


Figure 4. Buffer usage for two-phase algorithm: nonuniform case.

At 20 ms nonuniform delay jitters, all packet-scheduling al-gorithms synchronize equally, requiring the same amount ofbuffer space to play-out smoothly. With small amounts ofnonuniform network jitters, the network can easily counterthese delays by pushing delay to the upper most layers wheremultimedia packets are buffered by the BS and played-outonto the mobile client. At 40 ms nonuniform delay jitters thepacket scheduling algorithms all decrease and need more mul-timedia packets to play-out smoothly. WFQ and PQ decreasethe slowest and are not affected by the large delays in thenetwork, while both RR and FIFO need about twice the num-ber of multimedia packets to play-out smoothly. WFQ andPQ reduce delay for a high-priority class that is both burstyand low rate, relative to link bandwidth (video stream, inter-active traffic, etc.), without reducing the throughput of lowerpriority classes [12]. FIFO packet scheduling does not iso-late flows from misbehaving or bandwidth heavy flows. Mis-behaving or heavy bandwidth flows such as video can cap-ture the limited wireless bandwidth increasing delays of otherlower bandwidth multimedia packets that need to be playedout. Finally, when we introduce 60 ms nonuniform delay jit-ters, the scheduling algorithms all decrease quickly needingmore multimedia packets to play smoothly. PQ and WFQneed 15 less multimedia packets than FIFO and RR to play-out smoothly because under any delay, WFQ and PQ willguarantee play-out of heavy bandwidth multimedia packets,where RR and FIFO cannot because no priority is given to aparticular packet flow.

6.2. Overflow rate

Figures 5 and 6 illustrates the overflow rate for both thenonuniform and uniform cases. At the network’s applicationlayer multimedia packets are buffered and handle as continu-ous data stream to allow for play-out of multimedia packetswithout jitters. The object is not to overflow the mobile clientsbuffer, hence, causing data in buffers to be rendered useless.All packet scheduling algorithms perform well under the dif-ferent ranges of network delay jitters. It is much easier tocontrol overflow then to control underflow. We can introducea delay great enough to eliminate all overflow at the mobileclient, to the maximum 150 multimedia packets request asillustrated in figures 5 and 6. This works in theory, but in ap-plication, will under-utilize the network’s resources and notallow for smooth play-out of multimedia packets. The Mo-Sync algorithm is very efficient in preventing overflow. TheMoSync algorithm’s ability to adjust to network delay jittersand prevent mobile client’s buffer overflow is illustrated infigures 5 and 6.

6.3. Underflow rate

Figures 7 and 8 illustrate the underflow rate for both thenonuniform and uniform cases. In figure 7, at 20 ms nonuni-form delay jitters, all algorithms perform well and produce nounderflow because the network shifts the small delays to theapplication layer where the multimedia packets are buffered


Figure 5. Overflow for two-phase algorithm: nonuniform case. Figure 6. Overflow for two-phase algorithm: uniform case.


Figure 7. Underflow for two-phase algorithm: nonuniform case. Figure 8. Underflow for two-phase algorithm: uniform case.


at the BS and then played-out on the mobile client, withoutjitters. In figure 7, with nonuniform delay jitters of 40 msecdistributions, WFQ and PQ produce a great deal of underflowat higher number of requests, while FIFO and RR remain al-most constant. At 60 msec distributed nonuniform delay jit-ters, PQ and WFQ produce a lot of underflow at the mobileclient, while FIFO and RR remain almost constant because alarger number of mobile clients are able to play-out their mul-timedia packets. With WFQ and PQ, with a particular dataflow, only the mobile clients with higher priority requests areable to play their multimedia packets out on-time. At greaterdelays, the RR scheduling produces little or no, caused bythe ability of RR spread out a burst of multimedia packet re-quest among flows, fairly, allowing for more mobile clients toplay-out the multimedia packets smoothly. In figure 8, uni-form delays jitters of 20, 40, 60 and 80 msec are introduceinto the network. Figure 8 illustrates 80 ms delay jitters, RRand FIFO produce little or no underflow while PQ and WFQproduces a lot because with PQ we service the packets withthe smallest play-out time causing clients to wait on multi-media packets until all higher priority multimedia packets areplayed out. We can conclude that the RR packet scheduling isthe best among the four packet scheduling algorithms. WFQis used to ensure fairness among the various data streams butdoes not easily adjust to traffic flow. We can notice little un-derflow for the RR and FIFO packet scheduling for all uni-form and nonuniform delay jitters. This implies that RR andFIFO algorithms provide better intra-stream and inter-streamsynchronization. RR and FIFO based scheduling algorithmsisolate flows from one another. With RR, packets for a par-ticular flow are spread out and serviced one at a time. WithFIFO, packets for a particular flow received are transmitted ina similar burst. RR protects multimedia data flows from largebandwidth or misbehaving data flows by guaranteeing eachflow a certain amount of time to play-out.

6.4. Message complexity

Figures 9 and 10 illustrate the message complexity of eachpacket-scheduling algorithm. The message complexity is themessage overhead caused by the mobile client requestingmultimedia packets from the multimedia servers. The mes-sage complexity of MoSync is O(I (K + L)) where L is re-questing messages from mobile client, K is the number ofservers processing request and I is the number of rounds ofL + K messages. In the beginning round, the mobile clientssend a request message to the BS. The BS, upon receivingthe request message from the mobile client, forwards the re-quest message to each K server. The servers reply with a K

dummy message to the forwarding BS. The BS, upon receiv-ing the reply message from the K server, sends a ready mes-sage to each mobile client. No multimedia packets are sentduring the beginning round. In later rounds, there are at mostL request messages from the mobile client, K messages fromthe base station and L reply messages from the multimediaservers. There are I rounds which imply that the total num-ber of messages per request is I (2L+K) = O(I (K +L)). In

Figure 9. Message complexity for two-phase algorithm: nonuniform case.


Figure 10. Message complexity for two-phase algorithm: uniform case.

the nonuniform case, at 20 ms, 40 ms and 60 ms delay jitters,the message overhead per request is between 8 and 9 mes-sages for small numbers of requests and large numbers of re-quests. The four packet scheduling algorithms have no effecton the number of messages per multimedia packet requests inthe nonuniform case. The message complexity, in this case,implies that there is little or no buffer overflow at the mobileclients. The number of messages per request increase whenbuffer overflow at the mobile client occurs. All the packets inthe mobile clients buffer must be requested again, increasingthe rounds I, thus increasing the number of overhead mes-sages. In the uniform case at 20 ms, 40 ms, 60 ms and 80 ms,all of the four packet scheduling algorithms message com-plexity is between 8 and 9 messages for small numbers ofrequests and large numbers of requests. The message com-plexity in this case implies little or no buffer over flow at themobile client. In cases where buffer overflow does not occur,the mobile client need not request multimedia packets to re-place lost multimedia packets caused by buffer overflow at themobile client. Figures 9 and 10 illustrate the MoSync algo-rithm’s low message complexity rate and show that the play-out of multimedia packets at the BS onto the mobile clienthave little effect upon the MoSync algorithm’s message com-plexity rate.

6.5. Multimedia unit arrival rate

Figures 11 and 12 illustrate the multimedia packets arrivalrate (intra-stream synchronization) for both the nonuniformand uniform cases. The multimedia packets arrival rate tellsus much about the condition of the network in terms of thedelay between multimedia packets delivered to the mobileclient. The object is to have a constant rate of multime-dia packets, from the servers, that match the mobile client’sprocessing rate of multimedia packets. The MoSync algo-rithm maintains intra-synchronization when the servers sendrate matches the mobile client play-out (processing rate) andthere is no buffer underflow or buffer overflow at the mobileclient. At 20 ms uniform delay jitters, the four packet schedul-ing algorithms all ensure intra-stream synchronization. Atmost the delay is 6 ms between successive multimedia pack-ets. To play-out audio stream smoothly at the mobile client re-quires intra-stream synchronization within 11 ms, for tightlycoupled audio [21]. At 40 ms uniform delay jitters, the over-all delay increases in terms of arrive time because of the in-creased delays introduced into the network. Intra-stream syn-chronization remains tight in the 40 ms uniform case. Forlarge numbers of multimedia packet requests, the intra-streamsynchronization becomes erratic because no feedback mecha-nism is used for large numbers of multimedia packet requests.The multimedia servers send the multimedia packets directlyto the mobile client via the BS for large numbers of multime-dia packet requests. At 60 and 80 ms uniform delay jitters,intra-stream synchronization is guaranteed up to 100 multi-media packet requests. In the nonuniform case, at 20 ms de-lay jitters, the intra-stream synchronization is guaranteed upto 100 multimedia requests. The four packet scheduling algo-


Figure 11. Multimedia units arrival for two-phase algorithm: uniform case. Figure 12. Multimedia units arrival for two-phase algorithm: nonuniformcase.


rithms guaranteed intra-stream synchronization at 60 ms and80 ms uniform delay jitters. In the nonuniform case, a feed-back mechanism is used to calculate the expected arrival timeof the next multimedia packet for smaller numbers of mul-timedia packet requests. The next packet arrival calculationallows the BS to schedule multimedia packet for smooth play-out onto the mobile client. In the 40 ms and 60 ms nonuni-form cases, we can see that the calculations of the next multi-media packet arrival time allows for guaranteed intra-streamsynchronization up to 100 multimedia packet requests. Theon-time arrival of packets allow the BS to play-out the multi-media packets onto the mobile client smoothly. At large num-bers of requests, no feedback mechanism is used and mul-timedia servers send the multimedia packets directly to themobile client, without continuous updates being made by themobile client to the servers via the base station. The MoSyncalgorithm operates at the network’s application layer. The ap-plication layer absorbs network delay jitters by buffering dataand playing out multimedia packets as a stream of data ontothe mobile client. The MoSync algorithm maintains intra-stream synchronization using WFQ, RR, PQ and FIFO queu-ing during play-out of multimedia packets onto mobile client,by the BS. We can conclude that the order in which the mul-timedia packets are played-out onto the client has little effecton intra-stream synchronization because all of the four packetscheduling algorithms perform intra-stream synchronizationequally.

6.6. Delay between multiple data streams

Figures 13 and 14 illustrate the delay between multiple datastreams servicing the same mobile client concurrently (inter-stream synchronization) for both the nonuniform and uni-form case. In figure 13, under uniform delay jitters, theMoSync feedback mechanism works very well. The fourpacket scheduling algorithms average about 5 ms betweenthe data, text, audio and video data streams, servicing themobile clients. Notice, after 80, the number of requests forthe 4 algorithms increases in delay between data streams be-cause no feedback mechanism is used for large numbers ofrequests. For large numbers of multimedia packet requests,the server sends the multimedia packets directly to the mo-bile client to avoid feedback messages congestion at the BS.The ability for the concurrent data streams to remain within5 ms of packet delivery illustrates and proves that the Mo-Sync algorithm guarantees inter-stream synchronization. Infigure 14, we use nonuniform delay jitters. At 20 ms delay jit-ters, the four algorithms all performs extremely well becausesmall delays are compensated by shifting the small delay tothe higher layers of the network where they are buffered andplayed-out smoothly. After 80 requests, the delay becomeserratic because, for large numbers of multimedia packet re-quests no feedback mechanism is used but still remains in thebounds for smooth multimedia packet play-out onto the mo-bile client. Under nonuniform 40 ms delay jitters, RR outperforms PQ, WFQ and FIFO. RR out performs PQ, WFQand FIFO inter-stream synchronization because RR has the

Figure 13. Delay between data streams for two-phase algorithm: uniformcase.


Figure 14. Delay between data streams for two-phase algorithm: nonuniformcase.

ability to isolate flows by allowing all flows an equal amountof time to play-out. Under large network delay conditions,the network is unable to compensate for the large delay jittersusing FIFO, WFQ and PQ because only the flows with higherpriority are played-out resulting in very large delays for lowerpriority flows. FIFO, WFQ, PQ does not isolate higher band-width usage flows such as video from lower bandwidth usagedata flows such as text. When higher bandwidth usage flowover utilize the bandwidth, lower concurrent bandwidth us-age flows such as text, graphics and audio arrives to the mo-bile client late, hence inter-synchronization is not maintainedand the quality of service at the mobile client is decreased interms of play-out of multimedia packets because mobile clientcannot play-out multimedia packets until the latest packet hasarrived.

RR scheduling allows the bursty flows to be spread outamong flows and not utilize the majority of the network’sbandwidth. RR’s ability to utilize the bandwidth efficientlyallow each data flow to arrive to the mobile client on time.When the multimedia packets of concurrent streams arriveon time, the mobile client can give correct feedback to theBS per multimedia packet request. When correct feedbackis given to the BS by the mobile client, the BS can syn-chronize multimedia packets arrival times thus, improve theQoS at the mobile client in terms of multimedia packet play-out. Under 60 ms nonuniform delay jitters, RR out performsPQ, WFQ and FIFO. With RR scheduling, the delay time be-tween the concurrent data streams are half of the delay ofFIFO, WFQ and PQ scheduling algorithms. Because largenetwork delay jitters effect the inter-stream synchronizationdifferent scheduling algorithms need to be used accordinglywhen multimedia packets are played-out by the base stationonto the mobile client. We can conclude that RR clearlyout performs WFQ, PQ and FIFO scheduling under nonuni-form delay network jitters in terms of inter-stream synchro-nization. RR packet scheduling at the base station main-tains inter-stream synchronization best among the 4 packetscheduling algorithms. With RR scheduling, inter-streamsynchronization is maintain and on-time delivery of multime-dia packets is guaranteed, enhancing the quality of service atthe mobile client in terms of smooth multimedia packet play-out.


In this paper we make a correlation between the play-out ofmultimedia packets from the base station onto the mobileclient and wireless network multimedia synchronization. Weinvestigate four packet scheduling algorithms: First-Come-First-Out (FIFO), Highest-Priority-First (PQ), Weighted-FairQueuing (WFQ) and Round-Robin (RR). We show the dif-ferent algorithm under both uniform and nonuniform delayjitters. We analyze their behavior, based on the QoS of themobile client in terms of buffer usage at the mobile client,underflow at the mobile client, overflow at the mobile client,the message complexity and intra-stream synchronization and


inter-stream synchronization during mobile client request andplay-out of multimedia packets. The different packet schedul-ing algorithms affect the QoS of the mobile client, in termsof buffer overflow, buffer underflows, buffer usage and intra-stream synchronization and inter-stream synchronization. Weconclude that the RR packet scheduling is the most efficient inthe wireless multimedia in terms of synchronization betweenmultimedia servers, BS and mobile client. The RR packetscheduling algorithm allows for 50 percent less delay betweenconcurrent data streams. The decrease in delay of concurrentdata streams enhances the QoS at the mobile client and al-lowed for smooth multimedia packet play-out at the mobileclient. RR packet scheduling, with its fair packet schedul-ing and flow isolation technique, is the best choice in termsof buffer usage and inter-stream synchronization, at the ap-plication layer, for multimedia packet play-out. Through ourresearch, we discover that the play-out of multimedia pack-ets, by the BS onto the mobile client affects the overall net-work synchronization and can be used to enhance the QoS atthe mobile client by allowing on-time delivery of multimediapackets to the mobile client.

We see several directions for future research. We plan toextend our MoSync scheme and make them power aware [6].

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Azzedine Boukerche is a Canada Research Chairand an Associate Professor of Computer Sciences atthe School of Information Technology and Engineer-ing (SITE) at the University of Ottawa, Canada. Heis also the Founding and Director of PARADISE Re-search Laboratory (PARAllel, Distributed and Inter-active Simulation of LargE scale Systems and Wire-less&Mobile Networking). Prior to this he was anAssistant Professor of Computer Sciences at the Uni-versity of North Texas, and Director of the Parallel

Simulations, Distributed and Mobile Systems Research Laboratory) at UNT.He also worked as a Senior Scientist at the Simulation Sciences Division,Metron Corporation located in San Diego. He was employed as a Faculty atthe School of Computer Science (McGill University, Canada) from 1993 to1995. He spent the 1991–1992 academic year at the JPL-California Institute


of Technology where he contributed to a project centered about the specifica-tion and verification of the software used to control interplanetary spacecraftoperated by JPL/NASA Laboratory.

His current research interests include wireless and mobile networks, dis-tributed and mobile computing, distributed systems, parallel simulation, dis-tributed interactive simulation, and performance modeling. Dr. Boukerchehas published several research papers in these areas. He was the recip-ient of the best research paper award at IEE/ACM PADS’97, and the re-cipient of the 3rd National Award for Telecommunication Software in 1999for his work on a distributed security systems on mobile phone operations,and has been nominated for the best paper award at IEEE/ACM PADS’99and at the ACM Modeling, Analysis and Simulation of mobile and wire-less systems Conference 2001. He served the General co-Chair of the prin-ciple Symposium on Modeling Analysis, and Simulation of Computer andTelecommunication Systems (MASCOTS), in 1998, General Chair of theFourth IEEE International Conference on Distributed Interactive Simula-tion and Real Time Applications (DS-RT2000), General Chair for the 3rdACM Conference on Modeling Analysis, and Simulation of Wireless andMobile Systems (MSWiM’2000), as Program Co-Chair for the 5th IEEE In-ternational Conference on Mobile and Wireless Computing and Communica-tion (MWCN’03), ACM/IFIP Europar 2003, IEEE Wireless Local Networks(WLN’03), the 35th SCS/IEEE Annual Simulation Symposium ANSS2002,and the 10th IEEE/ACM Symposium on Modeling Analysis, and Simula-tion of Computer and Telecommunication Systems (MASCOTS’2002), the3rd International Conference on Distributed Interactive Simulation and RealTime Applications (DS-RT’99), and ACM MSWiM’2000, and as a DeputyVice Chair of Wireless and Mobilty Access Track for ACM WWW 2002, asa Guest Editor for VLSI Design, the Journal of Parallel and Distributed Com-

puting (JPDC), ACM Wireless Networks (WINET), ACM Mobile Networksand Applications (MONET), and the International Journal of Wireless andMobile Computing. He was the main organizer of a Special Session on Per-formance Analysis of Mobile and Wireless Communication systems at the7th IEEE HiPC Conference. He has been a member of the Program Commit-tee of several international conferences such as ICC, VTC, ICPP, MASCOTS,BioSP3, ICON, ICCI, MSWiM, PADS and WoWMoM, LWN, Networkingconferences.

Dr. A. Boukerche serves as a General Chair for the 1st International Con-ference on Quality of Service on Heterogenous Wired/Wireless Networks(QShine 2004), a Program Chair for the 5th IEEE Workshop on Wireless,Mobile Ad-Hoc and Sensors Networks, as an Associate Editor for the In-ternational Journal of Parallel and Distributed Computing (Mobile Comput-ing Area), ACM/Kluwer Wireless Networks, Wiley International Journal onWireless Communications and Mobile Computing, and SCS Transactions onSimulation, a Founding and a Steering Committee Chair of ACM MSWiMSymposium, ACM Performance Evaluation of Wireless Ad hoc and SensorsNetworks (PE-WASUN), and IEEE DS-RT Symposium, and on the IEEETask Force on Cluster Computing (TFCC) Executive Committee. He is amember of the IEEE and ACM.E-mail: [email protected]

Harold Owens II received his B.Sc. degree and M.Sc. degree from the De-partment of Computer Science at the University of North Texas. His mainareas of research interests include mobile ad hoc networks, and wireless mul-timedia systems.

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