ver

Article history:Received 3 August 2012Received in revised form 1 February 2013Accepted 9 March 2013Available online 19 March 2013

Keywords:Area coverage

systems, and wireless communications technology havegreatly promoted the emergence of modern WSNs, theycontinue to be networks with constrained resources interms of memory, power supply, and processing power[35]. Due to the severe resource limitations in WSNs, cov-

ell-known classico. The artcated to mean area co

problem [11] in which the satellites are placed in thto provide the maximum ocean monitoring also dethe coverage problem [9].

Depending on the subject is covered, the coverage prob-lem can be classied as area coverage [12,13], barrier (orpath) coverage [1416], and target (or point) coverage[13,17]. The area coverage deals with the problem of cov-ering all the points within the monitoring area. The area

1570-8705/$ - see front matter 2013 Elsevier B.V. All rights reserved.

Tel./fax: +98 861 3422292.E-mail address: j-akbari@iau-arak.ac.ir

Ad Hoc Networks 11 (2013) 16551666

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Ad Hoc Ne

.e lshttp://dx.doi.org/10.1016/j.adhoc.2013.03.002over a vast region for different purposes such as environ-ment monitoring, object or target tracking, industry auto-mation and control, and etc. [1,2]. Although the recentadvances in sensor technology, micro-electromechanical

[9]. Besides the WSN, coverage is a wproblem in computational geometry toproblem [10] in which the cameras are loevery point in the art gallery, and the ocgalleryonitorveragee orbital with1. Introduction

A wireless sensor network (WSN) is a multi-hop, infra-structureless, and self-organized network comprising agroup of small and power-constrained sensors deployed

erage is the most fundamental and challenging issue ofthese networks focusing on how well the sensors coverthe monitoring region [68]. WSN coverage problem aimsto minimize the number of sensor nodes to be activated,while maintaining the full coverage of the monitoring areaMinimum weight CDSDegree-constrained CDSWSNLearning automataThe connected dominating set (CDS) concept has recently emerged as a promisingapproach to the area coverage in wireless sensor network (WSN). However, the majorproblem affecting the performance of the existing CDS-based coverage protocols is thatthey aim at maximizing the number of sleep nodes to save more energy. This places aheavy load on the active sensors (dominators) for handling a large number of neighbors.The rapid exhaustion of the active sensors may disconnect the network topology and leavethe area uncovered. Therefore, to make a good trade-off between the network connectivity,coverage, and lifetime, a proper number of sensors must be activated. This paper presents adegree-constrained minimum-weight extension of the CDS problem called DCDS to modelthe area coverage in WSNs. The proper choice of the degree-constraint of DCDS balancesthe network load on the active sensors and signicantly improves the network coverageand lifetime. A learning automata-based heuristic named as LAEEC is proposed for ndinga near optimal solution to the proxy equivalent DCDS problem in WSN. The computationalcomplexity of the proposed algorithm to nd a 11 optimal solution of the area coverageproblem is approximated. Several simulation experiments are conducted to show the supe-riority of the proposed area coverage protocol over the existing CDS-based methods interms of the control message overhead, percentage of covered area, residual energy, num-ber of active nodes (CDS size), and network lifetime.

2013 Elsevier B.V. All rights reserved.a r t i c l e i n f o a b s t r a c tAn adaptive energy-efcient area cowireless sensor networks

Javad Akbari Torkestani Young Researchers Club, Arak Branch, Islamic Azad University, Arak, Iran

journal homepage: wwwage algorithm for

tworks

evier .com/locate /adhoc

1656 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666coverage problem aiming at minimizing the number of ac-tive nodes without failing to cover the entire area is themost common form of the coverage problem [9]. Barriercoverage is considered as monitoring a boundary region(or barrier) within the sensor eld aiming at minimizingthe probability of undetected penetration through the bar-rier. Barrier coverage is used to detect intruders attemptingto penetrate a protected region. The target coverage prob-lem intends to cover a set of stationary or moving pointswithin the sensor eld [18]. All types of the coverage prob-lem aim to minimize the required number of sensors forcovering the area, barrier or targets. A fundamental solu-tion to the coverage problem is to place the sensors atthe predetermined locations of the sensor eld determinis-tically. Deterministic sensor placement can be applied onlyto a relatively small sensor network deployed in a friendlyenvironment. However, when a large sensor network is de-ployed in a hostile, harsh, and hard-to-access eld, randomsensor deployment, generally scattered from an aircraft,might be the only choice [18,19]. In random deployment,to guarantee the complete coverage of the sensor eld,the number of sensors that must be scattered is signi-cantly more than that is required [18]. Under such circum-stances, the design of a coverage protocol minimizing therequired number of active sensors signicantly improvesthe performance of the WSNs in terms of energy consump-tion and network lifetime.

The connected dominating set (CDS) principle has re-cently emerged as a new solution to the energy-efcientcoverage in WSNs. A CDS of a given graph is a connectedsubset of the graph vertices such that every vertex of thegraph is either in the set or adjacent to at least one vertexof the set. Several approaches have been proposed to solvethe CDS problem. To name just a few, Li et al. [20] andAlzoubi et al. [21] proposed MIS (maximal independentset)-based greedy algorithms to construct the CDS. Daiand Wu [22] and Butenko et al. [23] presented prune-based heuristics for the CDS problem. In a CDS-based cov-erage protocol, a virtual backbone covering every pointwithin the sensor eld is formed [4,9]. Misra et al. [9] pro-posed an energy-efcient solution to maintain the cover-age problem in WSNs. The proposed method preservesthe network connectivity by formation of the networkbackbone. The proposed solution aims to cover the areaof interest and minimizing the number of active sensorstoo. Wightman and Labrador [24] proposed an approxi-mate CDS-based solution called A3 to the topology controlproblem in WSNs. A3 assumes that the sensors have noinformation about the position of the neighbors and soabout the network topology. In this method, the distancebetween the nodes is estimated based on the received sig-nal strength. A3 generally is composed of two processes,neighborhood discovery process and children selectionprocess. However, another process called the secondopportunity process is rarely used in special cases. Theresidual energy of the child node and its distance fromthe parent are two metrics that A3 uses to construct theCDS-based tree. This selection rule gives the higher priorityto the child nodes with higher energy and farther distancefrom the parent node. A3 uses four messages for topologyconstruction, hello message sent out by the parent, parentrecognition message including the residual energy and sig-nal strength that is sent back by the children, children rec-ognition message including the sorted list of all childrenand their timeouts sent out by the parent, and sleepingmessage that is sent by the active candidate node. Wight-man and Labrador [24] proposed an extended version of A3called A3lite that uses only (two) hello and parent recogni-tion messages for topology construction. A3lite avoidssending the large size children recognition message. A3sends a sleep message to all nodes, in the reduced treetopology, that are within the communication area of theother nodes. This may cause some points of the area re-main uncovered when the sensing range is considerablysmaller than the communication range. The same authorsin [6] proposed two solutions called A3Cov and A3CovLitefor the coverage problem of A3. A3Cov rst checks to see ifan unconnected node is sensing-covered by another activenode. If so, the node is sent directly to the sleeping mode.This is because it is not needed for connectivity nor cover-age. Otherwise, the node is asked to stay awake for an extraperiod of time. If it receives a sensing coverage message(indicating that sensing area of the node has been alreadycovered) from its neighbors before the timeout expires, itgoes to the sleeping mode. Otherwise, the node must re-main active. A3CovLite [6] is a combination of A3Lite thatreduces the required number of messages as compared toA3 original, and A3Cov that solves the area coverage prob-lem in A3. Rizvi et al. [4] proposed a distributed energy-efcient topology control algorithm referred to as A1 forconnected area coverage in WSNs. Similar to A3 familyprotocols, A1 uses the signal strength and residual energyas the criteria to select the dominator nodes. A1 only usesthe hello message to construct the CDS-based backbone.A1 constructs the topology in one phase. The starting noderst discovers its neighbors. Similarly, the neighbors of theinitiator node discover their neighbors and this processcontinues until the complete topology is formed with thebackbone nodes. Like A3Cove, A1 lets the children calculateand set the timeout value independently.

Literature review reveals that there are critical problemsaffecting the performance of the existing CDS-based con-nected area coverage protocols. These protocols generallyaim at covering the sensor led with the minimum numberof active nodes (i.e., constructing the minimum size CDS).This reduces the energy consumption by turning off a largenumber of sensors. On the other side, this signicantlyshortens the sensor lifetime (even for energetic nodes) be-cause of a heavy burden on the active sensors for handling alarge number of neighbors while the sensors severely sufferfrom the limited energy, and processing power. Exhaustingthe energy of the active sensors may leave the area uncov-ered. Though the redundant active nodes (i.e., large sizeCDS) extend the covered area, the overlapped sensing areasincrease the network energy consumption. This itself re-duces the network lifetime on the other hand. Therefore,the CDS must be constructed such that a good trade-off be-tween the coverage (covered area) and network lifetime ismade. In this paper, the degree-constrained minimum-weight version of the CDS, so called DCDS, is presented toalleviate the above mentioned problems with the CDS-based area coverage protocols. DCDS is the CDS having

J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666 16572. Learning automata theory

A learning automaton [25,26] is an adaptive decision-making unit that improves its performance by learninghow to choose the optimal action from a nite set of al-lowed actions through repeated interactions with a ran-dom environment. The action is chosen at random basedon a probability distribution kept over the action-set andat each instant the given action is served as the input tothe random environment. The environment responds thetaken action in turn with a reinforcement signal. The actionprobability vector is updated based on the reinforcementfeedback from the environment. The objective of a learningautomaton is to nd the optimal action from the action-setso that the average penalty received from the environmentis minimized. Learning automata have a wide variety ofapplications in combinatorial optimization problems[33,34,38,39], computer networks [36,37,4245], Gridcomputing [30,32,41], and Web engineering [31,35,40].

The environment can be described by a triple {a, b, c},where a {a1, a2, . . ., ar} represents the nite set of the in-puts, b {b1, b2, . . ., bm} denotes the set of the values thatcan be taken by the reinforcement signal, and c {c1, c2,. . ., cr} denotes the set of the penalty probabilities, wherethe element ci is associated with the given action ai. Ifthe penalty probabilities are constant, the random environ-ment is said to be a stationary random environment, and ifthey vary with time, the environment is called a non-sta-tionary environment. The environments depending onthe nature of the reinforcement signal b can be classiedthe minimum weight subject to a predened degree-con-straint. The weight associated with each node is denedas the inverse of its residual energy. Therefore, the DCDSmaximizes the network lifetime by selection of the sensorswith the maximum residual energy. A DCDS is a CDS inwhich no node has a degree greater than a predened de-gree-constraint. Therefore, by the proper choice of the de-gree-constraint the DCDS is able to make a trade-offbetween the percentage of the covered area and the net-work lifetime. This paper proposes a learning automata-based heuristic called LAEEC (short for learning automata-based energy-efcient coverage protocol) to construct theDCDS in the WSNs. The computational complexity of theproposed algorithm to nd a 11 optimal solution of the areacoverage problem is approximated. Extensive simulationexperiments are performed to show the performance ofthe proposed area coverage algorithm. The obtained resultsshow the superiority of the proposed algorithm over thebest existingmethods in terms of the control message over-head, percentage of covered area, residual energy, numberof active nodes (CDS size), and network lifetime.

The rest of the paper is organized as follows. Section 2briey reviews the learning automata theory. In Section 3,the proposed area coverage algorithm is presented. Sec-tion 4 approximates the time complexity of the proposedalgorithm to nd a 11 optimal solution of the area coverageproblem. Section 5 shows the performance of the proposedalgorithm through simulation experiments and comparisonwith the existing methods. Section 6 concludes the paper.into P-model, Q-model and S-model. The environments inwhich the reinforcement signal can only take two binaryvalues 0 and 1 are referred to as P-model environments.Another class of the environment allows a nite numberof the values in the interval [0,1] can be taken by the rein-forcement signal. Such an environment is referred to as Q-model environment. In S-model environments, the rein-forcement signal lies in the interval [a,b].

Learning automaton can be classied into two mainfamilies [25]: xed structure learning automata and vari-able structure learning automata. Variable structure learn-ing automata are represented by a triple hb, a, Li, where b isthe set of inputs, a is the set of actions, and L is learningalgorithm. The learning algorithm is a recurrence relationwhich is used to modify the action probability vector. Letai(k) 2 a and p(k) denote the action selected by learningautomaton and the probability vector dened over the ac-tion set at instant k, respectively. Let a and b denote the re-ward and penalty parameters and determine the amountof increases and decreases of the action probabilities,respectively. Let r be the number of actions that can be ta-ken by learning automaton. At each instant k, the actionprobability vector p(k) is updated by the linear learningalgorithm given in Eq. (1), if the selected action ai(k) is re-warded by the random environment, and it is updated asgiven in Eq. (2) if the taken action is penalized.

pjk 1 pjk a1 pjk j i1 apjk 8j i

(1

pjk 1 1 bpjk j i

br1 1 bpjk 8j i

(2

If a = b, the recurrence Eqs. (1) and (2) are called linearreward-penalty (LRP) algorithm, if a b the given equa-tions are called linear reward- penalty (LRP), and nallyif b = 0 they are called linear reward-Inaction (LRI). In thelatter case, the action probability vectors remainunchangedwhen the taken action is penalized by the environment.

A variable action-set learning automaton is an automa-ton in which the number of actions available at each in-stant changes with time. It has been shown in [26] that alearning automaton with a changing number of actions isabsolutely expedient and also -optimal, when the rein-forcement scheme is LRI. Such an automaton has a niteset of r actions, a = {a1, a2, . . ., ar}. A = {A1, A2, . . ., Am} de-notes the set of action subsets and A(k) # a is the subsetof all the actions can be chosen by the learning automaton,at each instant k. The selection of the particular action sub-sets is randomly made by an external agency according tothe probability distribution W(k) = {W1(k), W2(k), . . .,Wm(k)} dened over the possible subsets of the actions,where

Wik probAk AijAi 2 A;1 6 i 6 2r 1p^ik probak aijAk;ai 2 Ak denotes the proba-

bility of choosing action ai, conditioned on the event thatthe action subset A(k) has already been selected andai 2 A(k) too. The scaled probability p^ik is dened as

p^ik pikKk 3

based coverage protocols. On one hand, MCDS (minimum

the graph vertices such that every vertex of the graphcorresponds to the set of sensor nodes and the action-set

1658 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666either is in the set or is the adjacent to at least one vertexof the set. MCDS is the CDS with the minimum cardinality.The minimum weight CDS (MwCDS) is the CDS having theminimum total weight. LetDi be the degree of vertex vi 2 V.The degree of a given vertex is dened as its number ofneighboring vertices. The degree-constrained connecteddominating set of graph G is a CDS of G subject to Di 6 d(for all vi 2 V), where d is a positive integer number denot-ing the degree-constraint. The degree-constrained mini-mum-weight CDS (DCDS) problem seeks the CDS withthe minimum weight subject to a degree-constraint d.

Let triple hN; L; Ei denotes the topology graph of a WSN,where N fn1;n2; . . .g is the set of sensor nodes,L flni ;njg#NN is the set of communication links,and E fEni j8ni 2 Ng denotes the set of energies associ-ated with the sensor nodes. Let Eni be the residual energyof sensor node ni. LAEEC aims to construct the most-stableenergy-efcient sensor network covering the monitoringsize CDS)-based coverage results in saving more energyby maximizing the number of sleep nodes. On the otherhand, MCDS-based coverage rapidly drains the activenodes and shortens the network lifetime. Though theredundant active nodes extend the network lifetime, theyincrease the network energy consumption. DCDS aims atmaking a good trade-off between the network coverage,lifetime, and energy consumption by an additional con-straint on the node degree.

3.1. Problem statement

Let GhV, E,Wi be a weighted, connected, and undirectedgraph, where V denotes the vertex set, E denotes the edgeset, and W denotes the set of weights associated with thegraph nodes. A CDS of a graph is a connected subset ofwhere Kk Pai2Akpik is the sum of the probabilities ofthe actions in subset A(k), and pi (k) = prob[a(k) = ai].

The procedure of choosing an action and updating theaction probabilities in a variable action-set learningautomaton can be described as follows. Let A(k) be the ac-tion subset selected at instant k. Before choosing an action,the probabilities of all the actions in the selected subset arescaled as dened in Eq. (3). The automaton then randomlyselects one of its possible actions according to the scaledaction probability vector p^k. Depending on the responsereceived from the environment, the learning automatonupdates its scaled action probability vector. Note that theprobability of the available actions is only updated. Finally,the probability vector of the actions of the chosen subset isrescaled as pik 1 p^ik 1 Kk, for all ai 2 A(k). Theabsolute expediency and e optimality of the method de-scribed above have been proved in [26].

3. Energy-efcient area coverage algorithm

This paper proposes the degree-constrained minimum-weight connected dominating set (DCDS) problem formodeling the energy-efcient coverage problem in WSNs.The CDS size remains the primary concern of the CDS-corresponds to the set of communications links. Therefore,action-set ai(k) is time-variable and its number of actionsmay change at each instant k. Learning automaton is aprobabilistic learning tool that selects its actions accordingto an action probability vector (APV) at random. APV is themain component of a learning automaton thatmust be keptup-to-date. The action probability vector of learning

automaton Ai is dened as pik pjik8ajik 2 aikn o,

where pjik denotes the choice probability of action aji atstage k. In this algorithm, the APV of each learning autom-aton Ai is set to the energy level of its neighboring nodes ini-tially. Let eni k

P8lni ;nj 2Lk

Enj k denotes the total energylevel of the neighbors of sensor ni at stage k. Therefore, theprobability with which sensor ni selects sensor nj (i.e., link

lni ;nj is dened as pjik Enj keni

k at stage k. This activates

the sensor having the maximum energy level (in eachneighborhood) to cover the sensor eld.area by nding a near optimal solution to the DCDS prob-lem, where the weight of each node is dened as its resid-ual energy level. DCDS seeks for the set of most energeticconnected sensors whose maximum degree is boundedabove by d. Let C fC1; C2; . . .g denotes the set of all possi-ble degree-constrained CDSs covering the sensing area. Cis the optimal solution to the DCDS problem (i.e., the de-gree-constrained CDS with the minimum weight), if

EC min8Ci2C1

min8nj2Ci

fEnjg

264

375

where EC denotes the energy of optimal degree-con-strained CDS C; min8nj2CifEnjg denotes the energy of de-gree-constrained CDS Ci subject to constraint d. Energy ofa degree-constrained CDS is dened as the residual energylevel of the most energetic active sensor.

3.2. Degree-constrained CDS-based area coverage algorithm

In this section, a fully distributed learning automata-based algorithm is proposed for solving the area coverageproblem in WSN by nding a near optimal solution to thedegree-constrained minimum-weight CDS problem. In thisalgorithm, a group of learning automata, named as GoL, isconstituted by equipping each sensor node ni with a vari-able action-set learning automaton Ai. Duple GoL is denedas hA(k), a(k)i, where Ak fAij8ni 2 Nkg denotes the setof learning automata assigned to the sensor nodes, anda(k) = {aij"Ai} denotes the set of actions that can be takenby each learning automaton Ai. Due to the frequent topol-ogy changes in WSN, N; L, and E are time-variable parame-ters. In this paper, these parameters are shown asNk; Lk,and Ek for each instant k. Let ai denotes the set of actionsthat can be taken by learning automaton Ai 2 A(k). Eachautomaton Ai chooses the communication links incidentat the corresponding node ni as its actions. That is,

aik ajik8lni ;nj 2 Lkn o. GoL is isomorphic to the net-

work topology graph, where the set of learning automata

Let us assume that the sink node ni starts the coverageprocess. As mentioned earlier, LAEEC is a fully distributedalgorithm that is run at each sensor node independently.Flowchart of the proposed coverage procedure running atsensor node ni is shown in Fig. 1. The node that is runningthe algorithm is called the current node. At each instant k,each current node ni discovers its neighbors and forms itsaction-set by sending an ASF (action-set formation) mes-sage. Each node that receives the ASF message replies it.The reply message includes the residual energy level ofthe node. Current node ni forms its action-set based onthe received replies. Due to the network topology changes,one node may leave (or join to) the other node at eachstage. If link lni ;nj breaks at stage k + 1, its corresponding

action (i.e., aji) must be removed from the action-set ofautomaton Ai (or action aij from automaton Aj). Moreover,

the choice probability of the other actions (e.g., aj0

i ) must

be updated as pj0

i k 1 pj0

i k 1 pjik=1 pjikh ih i

in

automaton Ai. When a new link lni ;nj is established at stagek + 1, the choice probability of the new action is initializedto 1/jai(k + 1)j, and that of the other actions is updated as

node nj with average energy level eni k, and average de-gree dk with degree-constraint d. Then, ni updates theinternal state of its automaton according to the followingupdating rules. If the residual energy level of the node se-lected by ni is higher than the average energy level of theneighbors of ni (i.e., if nj is the most energetic neighbor ofni) and dk does not exceed degree-constraint d, learningautomaton Ai rewards the selected action aji by Eq. (1). Ifthe energy level of selected node nj is lower than the aver-age energy level eni k, and the average degree dk is largerthan degree-constraint dk, learning automaton Ai penalizesthe selected action aji by Eq. (2). Otherwise, the APV of Airemains unchanged.

After learning automaton Ai updates its APV, currentnode ni sends an ACT (activation) message including de-gree-constrained CDS Ck, dominatee set C0k, average degreedk, and CDS energy level ECk to activate the selected sensornode nj. Sensor nj checks to see if its ID number is equal tothe receiver ID, as soon as it receives an ACTmessage. If so,it adds the IDs of its one-hop neighbors to C0k. If C0k includesall the network nodes (i.e., if the constructed CDS covers allthe points within the sensor eld), the current iteration, k,

ge pro

J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666 1659pj0

i k 1 pj0

i k jaik 1j 1=jaik 1j.Let Ck denotes the degree-constrained CDS that is con-

structed at stage k. Let C0k be the set of sensor nodes cov-ered by Ck (or set of dominatees). Ck is initially set to ni,and C0k is initialized to ni and its one-hop neighbors. ECk isinitialized to Eni . Let dk denotes the average degree of Ck.dk is dened (and updated) as

P8ni2CkDik

h i=jCkj at each

stage k, where Di(k) denotes the degree of node ni at stagek. As shown in Fig. 1, current node ni selects one of its ac-tions at random. Let us assume that action aji is selected bynode ni. Sensor ni adds sensor nj (corresponding to selectedaction aji) to the constrained CDS Ck and updates dk. Energyof Ck (i.e., ECk ) is set to minfEnj ; ECkg. eni k eni k=Dik de-notes the average energy level of the neighbors of node niat stage k. Node ni compares the residual energy of sensor

Fig. 1. Flowchart of the proposed area coveraof the coverage process is over. Otherwise, node nj changesits state to the current node and does the same operationsas node ni did. Sensor node ni in which the coverage pro-cess completes, broadcasts a SLP (sleep) message withinthe network through backbone Ck. This message only in-cludes degree-constrained CDS Ck and energy level ECk .Each receiving sensor node ni goes to the sleep mode if itdoes not nd its ID in Ck. Otherwise, it goes to the activemode for sensing the area. The sensor network composedof the active nodes covers the monitoring area until theresidual energy of an active sensor node falls down a pre-dened threshold T ni or one or more active sensor nodesfail. The active node that nds its residual energy levellower than the energy threshold T ni , and the sensor nodethat detects the failure of an active node are responsiblefor initiating a new coverage process.

cedure running at sensor node ni at stage k.

1

probability vector according to LAEEC, the time required for

!k 1 k log

1a ; 5

1660 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666jCj

2 (0,1) is the error rate, a denotes the learning rate of thealgorithm, jCj denotes the cardinality of the optimal degree-constrained CDS, n denotes the best (i.e., most energetic)node that can be chosen by learning automaton Ai, and Di de-notes the degree of node ni.

Proof. Let each learning automaton Ai updates its actionprobability vector pi according to the updating rule of LAE-EC. Let C be the optimal area coverage and ai n denotesthe best action (active sensor) that can be selected byautomaton Ai (node ni). Before stating the proof of this the-orem, the following two lemmas are discussed. h

Lemma 1. If each automaton Ai updates its action probabilityvector according to the proposed updating algorithm, theupper bound on the running time for nding a 11 opti is

21 k jCj

log

jC j1k1a

where kP pi 1 aDi1.

Proof. Lemma 1 aims at computing the worst case runningtime of the proposed algorithm. At each node ni, the worstcase occurs if the optimal sensor node n (the node withthe maximum residual energy satisfying the degree-con-straint) is chosen before all the other nodes. In this case,the learning process is subdivided into two distinct phases:(1) the shrinking phase, and (2) the growing phase. In therst phase, it is assumed that all the other nodes, fromthe node with the minimum energy level to the most ener-getic node are chosen in turn and rewarded before ni acti-nding a 11 opti at node ni is

!En keni k

6 Ti 6 !En keni k

1 aDi1

; 4

where

2 jC j1kproblem is conned between the estimated lower andupper bounds. Finally, it is proved (in Theorem 2) thatthe convergence time of LAEEC to a 11

optimal area cov-

erage (i.e., 11 C) is bounded to the convergence time of thenetwork node with the maximum degree.

Theorem 1. Let opti denotes the optimal action that can bechosen by learning automaton Ai. If Ai updates its action4. Complexity analysis and convergence results

In this section, the computational complexity of theproposed area coverage algorithm, LAEEC, is analyzed. Todo so, an upper bound (Lemma 1) and a lower bound (Lem-ma 2) on the number of iterations of the algorithm for nd-ing a 11

optimal action at each node ni (i.e., learning

automaton Ai) is approximated, where is the error rate.Then, it is shown (in Theorem 1) that the time requiredfor nding a 1

optimal solution to the area coveragevates the optimal node n. Therefore, in the worst case, thechoice probability of the optimal node n at the end of theshrinking phase is computed as

pi Di 1P pi Di 2 1 a 6where pi Di 1 denotes the choice probability of the opti-mal node n (or optimal action ai ) at stage (Di 1), Di de-notes the degree of node ni, a is the learning rate of theproposed algorithm, and pi Di 1 denotes the choiceprobability of the optimal node at the end of the shrinkingphase. Repeatedly substituting the recurrence functionpi : on the right hand side of Inequality (6), we havepi Di 1P pi 1 aDi1, where pi denotes the initialchoice probability of the optimal node n. For the sake ofsimplicity in notation, pi Di 1 is substituted by qi .

The growing phase starts when the optimal node n ischosen for the rst time by node ni. Since the reinforce-ment scheme by which the proposed algorithm updatesthe probability vectors is LRI, the conditional expectationof qi k (i.e., the choice probability of the optimal node atstage k of the growing phase) remains unchanged whenthe other nodes are selected. It increases only when theoptimal node is selected. Therefore, during the growingphase, the changes in the conditional expectation of qi kis always non-negative and as follows

qi 1 qi a 1qi

qi 2 qi 1a 1qi 1 qi 1 1 a a

..

.

qi k1 qi k2a 1qi k2 qi k2 1a a

qi k qi k1 a 1qi k1 qi k1 1 aa

7where k denotes the number of times ni selects the optimalnode n before the following stop condition (derived fromthe Bonferroni correction [27] to achieve an error rate low-er than for the optimal area coverage C) is met.

qi k 1jCj 8

where jCj denotes the cardinality of the optimal areacoverage.

After substituting the recurrence function qi kwe haveqi k qi k 1 1 a a qi k 2 1 a a 1 a a

qi k 2 1 a2 a 1 a a qi k 3 1 a a 1 a2 a 1 a a

qi k 2 1 a3 a 1 a2 a 1 a a...

qi 1 1 ak1 a 1 ak2 a 1 a a qi 1 ak a 1 ak1 a 1 a a

Hence, we have

qi k qi 1 ak a 1 ak1 a 1 a a9

After algebraic simplications, we have

2 logjC j 1pi

1a :

!k 2 logjC j1k

J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666 1661qi k qi 1 ak a 1 1 a1 a2 1 ak1

and

qi k qi 1 ak a Xk1i0

1 ai 10

The second term on the right hand side of Eq. (10) is a

geometric series that sums up to a 11ak11a

, where

j1 aj < 1. Since the learning rate a 2 (0,1), we have

qi k qi 1 ak a 1 1 ak1 1 a

!11

and

qi k qi 1 ak 1 1 ak 12From Eqs. (8) and (12), we have

qi 1 ak 1 1 ak 1jCj 13

and

1 ak jCj 1 qi 14

Taking log1a of both sides of Eq. (14), we derive

k log

jC j 1qi

1a 15As mentioned earlier, during the growing phase, qi

remains unchanged when the other nodes are penalized.Therefore, k does not include the number of times theother nodes are selected. Let qi be the choice probabilityof the optimal node at the beginning of the growing phase.After k iterations qi reaches 1 . On the other hand, thechoice probability of all the other nodes is initially 1 qiand reaches after the same number of iterations. There-fore, the number of times the other nodes are selected,before the stop condition given in Eq. (8) is met, is

1 qi jCj1 qi jCj

k 16

Let K denotes the total number of iterations required tosatisfy the stop condition. From Eq. (16) we have

K 21 qi jCj

k

By substituting k from Eq. (15) we have

K 21 qi jCj

log

jC j 1qi 1a 17

From Inequality (7) and Eq. (17), it is concluded that thetime complexity of the LAEEC for nding a 11 opti is lessthan

21 qi jCj

log

jC j 1qi 1a 18

where qi P pi 1 aDi1, and hence the proof of Lemma

1. h1 k jCj 1a

Proof. As mentioned earlier, the proposed algorithm isindependently run at each node and each leaning automa-ton locally updates its internal state to converge to theoptimal action. Therefore, node nM requires the maximumnumber of iterations for nding 11 optimal action of1 pi jCj

Proof. Lemma 2 considers the running time of the pro-posed algorithm in the best case, when ni selects the opti-mal node n before the others. In this case, the learningprocess does not include the shrinking phase. Therefore,pi denotes the choice probability of the optimal node inthe beginning of the growing phase. Similar to the proofof Lemma 1, it can be easily proved that the minimumnumber of iterations required for satisfying the stop condi-tion (8) is

21 qi jCj

log

jC j 1qi

1a ; 19

where qi pi , which completes the proof of Lemma 2. h

From Inequalities (18) and (19), it can be concluded that

21 qi jCj

log

jC j 1qi 1a 6 Ti 6

21 qi jCj

log

jC j 1qi 1a ;

where qi P pi 1 aDi1.

As described in Section 3, for each action aji, the initial

probability pji is set toEnj keni

k, where eni k P

ni ;nj2Lk

Enj k. Therefore, the initial probability pi is set to En keni k.Therefore, we have

!En keni k

6 Ti 6 !En keni k

1 aDi1

;

where

!k 21 k jCj

log

jC j1k1a ;

which completes the proof of the theorem.

Theorem 2. Let nM denotes the network node with themaximum degree D. The time complexity of the proposedalgorithm for nding a 11 optimal solution to the coverageproblem is

!En kenMk

6 T 6 ! En kenMk

1 aD1

;

where

Lemma 2. If the action probability vector pi (of each autom-aton Ai) is updated according to the updating rules of LAEEC,the lower bound to the running time of LAEEC for nding a1

1 opti is greater than

bittransmitter or the receiver circuitry. Each sensor node

In these experiments, the proposed area coverage algo-rithm, LAEEC, is congured as follows. The environmentin which the learning automata perform is assumed to beP-model. Each learning automaton updates its action prob-ability vector according to reinforcement scheme LRI. Inthese experiments, LAEEC is calibrated by tuning degree-constraint d and learning rate a as follows. The coveredarea and network lifetime are measured, where degree-constraint d changes from 2 to 15. The obtained resultsshow that the best trade-off between the covered areaand network lifetime is made when degree-constraint dis set to 7. The same experiment is conducted to adjustthe learning rate, where a changes from 0.05 to 0.5. The re-sults show that LAEEC has the best performance when thelearning rate is set to 0.15. In The energy threshold T ni isdened as 0:5 ECk . That is, a new coverage process initiates

1662 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666also consumes 100 pJbit =m2

for handling the transmit

amplier. Therefore, the energy amount required forreceiving a k-bit data packet is estimated as

kbit 50 nJbit

50knJ

The energy amount that is consumed to transmit a mes-sage of length k to a destination node located x(m) far fromthe transmitter is computed as

kbit 50 nJbit

kbit 100 pJ

bit

m2

x2m2

50knJ 100 kx2pJlearning automaton AM. On the other hand, from Lemmas1 and 2, the running time of the proposed algorithm fornding 11 optimal coverage is limited by the upper boundand lower bound on the running time of the algorithm forthe node with the maximum degree D. Therefore, it is con-cluded that the time taken by the proposed algorithm fornding a 11 optimal coverage is

!En kenMk

6 T 6 ! En kenMk

1 aD1

;

where!k 21k jC j log

jC j1k1a , that completes the proof of

Theorem 2. h

5. Experiments

In this section, several simulation experiments areconducted to show the performance of the proposedCDS-based area coverage algorithm. The results of theproposed method are compared with those of threeCDS-based energy efcient area coverage protocol A3[24], A3CovLite [6], and A1 [4] in terms of control mes-sage overhead, percentage of covered area, residual en-ergy, number of active nodes (CDS size), and networklifetime. In these experiments, the wireless sensor net-work is setup as follows. The wireless sensor nodes areuniformly and randomly distributed within a square sen-sor deployment area of size 150(m) 150(m) at random.The number of sensor nodes ranges from 50 to 250 withincrement step 50. The radio transmission range of eachsensor node is set to 20(m), and the sensing range of eachnode is set to 10(m). The size of each data packet is 100bytes. The simulation time of each experiment is 1500(s). Each sensor node has an omnidirectional antennawith a xed radio propagation range. IEEE 802.11 [28](Distributed Coordination Function) with CSMA/CA (Car-rier Sense Multiple Access/Collision Avoidance) is usedas the medium access control protocol, and two rayground as the radio propagation model. The maximumenergy level of each sensor node is 2.0(J), and the initialenergy level the sensors is randomly selected from theuniform distribution dened over interval [1.5(J), 2.0(J)].The energy model presented by Heinzelman et al. [29]is used for estimating the amount of energy consumption.In this energy model that is based on the rst order radiomodel, each sensor node consumes 50 nJ

to run thewhen the energy level ECk falls to 50% of its initial value. Alltimeouts are set to 100 ms. In class A3, the weights are setas follows: WE denoting the weight for the remaining en-ergy in the node is set to 0.5, and WD denoting the weightfor the distance from the parent node is set to 0.5 too,where WD +WE = 1. For A3-based protocols, the timersare set as t0 = 1.5, t1 = 30.0, t2 = 15.0, and t3 = 60.0.

5.1. Number of active nodes

This metric is dened as the average number of nodesthat are activated to cover the sensor eld (i.e., the averageCDS size). This metric implicitly shows the number ofdominators in the CDS. The residual energy of the networkis inversely proportional to the number of active nodes.Therefore, the energy-efcient protocols try to minimizethe number of active nodes. Fig. 2 shows the number of ac-tive nodes (dominators in CDS) against the total number ofnetwork nodes. From the results shown in this gure it canbe seen that A3 has the minimum number of active nodesas compared to the other protocols. This is because A3algorithm uses a selection metric giving the priority tothe farther nodes from the parent having higher energy le-vel. This method considerably reduces the CDS size. Asshown in Fig. 3, this results in the lower coverage rate ofA3. A3CovLite uses extra active nodes to cover the pointsof the sensor eld leaved uncovered in A3. So, it requiresmore active nodes than A3. Though A1 provides a higher

Fig. 2. The number of active nodes vs. the total number of network nodes.

that is covered by the active sensor nodes. The coverage

5.3. Residual energy

The residual energy is dened as the average remainingenergy of the active sensor nodes at the end of each simu-lation experiment. Fig. 4 shows the average residual energyof the active sensor nodes as a function of the network size.From the results shown in this gure, it is observed thatthe average residual energy level of the proposed area cov-erage algorithm is signicantly higher than the othermethods. This is due to the fact that the proposed methodmakes a good trade-off between the number of activenodes (required to cover the area) and the amount of en-ergy consumption in each active node by selection of aproper degree-constraint. This signicantly reduces thenumber of nodes covering the same points of the area,while avoids the rapid exhaustion of the active sensorsfor handling a huge number of neighbors. The results alsoshow that A3 has the lowest residual energy level, and

J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666 1663percentage is a measure of the quality of service (QoS) ofthe coverage protocol. Fig. 3 shows the percentage of thesensor eld covered by the selected active nodes in differ-ent algorithms as a function of the network size. From theresults shown in this gure, it can be seen that the coveredarea signicantly increases as the network density (i.e., thenumber of nodes in the network) increases. This is becausethe number of scattered nodes within the simulation areaconsiderably exceeds the number of active nodes requiredin optimal deployment. The obtained results depicted incoverage rate than A3 and A3CovLite, it suffers from themany redundant active nodes. This is due to the fact thatA1 forms the reduced topology without any metric desiredfor the reduction in the size of the CDS. From the resultsshown in Fig. 2, it can be seen that the number of activenodes in LAEEC is larger than that of A3-based approachesand smaller than that of A1. In LAEEC, the number of activenodes is controlled by degree-constraint d. Larger values ofd reduces the number of active nodes and smaller values ofd leads to very large CDS.

5.2. Covered area

This metric shows the percentage of the sensing area

Fig. 3. The percentage of the covered area vs. the network size.Fig. 3 also show that A3 provides the minimum area cover-age as compared to the other protocols. This is due to thefact that A3 sends a sleep message to the nodes withinthe communication area of the other nodes in the reducedtree topology. This may cause some points of the area re-main uncovered when the sensing range is smaller thanthe communication range. A3CovLite solves the coverageproblem with A3 by sending a node to the sleep mode ifit is sensing-covered by another active node. That is why,the A3CovLite has a higher rate of area coverage than A3.Comparing the results shown in Fig. 3, it is observed thatthe proposed area coverage protocol LAEEC always coversthe whole monitoring area. This is because, LAEEC rstconstructs a degree-constrained CDS-based backbone cov-ering all the network points. Then, it sends all the non-dominators to the sleep mode. A1 outperforms A3 andA3CovLite in terms of covered area. This can be due tousing a larger number of active nodes to cover the area.A3CovLite slightly outperforms A3. This is because A3-based approaches reduce the number of backbone nodesby activating the farther nodes from the parent node. Thiscauses a non-uniform distribution of the communicationoverhead and places a heavy load on the active nodes.Therefore, A3-based approaches result in the imbalancedenergy consumption within the network. Comparing theresults of A1 and A3, it can be seen that A1 provides a sig-nicant higher residual energy level as compared to A3.This is due to the fact that in A1 protocol the nodes calcu-late the timeout with the selection criteria resulting in abalanced virtual backbone.

5.4. Network lifetime

Network lifetime is dened as the average period oftime during which the set of active sensors remain con-nected. Minimizing the energy consumption and maximiz-ing the network lifetime are the major concerns of thedesign of the coverage protocols. Network lifetime implic-itly shows the energy-efciency and load balancing of thecoverage protocol. Fig. 5 shows the changes in the networklifetime as the number of network nodes changes from 50to 250 with increment step 50. From the results shown in

Fig. 4. The average residual energy of the active nodes as a function of thenumber of nodes.

1664 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666Fig. 5, it can be seen that for all coverage algorithms, thenetwork lifetime reduces as the network size increases.This can be due to the fact that the backbone size growsand it makes harder evenly distribution of the networkload on the backbone nodes. Comparing the curves de-picted in Fig. 5, it is observed that A3 has the shortest life-time and the proposed area coverage algorithm has thelongest lifetime. The main objective of the proposed cover-age algorithm is to extend the lifetime of the network de-ployed to monitor the area as much as possible. To do so, ituses the degree-constrained minimumweight CDS conceptto activate the nodes having the maximum residual energylevel, and to evenly distribute the network load on the ac-tive nodes. This signicantly extends the lifetime of the ac-tive nodes. As mentioned earlier, A3 tries to reduce therequired number of active nodes to cover the area. Onone hand, this reduces the total energy consumption ofthe network by keeping a larger number of sensors in sleepmode. However, on the other hand, the heavy burdenplaced on the small set of active nodes drains them sooner.This signicantly shortens the network lifetime. A3CovLiteshows a better performance in Fig. 5 as compared to A3. Asshown in Fig. 2, this is achieved by adding redundant activenodes (dominators) to the CDS. However, as the curvesshow in Fig. 4, there is no signicant gap between the aver-age residual energy level of A3 and A3CovLite. A1 uses the

Fig. 5. Network lifetime vs. the number of nodes.largest set of (redundant) active nodes to cover the area.The results given in Fig. 5 show its superiority over A3and A3CovLite.

5.5. Control message overhead

In this experiment, the control message overhead isdened as the number of (extra) control messages re-quired for coverage (degree-constrained CDS formation)process. The extra messages are the control messages thatare used to construct the CDS-based backbone (i.e., themessage overhead of the coverage protocol). This metricis measured as the number of control messages that mustbe sent per second. Fig. 6 depicts the control messageoverhead of the coverage algorithms vs. the number ofnodes. The results show that LAEEC has the lowest controlmessage overhead and A3 has the highest one. The resultsalso reveal that A3CovLite lags far behind A1. The reasonfor the highly message overhead of A3 is that this proto-col uses four messages to construct the CDS backbone.The message complexity of A3 (in worst case) is 4n,where n is the number of network nodes. A3 uses a chil-dren recognition message of size 100 bytes as well asthree other messages of size 25 bytes. A3CovLite onlyuses two messages of size 25 bytes. The message com-plexity of A3CovLite is at most 2n. Therefore, it has ameaningfully lower message overhead than A3. A1 usesonly one type of message (a hello message of size 25 by-tes) for CDS formation (having message complexity n).That is why, A1 outperforms A3 and A3CovLite in termsof control message overhead. The proposed area coveragealgorithm uses only an activation (ACT) message to con-struct the CDS structure. ACT is a variable-length messagewhose size is in the interval 1; jCkj bytes. The number oftimes this message is exchanged between the activenodes is jCkj. Therefore, the average message complexityof LAEEC is jCkj2=2 bytes that is signicantly lower thanthat of A1.

6. Conclusion

Fig. 6. The control message overhead vs. the number of network nodes.Over the past couple of decades, CDS has received a loof attention and found many applications in wireless networking such as routing, clustering, backbone formationand multicasting. CDS has recently emerged as an innovative approach to model the area coverage problem inwireless sensor networks and several CDS-based area coverage protocols have been proposed. However, the majoproblem affecting the performance of the existing CDSbased coverage protocols is that they aim at maximizingthe number of sleep nodes to save more energy. This imposes a heavy burden on the active nodes for handling alarge number of neighbors. The rapid exhaustion of theactive nodes may disconnect the network topology andleave the area uncovered. This paper proposed a degreeconstrained minimum-weight extension of the CDS probt-,-

-r-

-

--

lem called DCDS to model the area coverage problem in

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Javad Akbari Torkestani received the B.S. andM.S. degrees in Computer Engineering in Iran,in 2001 and 2004, respectively. He alsoreceived the Ph.D. degree in Computer Engi-neering from Science and Research University,Iran, in 2009. Currently, he is an assistantprofessor in Computer Engineering Depart-ment at Arak Azad University, Arak, Iran. Priorto the current position, he joined the facultyof the Computer Engineering Department atArak Azad University as a lecturer. Hisresearch interests include wireless networks,

multi-hop networks, fault tolerant systems, grid computing, learningsystems, parallel algorithms, and soft computing.

1666 J. Akbari Torkestani / Ad Hoc Networks 11 (2013) 16551666

An adaptive energy-efficient area coverage algorithm for wireless sensor networks1 Introduction2 Learning automata theory3 Energy-efficient area coverage algorithm3.1 Problem statement3.2 Degree-constrained CDS-based area coverage algorithm

4 Complexity analysis and convergence results5 Experiments5.1 Number of active nodes5.2 Covered area5.3 Residual energy5.4 Network lifetime5.5 Control message overhead

6 ConclusionReferences