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Constructing A Shortest Path Overhearing Tree With Maximum Lifetime In WSNs
Wael Y. Alghamdi, Hui Wu, Wenguang Zheng, Salil S. KanhereSchool of Computer Science and Engineering
The University of New South WalesSydney, Australia
Email: {waela, huiw, wenguangz, salilk}@cse.unsw.edu.au
Abstract—Secure data collection is an important problemin wireless sensor networks. Different approaches have beenproposed. One of them is overhearing. We investigate theproblem of constructing a shortest path overhearing treewith the maximum lifetime. We propose three approaches.The first one is a polynomial-time heuristic. The second oneuses ILP (Integer Linear Programming) to iteratively find amonitoring node and a parent for each sensor node. The lastone optimally solves the problem by using MINLP (Mixed-Integer Non-Linear Programming). We have implementedthe three approaches using MIDACO solver and MATLABIntlinprog, and performed extensive simulations using NS2.35.The simulation results show that the average lifetime of all thenetwork instances achieved by the heuristic approach is 85.69%of that achieved by the ILP-based approach and 81.05% of thatobtained by the MINLP-based approach, and the performanceof the ILP-based approach is almost equivalent to that of theMINLP-based approach.
Keywords-Overhearing; routing tree; network lifetime; se-cure data collection; integer linear programming; mixed-integer non-linear programming
I. INTRODUCTION
A WSN (Wireless Sensor Network) consists of a largenumber of autonomous sensors nodes and a single or multi-ple base stations. Each sensor node delivers the data sensedfrom the physical environment to its designated base station.Typically, sensor nodes are battery powered. Most of theenergy of a sensor node is consumed by communication. Inorder to save energy, the transmit power of each sensor nodeis kept low, leading to a short transmission range. Thus, datacollection is performed in a multi-hop way.
In WSNs, various attacks may exist. Among them areselective forwarding attacks and modification attacks. In theselective forwarding attacks, a malicious sensor node maydeliberately drop some packets received from other sensornodes, resulting in packet loss. In the modification attacks,a malicious sensor node may modify some packets receivedfrom other sensor nodes and forward the incorrect packetsto the base station. In order to ensure that the data sensedby each sensor are delivered to the base station correctly,the protocols for secure routing are required.
The secure and reliable data collection problems in WSNshave been extensively investigated. Many approaches arebased on the overhearing technique [1]. When a sensor nodereceives a packet and forwards it to another sensor node,
a third sensor node will overhear the packet reception andtransmission. Therefore, the overhearing technique can beused to detect the modification attacks and the selectiveforwarding attacks.
In this paper, we investigate the problem of constructinga routing and overhearing topology with the maximumnetwork lifetime in a WSN with a single base station.Specifically, we construct a shortest path tree for routingand select a monitoring sensor node for each sensor nodesuch that the network lifetime is maximized. For each sensornode vi, its monitoring sensor node overhears the receptionand transmission of each packet that the parent of vi receivesfrom vi. The network lifetime is defined as the time whenthe first sensor node depletes its energy. Even without con-sidering overhearing, the problem of constructing a shortestpath tree with the maximum lifetime is NP-Complete [2].
We make the following major contributions.
• We propose a polynomial-time heuristic approach, anILP-based approach, and a MINLP-based approach tothe problem of constructing a shortest path overhearingtree with the maximum network lifetime. To the bestof our knowledge, our work is the first attempt to con-struct such an overhearing topology with the maximumlifetime for security purposes.
• We have implemented our approaches using MIDACOsolver and MATLAB Intlinprog, and performed exten-sive simulations on 150 network instances with threedifferent distributions, namely, uniform, grid, and ran-dom distributions. The simulation results show that theaverage lifetime of all the network instances achievedby the heuristic approach is 85.69% of that achieved bythe ILP-based approach and 81.05% of that obtained bythe MINLP-based approach, and the average networklifetime achieved by the ILP-based approach is 96.2%of that obtained by the MINLP-based approach.
The rest of the paper is organized as follows. SectionII gives a brief survey of the related work. Section IIIprovides the network model and definitions. Section IVdescribes our heuristic approach in detail. Section V presentsour MINLP-based approach. Section VI proposes our ILP-based approach. Section VII shows the simulation resultsand analyses. Lastly, Section VIII concludes this paper.
II. RELATED WORK
The overhearing technique has been widely used to im-prove the security and the reliability of data collection inWSNs. The problem of constructing a routing tree with themaximum network lifetime has also been extensively inves-tigated. Next, we give a survey of the major work relatedto secure and reliable data collection by using overhearingand the lifetime-aware routing tree construction problem inWSNs.
A. Overhearing-Based Secure and Reliable Data Collection
[1] uses an overhearing technique to detect modificationattacks in WSNs. By the overhearing technique, a committeestructure is constructed for each sensor node. The committeestructure includes several committee sets, and each commit-tee set is designed for a specific communication link. Due tothe microwave nature of the wireless channel, neighbouringsensor nodes within a sender’s radio range can overhear thepacket the sender is transmitting. Therefore, each packetcan be examined by the sensor nodes of the committee setduring forwarding. If a packet is modified by a maliciousnode, the committee will detect the anomaly. [3] providesseveral drawbacks of the security mechanism used by [1].Firstly, there is no mechanism implemented for neighbournodes authentication within the construction phase. Hence,the malicious node may penetrate into the network to sendfake information about their neighbours and contribute invoting. Secondly, in the last phase, the proposed mechanismdoes not isolate the malicious nodes from the network.
[4] proposes an efficient overhearing-based reliable trans-fer protocol for WSNs. The monitoring sensor node over-hears a packet being transmitted by the monitored sensornode to determine whether the monitored node is maliciousor not. To make such a decision, a reputation value iscalculated for each sensor node in the designated area. Thesensor node with a reputation value below the threshold isconsidered as a malicious sensor node.
[5] proposes an efficient, reliable transfer protocol forWSNs by introducing implicit and selective acknowledge-ment. The selective acknowledgement mechanism is ex-ecuted by comparing the current path reliability and thebase reliability. Overhearing is also used in the implicitacknowledgement mechanism.
Several approaches [6]–[9] use the overhearing techniqueto avoid redundant information to be transmitted towards thebase station, improving the energy efficiency.
[6] proposes an energy-efficient data transmission re-duction approach for periodical data gathering in WSNsby using the overhearing technique. In this approach, eachsensor node in a WSN autonomously determines whetherits own reading is redundant or not by using the overheardpackets transmitted by its neighbors. If the reading is de-termined as redundant, the sensor node stops transmitting
it. [8] extends this approach by proposing an overhearing-based data aggregation method using spatial and temporalinterpolations.
[7] presents OBMAC, an enhancement for MAC protocolbased on the overhearing technique in WSNs. The objectiveof the proposed protocol is to reduce the number of redun-dant packets using the overhearing technique. In OBMAC,every sensor node verifies each overheard packet and com-pares it to its own in order to avoid transmitting the sameinformation to the base station. The notion of influentialrange is used to improve the efficiency of OBMAC.
[9] presents a method for evaluating the approach pro-posed in [6] by using a practical model of lossy links.The evaluation results show that the proposed approachsuppresses data transmissions and reduces total energy con-sumption even in a lossy environment.
[10] investigates the problem of improving the datapersistence. It introduces a distributed scheme based on LT(Luby Transform)-codes and an overhearing technique. Inthe proposed scheme, each sensor node uses overhearingto check if a packet has been transmitted by one of itsneighbors. When a sensor node needs to transmit a packet, itrandomly chooses one of its neighbors that does not transmitthe packet as the receiver. Each sensor node computes akey parameter of LT codes by using some properties ofthe packet transmission mechanism, and then stores thedata accordingly. After the process of storage is finished, acollector will recover all the data by visiting a small subsetof sensor nodes.
B. Lifetime-Aware Routing Trees
A routing tree of a WSN with a single base station is aspanning tree rooted at the base station. Each sensor nodesends its own data and the data received from its childrento its parent. Since each sensor node is typically battery-powered, it is important to construct a routing tree such thatthe network lifetime is maximized. Many approaches to thelifetime-aware routing tree construction problem have beenproposed.
[11] shows that the problem of constructing a span-ning tree with the maximum lifetime is NP-complete andproposes a polynomial-time approximation algorithm. Theapproximation algorithm starts with an arbitrary tree anditeratively reduces the loads of bottleneck nodes. [12] studiesan on-line data gathering problem, and proves that theproblem is NP-complete. It presents a generic cost model ofenergy consumption for data gathering queries and severalheuristics.
[13] and [14] take into account the remaining energy andthe load of each node, and propose top-level load balancingalgorithms with dynamic modifications. [15] constructs amulti-tree topology to allow more choices for the nexthop when routing messages. [16] proposes a distributedprobabilistic load-balancing converge cast tree algorithm to
address the heterogeneity issues in terms of nodal traffic bur-den and residual energy by dynamically forming convergecast routing trees.
[17] proposes a new weighted path cost function im-proved from the shortest path tree approach. In this ap-proach, links are assigned weights according to their pathlengths to the root, and those close to the root have largerweights. By balancing loads according to the link weights,this approach increases the network lifetime compared withthose randomly constructed shortest path trees.
[2] investigates the problem of finding a shortest pathaggregation tree with the maximum lifetime in a WSN. Theproposed algorithm first builds a fat tree which contains allthe shortest path trees. Then, it converts the problem into asequence of semi-matching problems each considering twoadjacent levels of the fat tree, and solves each semi-matchingproblem by using the min-cost max-flow approach in poly-nomial time. [2] also proposes a distributed algorithm forconstructing a maximum lifetime shortest path aggregationtree, and proves that if no data aggregation is performed,it is NP-complete to construct a shortest path tree with themaximum lifetime.
[18] investigates the lifetime-aware data collection prob-lem without data aggregation, and proposes an approxi-mation algorithm for constructing a routing tree with themaximum network lifetime. The approximation algorithmiteratively transfers some of the descendants of the nodewith the largest weight to a node with a smaller weight, andstops when no more descendants of a bottleneck node canbe transferred.
[19] studies the load balance problem in a grid topology.It focuses on the energy consumptions of the nodes whichcan communicate with the base station directly. Firstly, thealgorithm selects the most lightly loaded and most con-fined branches for growth. Secondly, it selects the heaviestnodes with the maximum growth space. After establishinga loosely balanced tree, the algorithm re-balances the treeby moving nodes from the heaviest loaded branches to morelightly loaded neighbouring branches. The simulation resultsshow that the routing trees constructed by their algorithmare more balanced than the shortest path tree constructed byDijkstra’s algorithm.
[20] investigates the problem of network lifetime maxi-mization of WSNs in the context of data collection trees. Itproposes an efficient algorithm, called Randomized Switch-ing for Maximizing Lifetime (RaSMaLai) that aims atmaximizing the lifetime of WSNs through load balancingwith a low time complexity, and a distributed version of thealgorithm.
[21] investigates the problem of lifetime and latency-aware data collection in WSNs with one base station. Itproposes a new routing structure, namely k-tree, and adistributed algorithm for constructing a lifetime-aware k-tree. A unique feature of the k-tree is that it provides the
maximum latency guarantee for data collection.
III. NETWORK MODEL AND DEFINITIONS
The target WSN consists of a set V = {v1, v2, · · · .vn}of n static sensors. There is only one base station. Eachsensor generates one packet of data per unit time and sendsthe packet to the base station without performing any dataaggregation. All the sensor nodes are identical with the sametransmission range and the same initial energy level.
We use an undirected graph, named as connectivity graph,to represents the connectivity between sensor nodes in theWSN. The connectivity graph G is defined as follows:G = (V ∪ {BS}, E), where, BS is the base station, andE = {(vi, vj): vi, vj ∈ V ∪ {BS} and vi and vj cancommunicate with each other directly }. A communicationlink between two sensor nodes indicates that the two sensornodes can communicate with each other. We assume that theconnectivity graph is connected. The base station can collectthe connectivity graph from all the sensor nodes.
Some sensor nodes may be compromised. A compromisedsensor node is called a malicious sensor node. A malicioussensor node may drop, or modify the packets it receivesfrom other sensor nodes.
Given the connectivity graph G = (V,E) of a targetWSN, the problem we investigate is to construct a shortestpath routing tree rooted at the base station and assign amonitoring sensor node to each sensor node such that thenetwork lifetime is maximized. Hereinafter, a monitoringsensor node is called a monitor. A monitor vi of a sensornode vk is used to detect if the data sent by vk to its parent vjwill be forwarded correctly by vj to its parent in the shortestpath tree. Therefore, the monitor vi needs to overhear thetransmission of each packet vk sends to its parent vj and theforwarding of each packet vj receives from vk. If vj dropsor modifies any packet received from vk, vi will detect itand report it to the base station.
For each sensor node, its parent and monitor must satisfythe following requirements:
1) There is a communication link between the parent andthe monitor.
2) There is a communication link between the monitorand the sensor node.
3) There is a communication link between the parent andthe sensor node.
4) The parent and the monitor have the same depth inthe routing tree.
When constructing a shortest path tree, we mainly con-sider the energy consumption of data receptions and trans-missions for each sensor node. The energy consumption oflistening and communication session setup is ignored. Foreach sensor node, α is the energy consumed to receive onepacket, and β is the energy consumed to transmit one packet.Given a set S, |S| denotes the size of S.
IV. HEURISTIC APPROACH
Our heuristic approach consists of three phases, namely,partitioning phase, initial monitor and parent selection phase,and energy balancing phase.
In the partitioning phase, our heuristic approach parti-tions all the sensor nodes into m disjoint groups Ci(i =1, 2, · · · ,m) such that the shortest path length betweeneach sensor node in the group Ci to the base station inthe connectivity graph is equal to i. In the initial mon-itor and parent selection phase, for each group Ci(i =m,m − 1, · · · , 2), our heuristic approach assigns a parentand a monitor from the group Ci−1 to each sensor node inthe group Ci, aiming at minimizing the maximum energyconsumption of all the individual sensor nodes in the groupCi−1. In the energy balancing phase, our heuristic approachperforms the monitor and parent adjustment for each groupGi(i = m,m − 1, · · · , 2) such that all the sensor nodes ineach group almost consume the same amount of energy perunit time.
Next, we describe how to assign a monitor and a parentfrom Cl−1 to each sensor node in Cl(l = m,m−1, · · · , 2).Recall that for each sensor node, its parent and its monitormust satisfy the four requirements stated in the previoussection.
For each sensor node vi ∈ Cl, we use a 3-tuple(vi, vj , vk)(j < k) to uniquely represent a triangle formedby vi, vj and vk in the connectivity graph, where vj and vkare in Cl−1.
For each sensor node vi ∈ Cl, we introduce the followingnotations.• Si: a set of all the triangles in the connectivity graph
each of which contains vi and two sensor nodes inCl−1. Notice that if Si is empty, no shortest pathoverhearing tree exists.
• Pi: a set of all the sensor nodes in Cl−1 each of which isadjacent to vi in the connectivity graph and not selectedas the monitor or the parent of vi.
• Mi: a set of all the sensor nodes in Cl+1 for which viis the monitor. Initially, Mi is ∅. Each time when vi isselected as the monitor of a sensor node vj in Cl+1,Mi is updated as follows: Mi =Mi ∪ {vj}.
• CHi: a set of all the children of vi in the partial shortestpath tree rooted at vi currently constructed.
• subtree− size(vi): the number of sensor nodes in thesubtree rooted at the sensor node vi constructed so far.
• ei: the energy consumed by vi per unit time under thecurrent assignment of monitors and parents. Initially,ei is equal to β. Each time when vi is selected as themonitor of a sensor node vj , ei is updated as follows:ei = ei +2tree− size(vj)α. Each time when vi is se-lected as the parent of a sensor node vj , ei is updated asfollows: ei = ei+tree−size(vj)α+tree−size(vj)β.
We define two types of priorities: a priority for each sensor
node in Cl and a priority for each sensor node in Cl−1. Thepriority P1(vi) of each sensor node vi in Cl is a 3-tupledefined as follows:
P1(vi) = (|Si|, |Pi|, 1/tree− size(vi)) (1)
The priority P2(vj) of each sensor node vj in Cl−1 is a2-tuple defined as follows:
P2(vj) = (ej , |Mj ∪ CHj |) (2)
For both types of priorities, a smaller tuple implies ahigher priority. Note that the priority of each sensor nodemay be changed dynamically during the initial monitor andparent selection phase.
For each group Cl(l = m,m − 1, · · · , 2), the initialmonitor and parent selection algorithm works as follows:
1) For each sensor node vi in Cl, compute Si and Pi.2) For each sensor node vj in Cl−1, do the following.
a) Set CHj and Mj to ∅.b) Set ej to β.
3) For each sensor node vi ∈ Cl, compute the priorityP1(vi).
4) For each sensor node vj ∈ Cl−1, compute the priorityP2(vj).
5) A = Cl.6) Repeat the following until A is ∅.
a) Select a sensor node vi with the highest priorityfrom A.
b) If Si is equal to ∅, no parent and monitor existfor vi, and the algorithm terminates.
c) Assign a 2-tuple rank R(X) to each triangle X ∈Si as follows:i) Let vj and vk be the two sensor nodes other
than vi in X such that P2(vj) ≤ P2(vk)holds.
ii) R(X) = (P2(vj), P2(vk)).d) Find a triangle Xmin in Si with the smallest
rank.e) Let R(Xmin) = (P2(vs), P2(vt)).f) If es < et holds, do the following.
i) Select vs and vt as the parent and the monitorof vi, respectively.
ii) Mt =Mt ∪ {vi}.iii) CHs = CHs ∪ {vi}.iv) es = es+tree−size(vi)α+tree−size(vi)β.v) et = et + 2tree− size(vi)α.
vi) Re-compute P2(vs) and P2(vt).Otherwise, do the following.i) Select vt and vs as the parent and the monitor
of vi, respectively.ii) Ms =Ms ∪ {vi}.
iii) CHt = CHt ∪ {vi}.
iv) et = et+tree−size(vi)α+tree−size(vi)β.v) es = es + 2tree− size(vi)α.
vi) Re-compute P2(vs) and P2(vt).g) A = A− {vi}.h) For each sensor node vj ∈ A that is adjacent to
vs in the connectivity graph, set Pj to Pj−{vs}.i) For each sensor node vj ∈ A that is adjacent tovt in the connectivity graph, set Pj to Pj−{vt}.
j) For each sensor node vj ∈ A that is adjacent tovs or vt in the connectivity graph, re-computeP1(vj).
After the initial monitor and parent selection phase, theenergy balancing phase starts. The energy balancing phaseworks from the group Cm−1 to the group C1. For eachgroup Cl(l = m − 1,m − 2, · · · , 1), the energy balancingalgorithm selects a sensor node vi with the maximum energyconsumption per unit time, and shifts the role of vi as theparent or the monitor of a sensor node vj to another sensornode with lower energy consumption. In order to find sucha sensor node vj , we define a new rank for each sensor nodevr in Mi ∪ CHi as follows:
W (vr) = (W1(vr),W2(vr), · · · ,WN (vr)) (3)
where N is equal to |Cl|, and W1(vr), W2(vr), · · · ,WN (vr) are the energy consumptions per time unit of allthe sensor nodes in Cl sorted in non-increasing order aftervi’s role as vr’s monitor or parent is switched to a candidatesensor node vp in Cl.
Let vr be a sensor node in Mi ∪ CHi such that the roleof vi as the monitor or the parent of vr will be replaced bya candidate sensor node vp. vp is found as follows:
1) Let B be a set of all the triangles of the form(vr, vi, vs)(i < s) or (vr, vs, vi)(s < i).
2) Let E be a set of all the sensor nodes in B that aredifferent from vr and vi.
3) vp is the sensor node in E that has the smallest energyconsumption per time unit.
If such a sensor node vp is not found, W (vr) is set to(+∞,+∞, · · · ,+∞). A role switch is allowed only if theswitch results in better energy balancing.
The energy balancing algorithm works for each groupCl(l = m− 1,m− 2, · · · , 1) as follows.
1) Mark each sensor node in Cl as switchable.2) Repeat the following until no sensor node in Cl is
switchable.a) Let e1, e2, · · · , eN be the energy consumptions
per time unit of all the sensor nodes in Cl sortedin non-increasing order.
b) W = (e1, e2, · · · , eN ).c) Pick a sensor node vi in Cl that is not marked
as unswitchable and has the maximum energyconsumption per unit time.
d) For each sensor node vs ∈ Mi ∪ CHi, computeW (vs).
e) T = min{W (vs) : vs ∈Mi ∪ CHi}.f) If T ≥W holds, mark vi as unswitchable.g) Otherwise, let vr be a sensor node in Mi ∪CHi
with the smallest rank W (vr), and vp the sensornode selected to replace vi’s role as the parent orthe monitor of vr when computing W (vr). Dothe following.i) Switch vi’s role as vr’s monitor or parent tovp.
ii) Re-compute ei and ep.3) If l > 1 holds, for each sensor node vj in Cl−1, re-
calculate ej based on the current partial shortest pathoverhearing tree.
Next, we analyze the time complexity of our heuristicapproach. The time complexity is broken down into thefollowing three parts.
1) The partitioning phase. We can use breadth-first searchto compute the shortest path length from each sensornode to the base station. Therefore, the time complex-ity of this phase is O(e), where e is the number ofedges in the connectivity graph.
2) The initial monitor and parent selection phase. First,we assume that for each sensor node, the maximumnumber of sensor nodes it can communicate directlyis a constant. Under this assumption, for each sensornode vi, O(|Si|) = O(|Pi|) = O(|Mi∪CHi|) = O(1)holds. The breakdown of the time complexity of thisphase is as follows.
a) Steps 1-5. Notice that the connectivity graphis connected. Therefore, the time complexity ofsteps 1-5 for all the groups is O(e).
b) Step 6. The time complexity of this step for allthe groups is O(ne), where n is the number ofsensor nodes in the WSN.
As a result, the time complexity of this phase is O(ne).3) The energy balancing phase. Notice that each time
when a role switch occurs, W will decrease monoton-ically. For each sensor node vi in Cl, it is processed atmost |CHi| times by the energy balancing algorithm.Therefore, the energy balancing algorithm terminatesin∑
vi∈Cl(|CHi| = O(|Cl|) steps for each group Cl.
For each group Cl, each step takes O(|Cl| log |Cl|).As a result, the time complexity of this phase isO(n2 log n).
As discussed above, the time complexity of our heuristicapproach is O(e)+O(ne)+O(n2 log n) = O(ne+n2 log n).
V. MINLP-BASED APPROACH
The objective of MINLP-based approach is to construct ashortest path tree for routing and assign a monitor for each
sensor node such that the network lifetime is maximizedwhile the four requirements described in Section III are met.
The MIINLP-based approach consists of two phases. Inthe first phase, it partitions all the sensor nodes into m dis-joint groups Cl(l = 1, 2, · · · ,m) such that the shortest pathlength between each sensor node in the group Cl to the basestation in the connectivity graph is equal to l. In the secondphase, for each sensor node vi ∈ Cl(l = 2, 3, · · · ,m), itassigns a monitor and a parent in Cl−1 to vi such thatthe maximum energy consumption per unit time of all theindividual sensor nodes is minimized by using MINLP.
As in our heuristic approach, we use (vi, vj , vk)(j < k)to uniquely represent a triangle formed by vi, vj and vkin the connectivity graph, where vi is in Cl, vj and vk arein Cl−1. For each sensor node vi ∈ Cl, we also use Si todenote a set of all the triangles in the connectivity grapheach of which contains vi and two sensor nodes in Cl−1 asin Section IV.
For each triangle (vi, vj , vk), we introduce two binarydecision variables x(i,j,k) and y(i,j,k) as follows:
x(i,j,k) =
{1 vj is the monitor & vk is the parent of vi0 otherwise
(4)
y(i,j,k) =
{1 vk is the monitor & vj is the parent of vi0 otherwise
(5)Therefore, for each sensor node vi ∈ Cl(l = 1, 2, · · · ,m),
we have the following monitor and parent selection con-straint: ∑
(vi,vj ,vk)∈Si
x(i,j,k) + y(i,j,k) = 1 (6)
The above constraint implies that among all the sensornodes in Cl−1 with which vi can communicate directly,only two sensor nodes are selected as the monitor and theparent of vi, respectively, and the monitor and the parentcan communicate with each other.
Next, we derive the energy constraint and other relatedconstraints for each sensor node. For each sensor vi ∈Cl(l = 1, 2, · · · ,m), we further introduce the followingnotations:
1) li: the number of packets vi receives from all itschildren in the shortest path tree per unit time.
2) mi: the total energy consumption per time unit by vibeing a monitor.
3) pi: the total energy consumption per time unit by vibeing a parent.
4) ei: the total energy consumption per unit time by vi.
For each sensor vi, if it is in Cm, it is a leaf node inthe shortest path tree. Therefore, we have the followingconstraint on li:
li =
0 vi is in Cm∑vk∈CHi
∑(vk,vi,vj)∈Sk
y(k,i,j) ∗ lk+∑vk∈CHi
∑(vk,vj ,vi)∈Sk
x(k,j,i) ∗ lkotherwise
(7)If a sensor node vi is the monitor of a sensor node vk, it
needs to overhear the packet transmission when vk sends itspacket to its parent vj and when vj forwards the packet toits own parent. Therefore, we have the following constrainton mi:
mi =
∑vk∈Mi
∑(vk,vi,vj)∈Sk
x(k,i,j) ∗ 2(lk + 1)α+
∑vk∈Mi
∑(vk,vj ,vi)∈Sk
y(k,j,i) ∗ 2(lk + 1)α(8)
If a sensor node vi is a parent of a sensor node vk, itneeds to not only receive the data from vk, but also forwardthe data to its parent. Therefore, we have the followingconstraint on pi:
pi =
∑vk∈CHi
∑(vk,vi,vj)∈Sk
y(k,i,j) ∗ lk(α+ β)+
∑vk∈CHi
∑(vk,vj ,vi)∈Sk
x(k,j,i) ∗ lk(α+ β)(9)
For each sensor node vi, its energy consumption pertime unit consists of three parts: mi, pi and the energyfor transmitting its own packet. Therefore, we have thefollowing constraint on ei:
ei = mi + pi + β (10)
Our optimization objective function is as follows:
minmaxvi∈V{ei} (11)
VI. ILP-BASED APPROACH
The ILP-based approach consists of two phases. In thefirst phase, it partitions all the sensor nodes into m disjointgroups Cl(l = 1, 2, · · · ,m) such that the shortest pathlength between each sensor node in the group Cl to thebase station in the connectivity graph is equal to l. In thesecond phase, for each group Cl(l = m,m − 1, · · · , 2), itassigns a monitor and a parent in Cl−1 to each sensor in Cl
such that the maximum energy consumption per time unitof all the individual sensor nodes in Cl−1 is minimized byusing ILP.
Next, we show how to use ILP to find a locally optimalassignment of a monitor and a parent for each sensor nodein Cl(l = m,m− 1, · · · , 2).
Similar to the MINLP approach, for each sensor nodevi ∈ Cl, we have the following monitor and parent selectionconstraint: ∑
(vi,vj ,vk)∈Si
x(i,j,k) + y(i,j,k) = 1 (12)
The binary decision variables x(i,j,k) and y(i,j,k) aredefined in the same way as in the MINLP approach.
For each sensor node vi ∈ Cl−1, we have the followingenergy constraint on mi:
mi =
∑vk∈Mi
∑(vk,vi,vj)∈Sk
x(k,i,j) ∗ 2(lk + 1)α+
∑vk∈Mi
∑(vk,vj ,vi)∈Sk
y(k,j,i) ∗ 2(lk + 1)α(13)
If vk is a sensor node in Cm, lk is equal to 0. Noticethat each lk is a constant as our ILP-based approach hasfinished the monitor and parent assignment for each groupCj(j > l).
For each sensor node vi ∈ Cl−1, we have the followingconstraint on pi:
pi =
∑vk∈CHi
∑(vk,vi,vj)∈Sk
y(k,i,j) ∗ lk(α+ β)+
∑vk∈CHi
∑(vk,vj ,vi)∈Sk
x(k,j,i) ∗ lk(α+ β)(14)
For each sensor node vi ∈ Cl−1, we have the followingconstraint on ei:
ei = mi + pi + β (15)
Our optimization objective function is as follows:
min maxvi∈Cl−1
{ei} (16)
After selecting the parent and the monitor of each sensornode in Cl, our ILP-based approach computes lk for eachsensor node vk in Cl−1.
VII. SIMULATION RESULTS
A. Setup
In order to evaluate our approaches, we use NS 2.35to generate 150 instances with three different distributions,namely, uniform, grid, and random distributions. We vary thenumber of sensor nodes from 100 to 300 with an incrementof 50. For each scenario with a fixed number of sensornodes and a particular distribution, we generate 10 instances.For each instance, sensor nodes are deployed in a 300 m x400 m rectangular area, and the base station is deployed at
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Figure 1: Network lifetimes for all the instances
the middle of the upper boundary of the rectangular area.The initial energy of every sensor node is 0.5 KJ. Thetransmission range is fixed to 70 m. The energy consumptionfor receiving one packet of data is α = 0.001 KJ while theenergy consumed to transmit one packet of data is β = 0.002KJ. MIDACO Solver is used to solve the MINLP problemsand the Intlinprog Solver of MATLAB is used for the ILPproblems.
The hardware platform is Intel Core i5-3470 with a clockfrequency of 3.20 Ghz, a memory size of 8 GB and a cachesize of 8134 MB.
B. Simulation Results
Figure 1. shows the network lifetimes of the shortest pathoverhearing trees constructed by the three approaches for allthe instances in the three different distributions.
The simulation results show that the heuristic approachobtains comparable network lifetimes compared with theILP-based approach and the MINLP-based approach. Forexample, for the instance having 150 sensor nodes withuniform distribution as shown in Figure 1. (b), the networklifetime obtained by the heuristic approach is 3.4, compa-rable to the network lifetime 4.42 obtained by the ILP-based approach, and the network lifetime 4.5 achieved bythe MINLP-based approach.
The relative maximum network lifetime obtained by theheuristic approach occurs in the instance of 100 randomlydistributed sensor nodes, as shown in Figure 1.(c), and thenetwork lifetime is 7.29. The relative minimum networklifetime obtained by the heuristic approach is 3.4, whichoccurs in the instance of 150 uniformly distributed nodes,as shown in Figure 1. (b).
Comparing the heuristic approach with the MINLP-basedapproach, the minimum ratio between the network lifetimeachieved by the heuristic approach and the network life-time obtained by the MINLP-based approach is 76.6%,which occurs in the instance of 150 nodes as shown inFigure 1.(b) while the maximum ratio is 85.62% as shown inFigure 1.(b). The average ratios for the grid distribution, theuniform distribution and the random distribution are 83.68%,81.13%, and 78.25%, respectively. The average ratio for allthe instances with the three different distributions is 81.05%.
In comparison between the ILP-based approach and theMINLP-based approach, the minimum ratio between thenetwork lifetime achieved by the ILP-based approach andthe network lifetime obtained by the MINLP-based approachis 94.43%. The maximum ratio is 98.24%. The averageratios for the grid distribution, the uniform distribution andthe random distribution are 94.43%, 98.23%, and 93.7%,respectively. The average ratio for all the instances with thethree different distributions is 96.2%.
In comparison between the heuristic approach and theILP-based approach, the minimum ratio between the net-work lifetime achieved by the heuristic approach and the
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Figure 2: Running times for all the instances
network lifetime obtained by the ILP-based approach is78.04% while the maximum ratio is 93.1%. The averageratios for the grid distribution, the uniform distribution andthe random distribution are 85.6%, 83.94%, and 87.5%,respectively. The average ratio for all the instances with thethree different distributions is 85.69%.
Figure 2 shows the running times of all the three ap-proaches for all the instances with the three different distri-butions.
The simulation results show that the heuristic approachalways constructs a shortest path overhearing tree in areasonable amount of time for all the instances while boththe MINLP-based approach and the ILP-based approach donot scale. For example, the ILP-based approach fails toconstruct a shortest path overhearing tree for the instanceswith more than 200 sensor nodes in 3 hours while theMINLP-based approach fails to construct a shortest pathoverhearing tree for the instances with more than 100 sensornodes in 3 days.
VIII. CONCLUSION
We investigate the problem of constructing a shortestpath overhearing tree with maximum network lifetime, andpropose three approaches, a polynomial-time heuristic ap-proach, a MINLP-based approach and an ILP-based ap-proach. We have implemented our approaches using MI-DACO solver and MATLAB Intlinprog, and performed ex-tensive simulations on the 150 network instances with threedifferent distributions, namely, uniform, grid, and randomdistributions. The simulation results show that the heuristicapproach performs very well. The average lifetime of allthe network instances achieved by the heuristic approachis 85.69% of that achieved by the ILP-based approach and81.05% of that achieved by the MINLP-based approach.
All the approaches proposed in this paper are centralizedones. A centralized approach requires that the base stationcollect the connectivity graph from all the sensor nodes, andsend the shortest path overhearing tree constructed to allthe sensor nodes. We can construct a shortest path spanningtree with a special naming scheme as proposed in [22] forcollecting the connectivity graph and sending the shortestpath overhearing tree. A distributed heuristic can avoidthose costs. Nevertheless, a distributed heuristic introducesadditional message complexity. It is interesting to know ifthere is a distributed heuristic for the problem with betteroverall performance than our centralized heuristic.
One limitation with our shortest path overhearing tree isthat if a monitor colludes with the parent of the monitoredsensor node, the attacks by the parent of the monitoredsensor node may not be detected. Our future work includesconstructing a more secure shortest path overhearing treewhere there are multiple monitors for each sensor node.
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