Energy efficient all-to-all broadcast in all-wireless networks

Information Sciences 180 (2010) 1781–1792

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Information Sciences

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Energy efficient all-to-all broadcast in all-wireless networks

Doina Bein *, S.Q. ZhengDepartment of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083-0688, USA

a r t i c l e i n f o

Article history:Received 20 November 2008Received in revised form 1 August 2009Accepted 6 November 2009

Keywords:All-to-all broadcastBroadcast treeConvergecastEnergy efficiencyMinimum powerWireless network

0020-0255/$ - see front matter � 2009 Published bdoi:10.1016/j.ins.2009.11.013

* Corresponding author. Tel.: +1 972 883 2184; faE-mail addresses: [email protected] (D. Bein), s

a b s t r a c t

All-to-all broadcast is a communication pattern in which every node initiates a broadcast.In this paper, we investigate the problem of building a unique cast tree of minimum totalenergy, which we call Minimum Unique Cast (MUC) tree, to be used for all-to-all broadcast.The MUC tree is unoriented and unrooted. We study three known heuristics for the mini-mum-energy broadcast problem: the Broadcast Incremental Power (BIP) algorithm, theWireless Multicast Advantage-conforming Minimum Spanning Tree (WMA-conformingMST) algorithm, and the Iterative Maximum-Branch Minimization (IMBM) algorithm.Experimental results conducted on various types of networks are reported. We show thatneither of these methods is best overall for building all-to-all broadcast trees.

� 2009 Published by Elsevier Inc.

1. Introduction

In wireless ad hoc networks basic tradeoffs exist between energy and information, and their time critical effect on oper-ations have been studied extensively. Architectural design issues with regards to the number of nodes needed to cover a cer-tain region, placement of such nodes, clustering, and routing are generally treated as emergent behaviors. Worst-casescenarios are studied in order to obtain either exact or approximate solutions.

One calls a communication the process of sending a message in a single transmission. Broadcast (one-to-all), multicast(one-to-some), convergecast (all-to-one) and anycast (all-to-all) are important communication mechanisms for diffusinginformation in the network. A broadcast is the mechanism through which a message sent by a fixed node, called initiatoror source, reaches the rest of the nodes. A multicast is similar to a broadcast, with the restriction that the message needsto reach only a subset of nodes. Many routing protocols for wireless networks need a broadcast or multicast mechanismto update the routing tables in order to maintain routes between nodes. Given a fixed node called sink, a convergecast isthe mechanism through which each node sends a message to that sink. Thus a convergecast is the dual of a broadcast sincethe data flows back to a single node. For an example of convergecast, consider acquiring data in a wireless sensor networkfrom the leaf nodes (i.e. the sensors), to the root node (referred to as the ‘‘source” or ‘‘initiator”) for collection and analysis.Indeed, this is the most common type of communication in a wireless sensor network. If the convergecast to a certain node ruses as communication paths a broadcast tree rooted at r, then the total cost of this convergecast can be different from thetotal cost of the broadcast initiated by r. The reason is that during a convergecast, each node has to send the data to its parentand not to its children. Thus the total energy spent during a convergecast, which uses communication paths arranged as atree rooted at the sink node, is the sum of the energy spent by each node to send that packet to its parent.

In all-to-all broadcast, every node is an initiator for broadcasting messages in the network. But constructing and main-taining n individual broadcast trees for an n-node network where each is rooted in an initiator is unfeasible since it requires

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Tree

Network

Cast Tree

Fig. 1. Minimum cast tree for all-to-all broadcast.

1782 D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792

large memory space and/or large and fast processing capabilities at each node. Large bandwidth is needed when such broad-cast trees are computed. Additionally, significant bandwidth is needed when a network topology change occurs. This is be-cause in the worst case, all nodes have to recompute their broadcast trees when a single topology change occurs. A simplesolution, albeit the most power-consuming, is to allow a node to send its data directly to all the nodes in the network. Thenthe transmission range of any node is proportional to the maximum distance between a pair of two nodes.

In a wireless network, a node equipped with an omnidirectional antenna can simultaneously communicate with multiplenodes located within some distance which is called the transmission range. A message sent by some node v with transmissionrange rv is received simultaneously by all nodes situated at a distance no more than rv . We call these nodes siblings or neigh-boring nodes.

The power required to support a communication link of length d between two nodes is divided into two components: thepower to send a packet and the power to receive a packet. To send a packet at distance d, a node uses the power

Pe ¼ da þ ce: ð1Þ
Here a is a constant parameter which depends on the medium and typically has a value between 2 and 4. The value ce is aparameter representing an overhead due to signal processing.
For any two nodes u and v, let duv be the physical distance between u and v. We use notation Puv to denote the minimumenergy that node u has to use for the transmission of one message in order to ensure that the message will reach node v.Consider an example of four nodes, a; b; c, and d, with arbitrary coordinates chosen in such a way that dab < dac < dad. To com-municate with the nodes b; c, and d, node a would spend at least Pab; Pac , and Pad energy to reach each individual node. But atransmission of power P1 ¼ maxfPab; Pac; Padg at node a with an omnidirectional antenna will reach the nodes b; c, and dsimultaneously, while a transmission of power P2 ¼maxfPab; Pacg at node a with an omnidirectional antenna will reach onlynodes b and c.

There is a tradeoff between the energy spent and the connectivity among nodes in the network. In order to save energy, anode tends to reduce its transmission range, in this way reducing also the number of possible neighbors. But in order to keepthe network connected, some nodes may have to spend more energy to reach other nodes beyond the nearest ones. If that isthe case, a node will attempt to minimize its transmission range and thus its power level, while some nodes will have toincrease their power level in order to maintain a connected network.

In this paper, we address the issue of minimizing total energy for the all-to-all broadcast when all nodes need to broad-cast one message each. We propose a power assignment for each node with the purpose of creating communication pathsform an unrooted spanning tree, with the additional requirement that the tree has the minimum total energy among all suchpossible trees. We call this tree T the minimum unique cast (MUC) tree (see Fig. 1).

We propose a heuristics, called MUCT, which determines for each node i in the network the minimum-power broadcasttree rooted at i using some approximation algorithm to build broadcast trees. (In our experiments we use the algorithmsproposed in [28,13,4].) Then MUCT selects the tree T which achieves the minimum value for the total power, powerT , givenby Eq. (3).

The paper is organized as follows. In Section 2 we present related work. In Section 3 we define the MUC problem. Wepropose an approximation algorithm for the MUC problem in Section 4. In Section 5 we show the relationship betweenthe broadcast, the convergecast, and the MUC problem, and we give upper and lower bounds for the total power requiredby the MUC problem. Experimental results are presented in Section 6. We conclude in Section 7.

2. Related work

Minimizing total energy consumption – the sum of the energy spent by each node – for broadcast and multicast has beenthen extensively studied in the literature under the assumption that the transmission range of a node can be adjusted (see[28,23,29,13,25,31,4,9,26,30,24,3]).

D. Bein, S.Q. Zheng / Information Sciences 180 (2010) 1781–1792 1783

The work of Wieselthier et al. [28] as well as Stojmenovic et al. [23] has given a new orientation in designing broadcastand multicast trees for wireless networks. Wieselthier et al. [28] introduced the notion of wireless multicast advantage andproposed the Broadcast Incremental Power (BIP) algorithm. Starting from the source node, BIP constructs the broadcast treeincrementally, by adding nodes to the tree based on the minimum additional cost. Empirical results showed the advantage ofBIP over shortest-path tree, minimum-spanning tree, and variations, but no performance analysis was provided. Stojmenovicet al. [23] proposed the concept of internal nodes as alternatives to clusterheads, to reduce the communication overhead ofbroadcasting. A yardstick metric was proposed in [29] to measure the efficiency of approximation algorithms, in terms of howmany destinations are reached per unit energy; the purpose is to reach a large fraction of the number of desired destination(for a multicast-tree). Li and Nikolaidis [13] proved that the minimum-energy broadcast problem (i.e. building a broadcasttree rooted at some given node) is NP-hard, and proposed the Iterative Maximum-Branch Minimization (IMBM) algorithm.Independently, Cagali et al. [4] and Liang [14] proved that the problem is NP-complete. Maric and Yates [17] proposed a newstrategy, called accumulative broadcast, in which a node collects energy from unreliable reception. Das et al. [7] proposedthree integer programming models for the broadcast/multicast tree problem, which can be solved optimally usingbranch-and-bound (exponential time), but also gave approximations using genetic algorithms. Montemanni and Gambard-ella [18,19] as well as Montemanni et al. [20] studied the problem of assigning transmission power to the nodes of a staticwireless network in such a way that all nodes are connected by bidirectional links and the total power is minimized.

Once the broadcast tree is constructed, it needs to adapt to unreliable communication from interference and to link fail-ures due to energy depletion of some nodes. Wang and Chao [27] proposed a backup mechanism at each node for quickrecovery of the routing data due to link failures. The algorithm proposed, called Dynamic Backup Routes Routing Protocol(DBR2P), creates alternative routes on-demand. Chang et al. [5] addressed the problem of degraded performance of an broad-cast tree due to node mobility or diminished energy, and proposed to search for alternative routes to compensate for thebroken links.

For building a broadcast tree of minimum total energy, the source node and the network are the instance of the problem.The total power used by the nodes in the tree rooted at some source node and spanning the entire network is to be mini-mized among all such trees. Hence, for an n-node wireless network, each node has to keep track of n rooted broadcast trees,including the one rooted at itself. Papadimitriou and Georgiadis [21] constructed a single broadcast tree for the entire net-work; broadcasting initiated by any source node takes place in a predetermined manner. They proposed a polynomial-timealgorithm for building such a tree, in which the total power consumed for broadcasting by any node is at most 2Hðn� 1Þtimes the optimal power. (The function HðnÞ is the harmonic function, HðnÞ ¼

Pnk¼1

1k). The algorithm works for any type

of weighted, general networks. They established that the minimum-spanning tree (MST) for Euclidean graphs is not alwaysthe broadcast tree of minimum total power. They also showed that the MST is within D times the optimal total power of anytree, where D is the maximum node degree in the network.

The convergecast problem proposed by Chlamtac and Kutten [6] has been studied from the point of view of collisiondetection [1,11,33], latency (total steps needed to collect the data) [32,12,8], and network lifetime (the time interval untilthe first node depletes its battery) [10]. For wireless sensor networks, Huang and Zhang [11] proposed a coordinated con-vergecast protocol to solve the collision problem. Zhang and Huang [33] studied data aggregation and duplication, whereas[32,22] focused on channel contention and packet collision, when a high-volume traffic occurs in a short period of time. Inthis context they designed a window-less block acknowledgment scheme. Gandham et al. [8] and Kesselman and Kowalski[12] studied the problem of minimizing the total time necessary to complete a convergecast. They proposed distributed con-vergecast scheduling algorithms as well as randomized distributed algorithms for the TDMA problem. He et al. [10] pre-sented an aggregation protocol with the objective of maximizing the network lifetime and minimizing the error of senseddata. The authors proposed to periodically modify a filter threshold for each sensor in a way that is optimal, by translatingthe problem into a mathematical programming formulation with constraints. Their work took into consideration the objec-tive of the user, the current characteristics of the sensor network (namely the power remained at each node and the connec-tivity) and the characteristics of the sensed data.

All-to-all broadcast consumes much energy. Few results exist for this type of communication. Existing solutions for single[21] or source-dependent broadcast trees [28] are not optimal for all-to-all broadcast, since back-communication to the ini-tiator is not considered. The effect on energy efficiency was studied by [15,16]. An earlier approach [15] is to select a centralnode that is the closest to all the nodes, let the central node collect the data and send then to everyone else. This approach isnot very expensive in power, but requires central coordination. It is extended to a distributed scheme ([16]), in which thenetwork is partitioned into clusters and some selected nodes (called clusterheads) play the role of a central node for theirclusters. Bauer et al. [2] proposed a data structure, called legend, which gathers and shares its contents with visited nodes;several traversal methods have been explored.

3. Models

A static n-node wireless network can be modeled as a pair ðV ;wÞ where V is the set of nodes, jV j ¼ n, and w is a non-neg-ative function, which is defined over V � V , and which measures the distance between the nodes. If the nodes have assignedcertain power values then we model the network as a weighted digraph G ¼ ðV ; E;wÞ, where E is the set of arcs that representunidirectional communication links: The power level of some node decides to which nodes it is connected. We note that if


the power value of some node changes, then the set E may change. Given any two nodes i and j in G, we define the functioncost to be costij ¼ wa

ij if ði; jÞ 2 E and 1 otherwise (where a is the constant defined in Eq. (1)).Nodes do not necessarily have the identical transmission ranges. The transmission range of each node dictates whether

there is an arc to some other node. Thus between two nodes u and v, there can be an arc ðu;vÞ 2 E, or ðv; uÞ 2 E, or both, orneither. Consider the example given in Fig. 2. The power values of each node are in written in the box under the node IDs. Thepower level of node a has is high enough for a transmission to cover the distance to node c. The power level of node b isenough to cover the distance to node d, and for node d to cover the distance to node c. But node c does not have enoughpower to reach any node. If we consider the function w to be the Euclidean distances between nodes, then there is a bidi-rectional communication between nodes b and d since wbd < wcd, and a unidirectional communication from a to b, aswbd < wab. Node c is isolated, since there is no node situated at a distance less than wcd.

For the rest of the paper we assume the Euclidean distance as the metric distance among the nodes. By selecting (n � 1)edges to connect the nodes in G, we are able to obtain an unoriented (or unrooted) tree T ¼ ðV ; ETÞ. The total number of pos-sible unoriented trees buildable from an n-node network is no more than ðnþ 1Þn�1, by the Cayley’s tree enumeration. In anunoriented tree, there is no notion of parent or child for a node, only the notion of siblings – the set of nodes to which a nodeis connected. For unoriented tree T and for any node i, let Si be the set of siblings of i in T: Si ¼ fj 2 V : ði; jÞ 2 ETg. We proposeto select the transmitting energy at a node based on the distance to the farthest sibling. Given a tree T ¼ ðV ; ETÞ rooted atsome node r which spans the underlying graph G ¼ ðV ; EÞ, we propose a new measure for the power used by a node i,powerT

i , to be

powerTi ¼ maxj2Si

costij: ð2Þ

We denote by powerT the total power of all the nodes in the tree T:

powerT ¼X

i2V

powerTi : ð3Þ

Different power assignments at a node generate different network topologies. In Fig. 3, let jxyj be the distance betweennodes x and y, for x; y 2 fa; b; c; dg, in such a way that jcdj < jacj < jabj < jbcj < jbdj; jacj < jabj < jadj, and jcdj < jadj.

In Fig. 3a, if node c has power Pca (enough to reach nodes a and d but not node b), then the only possible spanning tree forthe entire network is T1, which is the tree rooted at node b and with the set of arcs fðb; aÞ; ða; cÞ; ðc; dÞg. The total power usedfor broadcasting with node b as the source node is given by Pba þ Pac þ Pcd. If the nodes set their power levels as indicated byEq. (2), then powerT1 ¼ Pab þ Pba þ Pca þ Pdc.

Pac

a< Pcd

Pbd

bPcd

c

d

Fig. 2. Asymmetric communication between nodes.

Pac

Pba

Pca

Pdc

b

a

c

d

Pac

Pba

Pcb

Pdc

b

a

c

d

a

b

Fig. 3. Selecting the transmission power.


In Fig. 3b, if node c has power Pcb (enough to reach all the nodes a, b, and d), then another tree T2, which is rooted at c andwith the arcs fðc; aÞ; ðc; bÞ; ðc; dÞg, can be considered. The total power used for broadcasting with node c as the source node isgiven by Pcb. If the nodes set their power levels as indicated by Eq. (2), then powerT2 ¼ Pac þ Pbc þ Pcb þ Pdc .

One can observe that T2 has total broadcast power smaller than T1 since jbcj < jbaj þ jacj (triangle inequality),jbaj þ jacj < jbaj þ jacj þ jcdj, and power at a node is defined as in Eq. (1). At the same time, powerT1 < powerT2 since we as-sume jbaj < jbcj. Thus selecting the tree with the minimum total broadcast power is not necessarily a good heuristic forselecting the tree with the minimum total power for all-to-all broadcast.

During a single-source broadcast a node may receive the same message more than once. Therefore, the broadcast does notinduce an oriented tree, but a connected graph which contains an oriented tree. Then, by eliminating the redundant mes-sages, we can consider the induced graph to be a tree. We formulate the minimum unique cast (MUC) graph problem asfollows:

MUC: Given a wireless network ðV ;wÞ, assign power levels to the nodes such that the corresponding unidirectional links(arcs) formed between nodes induce a strongly connected graph which can be used for all-to-all broadcast and such thatthe sum of power of all nodes is minimum.

By choosing some node in the network as the root (or source node), an unoriented tree can become oriented: select theparent of each node as the neighboring node with the shortest distance to the root.

4. Minimizing total energy for all-to-all broadcast

Since the value powerT is an upper bound for the total power to be spent during any single-source broadcast or any single-sink convergecast in T (Theorem 1), we need to select the tree with the minimum such value. We also show that for any treeT, in case of a broadcast followed by convergecast towards the same node, we cannot do better than powerT . In other words,the total energy spent is more than powerT . Thus powerT is a lower bound for any broadcast followed by a convergecast (Lem-ma 5)).

Consider the following algorithm MUCT for a given wireless network ðV ;wÞ. For each node r in V, one can construct asingle-source broadcast tree Tr rooted at r using some approximation algorithm. (See [28,23,29,4,17,7,18–20] for approxima-tion algorithms). For each tree Tr , consider at the nodes the power levels given by Eq. (2) and the total power given by Eq. (3).

We apply the following heuristic: select the tree To that has the minimum value for the total power, powerTo ¼min8r2V

powerTr.

Algorithm 4.1. Algorithm MUCT (Minimum Unique Cast Tree))

Read the input: ðV ;wÞInitialization: Let powero ¼ 1 and OPT ¼ ;.Main Procedure:

Forall r 2 V doBuild a broadcast tree rooted at r;BT , using an approximation algorithm A.Let T to be the unoriented tree obtained by ignoring the orientation in BT.Define the power level of some node i as in Eq. (2).Compute powerT as in Eq. (3).If powerT < powero then set OPT to T and powero to powerT .

Endfor

The power level of any node in the tree To ensures a bidirectional communication with its siblings. Thus the tree can beused for all-to-all broadcast, and Algorithm MUCT is an approximation for the MUC problem.

We note that when a node sends data to its parent, all its children receive the same data since they are within the range:when node i sends its data back to its parent pi, the data is also automatically received by all i’s children at no additional cost.When the data is then forwarded to the parent of pi, all pi’s children receive it at no additional cost, and so on. These nodescould also send the packet along their own subtrees, eventually piggybacking with other data.

5. Proofs

Let T be an oriented tree spanning the underlying graph of G, rooted at some node r and constructed by some approxi-mation algorithm. We choose the power of some node i to be the maximum cost for reaching the siblings (Eq. (2)). We showthat between any two siblings in T there is a bidirectional communication link (Lemma 1). We also show that during a broad-cast from any node in the tree, the total power spent is less than the value of powerT given in Eq. (3) (Lemma 2). Furthermore,during a convergecast back to any node in the tree the total power spent is less than the value of powerT (Lemma 3). Thisconcludes a lower bound for powerT (Theorem 1).


Then, we show that the total power spent during a convergecast is greater than half of powerT (Lemma 4) and during abroadcast followed by a convergecast, the total power spent is at least powerT (Lemma 5). This concludes an upper bound forpowerT (Theorem 2).

Lemma 1. For any node i in the tree, if the power level at i is defined as in Eq. (2) then there is a communication path between anytwo nodes in the tree.

Proof. Recall that the power level of a node decides whether there is a unidirectional, bidirectional, or no communicationbetween the node and some other node. We show that there will be a bidirectional communication between any two neigh-boring nodes in T if we select the power level of some node to be the value given in Eq. (2). Let i be some node in T and j besome sibling of i in T, j 2 Si. From Eq. (2) it follows that powerT

i P costij thus there is an unidirectional communication linkfrom i to j. Similarly, powerT

j P costji for node j. Therefore, there is a unidirectional communication link from j to i. Thusbetween any two sibling nodes in T there is a bidirectional communication. This implies that there is a communication pathbetween any two nodes in the tree. h

Since between any two neighboring nodes there is a bidirectional communication link, we can consider T to be a graphinstead of a digraph.

When we consider the orientation of the tree T with respect to some node r as the root, the ancestor/descendant relation-ship between nodes induces a partial order: i<r j (or simply i < j if r is understood) denotes that node i is an ancestor of nodej.

We term r-broadcast to be the broadcast initiated by node r. Also, we denote by r-convergecast the following process: aleaf node ends a message to its parent and a non-leaf node collects all the messages from its children into a single messagethat is set to its parent.

Lemma 2. If any node i in the tree T has the power level as defined in Eq. (2) , then the total power used for a broadcast initiated bysome node r along the tree T is less than the value of powerT given by Eq. (3).

Proof. During the r-broadcast, a node i in T uses the power

pr;Ti ¼ max

8j2Sir6i<j

costij: ð4Þ

The total power spent for r-broadcast is pr;T

pr;T ¼X

i2V

pr;Ti : ð5Þ

Obviously, pr;Ti 6 powerT

i , for any nodes i and r. Moreover, if i is a leaf node of the oriented tree T rooted at r, then pr;Ti ¼ 0,

thus pr;Ti < powerT

i , for any node r. It follows then that the total power spent for r-broadcast, pr;T , given by Eq. (5), is strictlysmaller than the value of powerT . h

Lemma 3. If any node i in the tree T has the power level as defined in Eq. (2) , then the total power used for a convergecast towardssome node r along the tree T is less than the value of powerT given by Eq. (3).

Proof. Some node i other than r uses during the r-convergecast an amount of power, ppr;Ti , equal to

ppr;Ti ¼ costik; ð6Þ

where k 2 Si; r 6 k < i, and ppr;Tr ¼ 0.

The total power spent for r-convergecast is ppr;T

ppr;T ¼X

i2V

ppr;Ti ¼

X

ði;jÞ2T;i<j

costij: ð7Þ

Obviously, ppr;Ti 6 powerT

i , for any nodes i and r. Moreover, if i is a leaf node in the oriented tree T rooted at r thenppr;T

i ¼ powerTi . Note also that ppr;T

r ¼ 0 < powerTr .

It follows then that the total power spent for r-convergecast, ppr;T , given by Eq. (7), is strictly smaller than the value ofpowerT . h

We can then conclude:

Theorem 1. For any node r in the network G and any tree T which spans G, if the power level at r is the one defined in Eq. (2) then

powerT P maxðpr;T ;ppr;TÞ:

Lemma 4. The total power used for convergecast towards some node r in the tree given by Eq. (7) is greater or equal to half ofpowerT given by Eq. (3), for any node r.


Proof. Note that 2ppr;T ¼ 2Pði;jÞ2T costij ¼

Pi2V

Pj2SðiÞcostij.

For any set A, the sum of elements in A is greater or equal to the maximum element in A. It follows that, for any nodei;P

j2SðiÞcostij P maxj2SðiÞcostij. In this inequality, the right expression is powerTi . Thus 2ppr;T P powerT . h

Lemma 5. If for any node i in the tree T, the power level at i is the one defined in Eq. (2), then the total power used for a broadcastinitiated by some node r followed by a convergecast towards r along the tree T is greater or equal to the value of powerT given byEq. (3).

Proof. The power spent by a node i during r-broadcast and r-convergecast is pr;Ti þ ppr;T

i .The power spent by all nodes during the r-broadcast and r-convergecast is
X
i2V

ðpr;Ti þ ppr;T

i Þ ¼ pr;T þ ppr;T :

Let SðiÞ be the set of neighboring nodes, which includes the parent of i and its children. Call the parent of i as the node k.Using Eq. (2) it follows that

powerTi ¼ max

8j2Si

costij ¼ maxðcostik;max8j2Sir6i<j

costijÞ:

For any two non-negative values a and b;maxða; bÞ 6 aþ b; it follows that

maxðcostik;max8j2Sir6i<j

costijÞ 6 costik þmax8j2Sir6i<j

costij:

In this inequality, the left expression is powerTi (Eq. (3)), and the right expression is the sum of pr;T

i (Eq. (4)) and ppr;Ti (Eq. (6)).

Thus for any node i, powerTi 6 pr;T

i þ ppr;Ti . It follows then directly that powerT

6 pr;T þ ppr;T . h

We can then conclude:

Theorem 2. For any node r in the network G and any tree T that spans G, if the power level at r is the one defined as in Eq. (2) then

powerT6 ppr;T þminðppr;T ;pr;TÞ:

From Theorems 1 and 2 it follows that maxðpr;T ; ppr;TÞ 6 powerT6 pr;T þ ppr;T .

6. Comparative analysis and results

To validate the effectiveness of the proposed MUCT scheme, we have conducted extensive experiments to compare theminimum power used for all-to-all broadcast based on the broadcast trees generated by algorithms BIP, IMBM, WMA-con-forming MST, in terms of total energy. These experiments took into account different types of data sets and different networksizes. Experimental results conducted on various types of networks using BIP, IMBM, and WMA-conforming MST showedthat both BIP and WMA-conforming MST are good for building all-to-all broadcast trees, while IMBM produces trees of veryhigh energy cost.

We have used four sets of networks. The first set included general networks with arbitrary distance among nodes ran-domly generated in the range [0.01..4.99], and a randomly chosen source node. We call this set of networks random generalnetworks. The second set included Euclidean networks where the coordinates of the nodes were generated randomly in therange [0.01..4.99] and a source node was randomly chosen. We call this set of networks random Euclidean networks. The thirdset included Euclidean networks where nodes were placed on a square grid with distances between rows and columns ran-domly generated in the range of [0.05,1.04] and the source node was randomly chosen from grid nodes. We call this set ofnetworks random Euclidean grid networks. The fourth set included Euclidean networks with nodes placed on a square gridwhere the row and column distances were the same, and the source node was randomly chosen from grid nodes. We callthis set of networks perfect Euclidean grid networks.

Since computing the optimal broadcast tree for a network with more than 10 nodes with randomly generated node loca-tions would have taken too long, we measured the performance of BIP, IMBM, and WMA-conforming MST in terms of min-imum energy used for all-to-all broadcast based on the broadcast trees generated by the corresponding method.

For each network instance m, let PBIP; PIMBM , and PWMA-MST be the minimum total energy used for all-to-all broadcast usingthe broadcast trees generated by algorithms BIP, IMBM, and WMA-conforming MST, respectively. We select the minimumenergy tree among them,

Pmin ¼minfPBIP; PIMBM; PWMA-MSTg:

Then we normalize PBIP; PIMBM; PWMA-MST against Pmin : pðBIPÞ ¼ PGTPmin

; pðIMBMÞ ¼ PGTPmin

and pðWMA�MSTÞ ¼ PWMA-MSTPmin

. The normal-ized energy associated with the tree generated by an algorithm is independent on the size of distance scaling factor.


6.1. Experiments with a low node density

We considered first networks of low node density, within the range 4� 10�5 to 16� 10�4. The number of nodes for thefirst, second, third, and fourth set of networks was in the range of [5,400], [5,400], [9,400], and [9,400], respectively. Similarto [28,13], we ran 1000 simulations for each setup consisting of a network m of a specified size and node distances, param-eter a, and an algorithm.

For random general networks, BIP outperformed WMA-conforming MST when the number of nodes was greater than 10,and both outperformed IMBM. Fig. 4 summarizes the performance of the three algorithms. (The results were obtained bysampling 1000 such networks of 5, 10, 20, 30, . . ., and 400 nodes, respectively.) The reason is that BIP selected a smaller num-ber of internal nodes than WMA-conforming MST, of relatively small transmission radius. IMBM, even though it selected aneven smaller number of internal nodes, the transmission range of these internal nodes was relatively large.

For random Euclidean networks, WMA-conforming MST outperformed BIP very slightly, and both outperformed IMBM.Fig. 5 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networksof 5, 10, 20, 30, . . ., and 400 nodes, respectively.) The reason is that WMA-conforming MST selected a smaller number ofinternal nodes than BIP, of relatively small transmission radius. As for random general networks, IMBM, even though it se-lected an even smaller number of internal nodes, the transmission range of these internal nodes was relatively large.

For random Euclidean grid networks, WMA-conforming MST slightly outperformed BIP and both outperformed IMBM.Fig. 6 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networksof 9, 16, 25, 36, . . ., and 400 nodes, respectively.) The reason is the same as for random Euclidean networks: WMA-conform-ing MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius, while the transmissionradius of the internal nodes selected by IMBM was relatively large.

Fig. 4. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random general networks, a = 2.

Fig. 5. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random Euclidean networks, a = 2.

Fig. 6. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for random Euclidean grid networks, a = 2.

Fig. 7. Normalized total energy for all-to-all broadcast using BIP, IMBM, and WMA-conforming MST for perfect Euclidean grid networks, a = 2.

Fig. 8. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random general networks, a = 2.



For perfect Euclidean grid networks, WMA-conforming MST always outperformed BIP, and both outperformed IMBM.Fig. 7 summarizes the performance of the three algorithms. (The results were obtained by sampling 1000 such networksof 9, 16, 25, 36, . . ., and 400 nodes, respectively.) The reason is the same as for random Euclidean networks and randomEuclidean grid networks: WMA-conforming MST selected a smaller number of internal nodes than BIP, of relatively smalltransmission radius, while the transmission radius of the internal nodes selected by IMBM was relatively large.

6.2. Experiments with a higher node density

We considered then networks of higher node density, within the range 16� 10�4 to 4� 10�3. The number of nodes for thefirst, second, third, and fourth set of networks was in the range of [400, 2400], [400, 2400], [400, 2500], and [400, 2500],respectively. We ran 100 simulations for each setup consisting of a network m of a specified size and node distances, param-eter a, and an algorithm. We measured the performance of BIP, IMBM, and WMA-conforming MST in terms of minimum en-ergy used for all-to-all broadcast based on the broadcast trees generated by the corresponding method. The trees obtained byAlgorithm IMBM had a very large normalized total power (extremely large for general random networks), so we did not in-clude them in the charts. Thus the charts contain only a comparison between the BIP and WMA-MST trees. We note thatneither BIP nor WMA-MST generates the lowest minimum total energy trees overall.

For random general networks, BIP always outperformed WMA-conforming MST. Fig. 8 summarizes the performance ofthe two algorithms. (The results were obtained by sampling 100 such networks of 400, 450, up to 2400 nodes, respectively.)The reason is that BIP selected a smaller number of internal nodes than WMA-conforming MST, of smaller transmissionradius.

For random Euclidean networks, WMA-conforming MST always outperformed BIP. Fig. 9 summarizes the performance ofthe two algorithms. (The results were obtained by sampling 100 such networks of 400, 450, up to 2400 nodes, respectively.)

Fig. 9. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random Euclidean networks, a = 2.

Fig. 10. Normalized total energy for all-to-all broadcast using BIP and WMA-conforming MST for random Euclidean grid networks, a = 2.

Fig. 11. Normalized total energy for all-to-all braodcast using BIP and WMA-conforming MST for perfect Euclidean grid networks, a = 2.


The reason is that WMA-conforming MST selected a smaller number of internal nodes than BIP, of smaller transmissionradius.

For random Euclidean grid networks, BIP outperformed WMA-conforming MST for most instances. Fig. 10 summarizesthe performance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 441, up to2500 nodes, respectively.) The reason is the same as for random Euclidean networks: BIP selected a smaller number of inter-nal nodes than WMA-conforming MST, of relatively small transmission radius, while the transmission radius of the internalnodes selected by WMA-conforming MST was slightly larger.

For perfect Euclidean grid networks, WMA-conforming MST always outperformed BIP. Fig. 11 summarizes the perfor-mance of the two algorithms. (The results were obtained by sampling 100 such networks of 400, 441, up to 2500 nodes,respectively.) The reason is the same as for random Euclidean networks and random Euclidean grid networks: WMA-con-forming MST selected a smaller number of internal nodes than BIP, of relatively small transmission radius, while the trans-mission radius of the internal nodes selected by BIP were slightly larger.

7. Conclusion

We propose an approximation algorithm MUCT which builds a unique cast tree to be used for all-to-all broadcast; thetree is unoriented and has minimum total power. To validate the effectiveness of Algorithm MUCT, we conducted extensiveexperiments. Experimental results conducted on various types of networks using BIP, IMBM, and WMA-conforming MSTshow that neither of these methods is overall the best for building all-to-all broadcast trees. In all the experiments, IMBMproduces trees of much higher energy cost, compared with the trees produced by BIP and WMA-conforming MST.

We also give a number of lower and upper bounds for the total energy of any tree that can be used for all-to-all broadcast.Our proposed algorithm provides a sufficient approximation for the MUC problem, but not a necessary one. It is an interest-ing open problem to find a tree of minimum total power for all-to-all broadcast which does not use a broadcast tree as apreprocessing step.

Another future line of work would be to construct broadcast trees of minimum energy in which certain nodes can only beleaf nodes. This problem is motivated by the following aspect. In order to preserves its energy when is idle, namely it doesnot sense any event, a sensor node can go in a ‘‘listening” mode. In this mode, a sensor node only receives packets, but it doesnot send or forward packets. Thus, it can be part of the broadcast tree only as a leaf node, but not as an internal node.

Acknowledgment

This work is supported in part by National Science Foundation, Grant NSF-0714057.

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Energy efficient all-to-all broadcast in all-wireless networks