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Energy Efficient Distributed Connected Dominating SetsConstruction in Wireless Sensor NetworksZeng YuanyuanComputer School, Wuhan UniversityWuhan, Hubei, Chinazyywhu@gmail.comJia XiaohuaDep. of Computer Science, CityUniversity of Hong KongHong Kong, Chinajia@cs.cityu.edu.hkHe YanxiangComputer School, Wuhan UniversityWuhan, Hubei, Chinayxhe@whu.edu.cnABSTRACTOne important characteristic of wireless sensor networks is energystringency. Constructing a connected dominating set (CDS) hasbeen widely used as a topology control strategy to reduce thenetwork communication overhead. In the paper, a novel energyefficient distributed connected dominating set algorithm based oncoordinated reconstruction mechanism is presented to furtherprolong the network lifetime and balance energy consumption. Thealgorithm is with O(n) time complexity and O(n) messagecomplexity. The simulation results show that our algorithmoutperforms several existing algorithms in terms of network lifetimeand CDS performance.Categories and Subject DescriptorsC.2.1 [Computer-Communication]: Network Architecture andDesign -Wireless communicationGeneral TermsAlgorithms, Performance, Design.KeywordsWireless sensor networks, connected dominating set, energyefficient, approximation algorithm1. INTRODUCTIONThe research on wireless sensor networks has been fueled up bymany applications in various areas, such as environment monitoring,target or object tracking, industry automation and control, and so on[1, 2]. One important characteristic of sensor networks is thestringent power budget of wireless sensor nodes. Research hasshown that a great amount of energy of sensor nodes is consumedfor communications. Reducing the communication cost is animportant way to save energy of sensor nodes and to prolong thelifetime of sensor networks.A dominating set (DS) of a graph is a subset of nodes such thateach node in the graph is either in the subset or adjacent to at leastone node in that subset. A CDS is a DS, which induces a connectedsub graph. A connected dominating set (CDS) is a good candidateof a virtual backbone for wireless networks, because any node in thenetwork is less than 1-hop away from a CDS node. Only thebackbone nodes are responsible for relaying messages for thenetwork. The non-backbone nodes can thus turn off theircommunication module to save energy when they have no data to betransmitted out. One objective for constructing the backbone is tominimize the size of a backbone (i.e., the number of backbonenodes). We assume that every node has the same transmission rangeso that we can model the network topology using unit disk graphs,UDG in short. Unfortunately computing a minimal CDS (denotedby MCDS) of a UDG graph has been proved to be NP-hard [4]. Inthis paper, we propose a novel efficient distributed approximationalgorithm (ECDS) that computes a sub-optimal MCDS inpolynomial time. Since the backbone nodes have heavycommunication load for relaying messages for the network, theirenergy consumption is high. To balance the energy consumptionload, each time when we compute the MCDS, we also compute outthe length of time that this MCDS should work as a backbone. Thelength of time is dependent on the residual energy of the MCDSnodes. When this length of time expires, the sensor nodes coordinatewith each other again and compute the next MCDS as the backbonefor the next period of time. By doing so, the energy consumption ofall nodes in the network is balanced and the lifetime of the networkis extended. There are three major advantages of the proposedalgorithm: 1) the algorithm is fully distributed, which can be easilyimplemented in sensor networks; 2) the constructed CDS has asmall size, which reduces the overhead of maintaining the backboneand the cost in communication; 3) the constructed CDS achievesload balancing, which extends the lifetime of the network.The remainder of this paper is organized as follows. Section 2briefly introduces the related work in the literature. Section 3discusses out distributed algorithm for constructing the CDS.Section 4 is the performance analysis. Section 5 presents simulationresults. Section 6 is the conclusion and future work.2. RELATED WORKCurrent MCDS approximation algorithms include centralized anddistributed algorithms. Following the increased interest in wirelessad hoc and sensor networks, many distributed approaches have beenproposed because of no requirements for global network topologyknowledge. These algorithms contain two types. One type is to finda CDS first, then prune some redundant nodes to attain MCDS. Wuand Li proposed in [6] a distributed algorithm with Θ(m) messagecomplexity and O(Δ3) time complexity, the approximation factor atmost n/2. Butenko et al [7] constructs a CDS starting with a feasiblesolution, and recursively removes nodes from the solution until aMCDS is found. The other type is to form a maximal independentPermission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, orrepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.IWCMC’06, July 3–6, 2006, Vancouver, British Columbia, Canada.Copyright 2006 ACM 1-59593-306-9/06/0007...$5.00.797
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set (MIS) at first, and then find some connectors to make theindependent nodes connected together. P. J .Wan et al [4] proposes adistributed algorithm with performance ratio of 8. Min et al in [8]propose an improved algorithm by employing a Steiner tree in thesecond step to connect the nodes in the MIS with performance ratioof 6.8. Recently, there’s a great increasing focus on low cost andlow energy consumption in wireless networks. Wu et al in [9]present an algorithm for power aware connected dominating setbased on [7]. Acharya et al in [10] present a power aware MCDSconstruction when introducing a concept of threshold energy levelfor dominating nodes based on [7]. Those algorithms have someenergy demands, but do not consider the balance of energyconsumption in the network.3. DISTRIBUTED ALGORITHM FORENERGY EFFICIENT CDS BACKBONE3.1 System Models and Basic IdeasWe assume that all nodes in wireless sensor networks are distributedin a two-dimensional plane and have an equal maximumtransmission range of one unit. The network topology is modeled asa unit disk graph, UDG in Short. We use G = (V, E) to representsuch networks, where V is the set of sensor nodes and E is the set ofedges. The aim of our algorithm is to compute a sub-optimal MCDSas a backbone for wireless sensor networks. Our algorithm consistsof two phases. In the first phase, we compute a maximalindependent set (MIS) of the network graph. An independent set (IS)of a graph is a subset of V that no two nodes in the subset have anedge. An MIS of a graph is an independent set that cannot includeany more nodes in V. Thus an MIS is a DS of a graph. Note that thisDS (obtained as the MIS) may not be connected. The second phaseof the algorithm is to choose the minimal number of nodes (calledconnectors) to make the DS connected, i.e., a CDS. We takedynamic reconstruction strategy to balance energy consumption inthe networks. Each time when a CDS is constructed as backbone,the length of operating time of this CDS for this round is determinedaccording to the energy level of the CDS nodes. If the minimalresidual energy of nodes in CDS is cut down to a certain level (suchas 50%) of the initial energy of current round, the operating periodof this round is due. When this operating period expires, the nextCDS operating round is computed. We also make improvements onenergy efficiency and performance on CDS based networkbackbone each round. So our algorithm can extend lifetime of thenetwork dramatically. We assume that each node u has a weightw(u) of being in the backbone. Here w(u) can be the value computedbased on a combination of its remaining battery power energy(u)and its effective degree degree(u) in the communication graph, andso on. Thus, each time the constructed CDS backbone could hashigher energy level and smaller size under the condition of energybalance. We’ll give the definition of effective degree in Section 3.2and Section 3.3 respectively.3.2 MIS ConstructionThe algorithm always starts from a node that initiates the execution.We call this node initiator. The initiator can be designatedbeforehand. We use colors to indicate if a node is in MIS or not, i.e.,use black to indicate the nodes in MIS and grey to indicate non-MISnodes.Figure 1 can describe the algorithm execution of each node.Each node is in one of the four states: white, black, grey andtransition. Initially, all nodes are in white, and at the completion ofthe algorithm all nodes in the network must be either in black (MISnodes) or in grey (non-MIS nodes). The transition is an intermediatestate. There are three types of messages: 1) BLACK message，sentout when a node becomes a black node; 2) GREY message, sent outwhen a node becomes a grey node; 3) INQUIRY message, sent outwhen a node inquires the weights and states of its neighbors. Eachmessage contains node state and id, i.e., state(i), id(i) . The replies ofINQUIRY message contains the weight of nodes, i.e., w(i).Nodecalculates its w(i) by battery power and its effective degree. Theeffective degree is the number of its neighbors in white andtransition state.whitetransitiongrayblackRecv aGREYmsgtimeout andcompetitionfailtimeout and if hasthe highest weightamong neighborsin transitionRecv aBLACKmsgRecv aBLACKmsgAnymsgAnymsgRecv anINQUIRYmsgOthermsgFigure 1. State transition diagram of MIS constructionAt the start of the algorithm, all nodes are in white. Firstly, theinitiator colors itself in black. A node that colors itself in black willbroadcast a BLACK message to its neighbors immediately toindicate itself as an MIS node. A white neighbor that receives theBLACK becomes a grey node (i.e., a non-MIS node), andbroadcasts a GREY message to indicate its neighborssimultaneously. A white node that receives a GREY message is aneighbor of the non-MIS node. It needs to compete to become ablack node. So it broadcasts an INQUIRY message toward itsneighbors to inquire their states and weights. The node sets atimeout to wait for the replies of the INQUIRY message, and thenenters the transition state. It will stay in the transition state untiltimeout expires. During the timeout, it may receive a BLACK,INQUIRY or GREY message. If receives a BLACK message, thenode becomes grey and broadcast a GREY message. If receivesother messages, the node ignore them and only stays in thetransition. When timeout is due, if the node finds it has the highestweight among all neighbors in transition (or the node has noneighbors in transition state) based on the replies of INQUIRY, itwill become Black node. Then the node in black does the same asthe other black ones do. Otherwise, the node enters back into whitestate after the timeout. During the coloring process, each grey nodewill keep a list of all the adjacent black (MIS) nodes. The sameoperation continues from node to node as the BLACK or GREYmessages propagate, until nodes have entered the state of eitherblack or grey.The initiator starts the construction of a new CDS by executingthe following procedure:initiator () {Color itself black;Broadcast a BLACK message;}Each node i, performs the following function:MIS-algorithm () {Recv a msg;If state(i) is black/grey then //final state of phase 1798
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Ignore the msg and return;If state(i) is white then //initial stateSwitch on msg-typeBLACK: // the sender is an MIS nodestate(i) ← grey;Broadcast a GREY msg;GREY: // the sender is a non-MIS nodeBroadcast an INQUIRY msg;Set a timeout;state(i) ← transition;INQUIRY: // the sender is in transitionReply its state(i) and w(i);endSwitchIf state(i) is transition then //state during timeout periodSwitch on msg-typeBLACK: // the sender is an MIS nodestate(i) ← grey;Broadcast a GREY msg;INQUIRY://sender is in transitionReply its state(i) and w(i);GREY: //the sender is a non-MIS nodeIgnore the msg;endSwitch}Upon the expiration of timeout for transition state, it executesthe following procedure:transition-timeout() { //timeout expiresIf w(i) is highest among transition neighbors thenstate(i) ← black;Broadcast a BLACK msg;elsestate(i) ←white; //competition fails}Theorem 1: The set of black nodes that computed by the phase1 algorithm forms a maximal independent set of the network graph.Proof: We donate the set of black nodes that computed byphase 1 algorithm as B. The MIS algorithm colors the nodes of thegraph layer by layer, and propagates out from the initiator to reachall nodes in the network, with one layer of black and the next layeras grey. At each layer (except initiator), black nodes are selected bygrey nodes of previous layer and are marked black. The constructionincrementally enlarges the black node set by adding black nodes 2hops away from the previous black nodes set. Also the newlycolored black nodes could not be adjacent to each other, for theinterleaving coloring layer of black and grey nodes. Hence everyblack node is disjoint from other black nodes. This implies that Bforms an independent set. Further, the algorithm will end up withblack or grey nodes only. Each grey node must have at least oneblack neighbor, so if coloring any grey node black, B will not bedisjoint anymore. Thus, B is the maximal independent set.3.3 CDS ConstructionSince an MIS is a dominating set with non-adjacent edges, a CDScan be constructed by making the DS connected together, i.e., byconnecting the nodes in an MIS through some nodes (calledconnectors) not in the MIS. A localized approximation of minimumspanning tree may perform well enough. We call it dominating tree.So we take a greedy approximation algorithm that every MIS nodeselects the non-MIS node with the highest weight which areequivalent to 2-hop to interconnect two or more MIS nodes, as aconnector.Figure 2 can describe the CDS construction algorithm. An MISnode can be in one of the three states: black, b-transition and blue,as shown in Fig. 2(a). A non-MIS node can be in one of the threestates: grey, g-transition and blue, as shown in Fig. 2(b). When thealgorithm is complete, all the nodes in the network graph are in thefinal state of either blue or grey. And all the blue nodes are the CDSnodes. There are three types of messages: 1) BLUE message, sentout when a node becomes a CDS node. 2) INVITE message, sentout by an MIS node to invite a non-MIS neighbor to be as aconnector. 3) UPDATE message, sent out by a non-MIS node tonotify all its neighbors about its current weight. In UPDATEmessage, the node will calculate its weight w(i) by battery powerand its effective degree. Here the effective degree is the number ofits current black neighbors (i.e., the number of its MIS neighbors inblack state).Figure 2. State transition diagram of CDS algorithmAfter finishes phase 1 algorithm, a node in the network graphis either in black (i.e., an MIS node) or in grey state (i.e., a non-MISnode), and each grey node keep a list of its black neighbors. Thenthe node will check to see whether it could begin the second phaseof CDS algorithm. If all its neighbors become grey or black (i.e., allits neighbors terminate phase 1 and become MIS or non-MISnodes), the node will begin phase 2. Apparently, the CDS algorithmstarts from MIS initiator too because of message propagation order,so it’s the root of dominating tree. At this point, the initiator canstart phase 2 by coloring itself in blue (i.e., becoming a CDS node)as the following:initiator() {Color itself in blue;Broadcast a BLUE message;}A node that colors itself in blue will broadcast a BLUEmessage to indicate itself as a CDS node. We assume that allmessages are delivered in order.799
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When a non-MIS node in grey begins phase 2 algorithm, itresponses to the messages receives. A grey node that receives aBLUE message (sent out by a neighbor of CDS node) willrecalculate its weight, because its effective degree changes as thestate of its MIS neighbors does. And then the updating weightinformation will be sent out by UPDATE message to notify all itsneighbors. After sending out the UPDATE message, the node entersthe g-transition state. A grey node that receives an UPDTAEmessage only ignores the message. The intuition of g-transition statefor a non-MIS node is to probe the network and compete to see if itis suitable to behave as a connector. The node in g-transition statemay receive three types of message: UPDATE, BLUE and INVITE.If a node in g-transition state receives an UPDTE message, it onlyignores the message. If a node in g-transition state receives a BLUEmessage, it implies that the competition fails for other nodes havebeen selected as a connector already. Then the node will enter backinto grey state. Otherwise, if a node in g-transition state receives anINVITE message, it implies that the competition succeeds and thisnode is invited to be a connector. Then the node will enter into bluestate to become a CDS node. When the node enters blue state as aCDS node, it broadcasts a BLUE message.When an MIS node in black begins phase 2, it responses to themessages receives. A black node that receives a BLUE message(sent out by a connector) will enter blue state directly to become aCDS node. A black node that receives an UPDATE message fromits neighbors (i.e., non-MIS nodes in g-transition state), will set atimeout and then enter b-transition state. It will stay in the b-transition state overhearing UPDATE messages from its effectivenon-MIS neighbors until timeout expires. Only non-MIS nodes withat least one blue neighbor are effective, and this could guarantee theconstructed dominating tree is connected. The UPDATE message issent out with updating weight information, when a non-MIS nodereceives a BLUE message. When a node in b-transition, it mayreceives two types of messages: BLUE and UPDATE. If a node inb-transition state receives an UPDATE message, it only ignores themessage. If a node in b-transition state receives a BLUE message, itimplies that the node already has a connector, and the node willenter blue state directly to become a CDS node. When the nodeenters in blue, it broadcasts a BLUE message to indicate itself as aCDS node as the other blue nodes do. Otherwise, when b-transitiontimeout expires, the node will choose the non-MIS neighbor withthe highest weight as a connector among the UPDATE messagesenders, and then sends out an INVITE message to the selected one.We select the node with the highest weight to guarantee theconstructed CDS with higher energy and smaller size. After sendingout the INVITE message, the node enters back to black state.The CDS construction algorithm continues until: 1) Any MISnode colored blue (i.e., becoming CDS node) terminates thealgorithm. 2) Any non-MIS node terminates when it satisfies eitherof the following two conditions terminate the algorithm: it is coloredblue (i.e., chosen as a connector) or all its neighbors are colored blueand grey (i.e., all its neighbors are in the final state).The phase 2 algorithm after initiator sending out BLUEmessage can be presented as below. Each node i, performs thefollowing function based on its local information, in responses to themessages it receives:CDS-algorithm{Recv a msg;Switch on state-type{blue:Ignore the msg and return; //terminateblack:If recv a BLUE msg thenstate(i) ← blue;Broadcast a BLUE message;If recv an UPDATE msg thenSet a timeout;state(i) ← b-transition;grey:If all its neighbors are blue/grey thenIgnore the msg and return;If recv a BLUE msg thenSend out a UPDATE with w(i);state(i) ← g-transition;If recv an UPDATE msg thenIgnore the msg;b-transition: // during timeout periodIf recv a BLUE msg thenstate(i) ←blue;Broadcast a BLUE msg;If recv an UPDATE msg thenIgnore the msg;g-transition: //intermediate stateIf recv an INVITE msg thenstate(i) ← blue;Broadcast a BLUE msg;If recv a BLUE msg thenstate(i) ← grey ;If recv an UPDATE msg thenIgnore the msg;}endSwitch}Before a node enters b-transition state, it sets a timeoutoverhearing UPDATE message from its effective neighbors. Uponthe expiration of this timeout, it executes the following procedure:b-transition-timeout() { //timeout expiresfind the highest w(k) from the senders of UPDATE;send out an INVITE msg to k; //invite k to be a connectorstate(i) ← black;}Lemma 1: For an MIS node calculated by our algorithm in anetwork graph, there always exists that it has a non-MIS neighborconnecting with at least another MIS node.Proof: Considering the propagation layer of our MISalgorithm, let Bi and Gi be the set of MIS and non-MIS nodes at ithlayer. For any node g ∈Gi is a non-MIS node formed at the ithlayer. In the phase 1 algorithm, it is marked grey (non-MIS node)from white state on receiving a BLACK message from its blackneighbor (MIS node) in Bi. Next, after determining its state, the greynode g sends out a GREY message to all its neighbors in the i+1thlayer. The neighbor finds itself with the highest weight among all itstransition neighbors will become a black node in Bi+1. This impliesthat there always exists a non-MIS neighbor node g∈Gi has at leasttwo MIS neighbor nodes in Bi and Bi+1 respectively. So for an MISnode in Bi, there always exists that it has a neighbor in Giconnecting at least another MIS node in Bi+1.Theorem 2: The set of blue nodes computed by the phase 2algorithm is a CDS of the network graph.Proof: The set of blue nodes include MIS nodes andconnectors. MIS is a dominating set, so we only need to proof the800
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connectivity. Let {b0, b1 …bn} be the independent set, whichelements are arranged one by one in the construction order. Let Hibe the graph over {b0, b1…bi},(1≤i＜n) in which pairs of nodes areinterconnected by connectors. We prove connectivity by inductionon j that Hj is connected. Since H1 consists of a single node, it isconnected trivially. Assume that Hj-1 is connected for some j≥2 .Considering message propagation layer in phase 1 algorithm, let Bi-1and Gi-1 be the set of MIS and non-MIS nodes at the i-1thlayer,respectively. The non-MIS node in Gi-1 with maximal weight isselected as connecters in phase 2 algorithm. According to Lemma 1,it’s enough to find non-MIS nodes, which interconnect Bi-1 nodes ati-1thlayer with Bi nodes in the ithlayer. As Hj-1 is connected, so mustbe Hj. Therefore the set of blue nodes computed by phase 2algorithm is a CDS.4. PERFORMANCE EVALUATIONTheorem 3: Our distributed algorithm has O(n) message complexity,and O(n) time complexity.Proof: In phase 1, for BLACK, GREY, and INQUIRYmessage, each node at most sends out once this kind of messages.Thus, the total number of these messages is O(n). In phase 2, forBLUE, UPDATE, and INVITE messages, since each node sends aconstant number of messages, the total number of messages is alsoO(n). The time complexity of this algorithm is bounded by MISconstruction, which has the worst time complexity O(n). The worstcase occurs when all nodes are distributed in a line and in eitherascending or descending order of their weight. The rest of theprocess have time complexity at most O(n).The following two important property is listed in [11]:Lemma 2: In a unit disk graph, every node is adjacent to atmost five independent nodes.Lemma 3: In any unit disk graph, the size of every maximalindependent set is upper-bounded by 3.8opt+1.2 where opt is thesize of minimum connected dominating set in this unit disk graph.Lemma 4: In the phase 2 algorithm, the number of theconnectors will not exceed 3.8opt, where opt is the size of MCDS.Proof: Let B be the independent set and S be the connector setof a graph. From lemma2: |B|≤3.8opt+1.2. From lemma1 andlemma 2, it can be deduced that the number of MIS neighbors for aconnector is ranged from 2 to 5. Let T be the dominating treespanning black and blue nodes found by our algorithm. The worstcase for T’ size occurs when all nodes are distributed in a line. Byanalyzing the utmost situation, the number of connectors must beless than the number of MIS nodes, i.e., |S|≤|B|-1≤3.8opt. Sothe number of output connecting nodes will not exceed 3.8opt.Theorem 4: Our distributed algorithm has an approximationfactor of not exceeding 7.6.Proof: Our distributed algorithm includes two phases. Phase 1is MIS construction, and phase 2 is CDS construction. FromLemma3, the performance ratio in the first phase is 3.8. FromLemma 4, the performance ratio is 3.8 in the second phase, so theresulting CDS will have size bounded by 7.6.In our algorithm, the residual energy and size of nodes are bothconsidered into CDS based backbone construction. Thereconstruction makes the balance of energy consumption innetworks as energy level changes. Our algorithm guarantees that theCDS nodes have good energy efficiency and extend the networklifetime.5. SIMULATIONSIn this section, we verify our ECDS by evaluate its performance onrandom networks in terms of CDS size and performance. And wemake comparison with Wu and Li’s algorithm in [6] (WLA) andWan et al’s level ranking based algorithm in [4] (WAA). Thesimulation network size is 100-300 numbers of nodes in incrementsof 50 nodes respectively, which are randomly placed in a 160X160square area to generate connected graphs. Radio transmission rangeis 30 to 50m. For the purpose to balance the choice of nodes indominating set between those with high degree and high remainingenergy, giving importance to both. We investigate the weightfunction as: w(u)= )(deg uree * energy(u). Each node isassigned initial energy level 1 J. Eelec is energy of actuation, sensingand signal emission/reception. Eamp is energy for communication.Eamp=ξfs, when distance d<dcrossover, and Eamp=ξmp when d≥dcrossover. The transceiver energy model: mimics a “sensor radio”with Eelec50nJ/bit, ξ fs 10pJ/bit/m2, ξ mp 0.0013pJ/bit/m4. Datafusion is omitted. The broadcast packet size is 25 bytes, data packetsize 100 bytes, and packet header size is 25 bytes. The data routingtakes flooding protocol. We take parameter all timeout in ouralgorithm as 100ms, and when minimal energy of backbone cutdown to 50% incurs a new backbone construction.The simulation makes average solutions over 50 iterations ofrandom generating scenes. Figure 3,4 shows the size of thedominating set with the increasing number of nodes in the networkfor a certain transmission radius. ECDS has a good performancewith smaller CDS size when comparing with WAA and WLA as thenetwork size increases. Figure 5 shows the average CDS residualenergy as the network size increases with r=50m for 300 workingperiods. ECDS achieves better energy efficiency with much higherresidual power comparing with WAA and WLA. And WLA has theworst energy efficiency for its big size of dominating set. Figure 6shows the network lifetime (the number of periods that the networksurvives until can’t construct a backbone for the network) as thenetwork size increases from 100 to 300 nodes when r=50m. ECDShas much better performance comparing with WAA and WLA. Itcan work with longest time until can’t construct a backbone anymore. Apparently, ECDS has better and more proper energyconsumption when compared with the other two algorithms.253035404550556065100 150 200 250 300SizeofCDSNetwork sizeWAAWLAECDSFigure 3. Size of CDS as network size increases (r=30m)801
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10152025303540100 150 200 250 300SizeofCDSNetwork sizeWAAWLAECDSFigure 4. Size of CDS as network size increases (r=50m)0.50.550.60.650.70.750.80.850.9100 150 200 250 300AveCDSenergy(J)Network sizeWAAWLAECDSFigure 5. Average residual energy as network size increases(r=50m)60070080090010001100120013001400100 150 200 250 300NetworklifetimeNetwork sizeWAAWLAECDSFigure 6. Network lifetime as network size increases(r=50m)6. CONCLUSTION AND FUTURE WORKIn this paper, a distributed energy efficient sub-optimal connecteddominating set algorithm to construct a sparse structure for wirelesssensor networks is presented. We assign a weight to nodes, which isa combination of its remaining battery power and node effectivedegree in the communication graph. The algorithm takesreconstruction mechanism to compute a new CDS when nodsresidual energy of network cut down to a certain level. Ourdesignation can quickly construct a small-scaled backbone size. Thetime complexity and message complexity of this algorithm are bothO(n). The performance ratio is 7.6. Moreover, the algorithm is fullydistributed, only uses simple local node behavior to achieve adesired global objective. The simulation results show that thealgorithm can efficiently prolong network lifetime and balance nodeenergy consumption with a smaller backbone size, comparing withexisting classic algorithms. The future work will focus onsimulations under various settings and exploit how to interleave theconstruction of virtual backbones and route discovery.7. REFERENCES[1] Pottie, G. J. and Kaiser, W .J. Wireless integrated networksensors. Communications of ACM, Vol. 43, No. 5, May 2000,51-58..[2] Akyildiz, IF, Su, W., Sankarasubramaniam, Y., and Cayirci, E.A survey on sensor networks. IEEE CommunicationsMagazine, vol. 40, no.8, 2002, 102-114.[3] Clark, B. N., Colbourn, C.J., and Johnson, D. S. Unit diskgraphs. Discrete Mathematics, Vol. 86, 1990, 165-177.[4] Wan, P. J., Alzoubi, K. and Frieder, O. Distributed wellconnected dominating set in wireless ad hoc networks. In Proc.IEEE INFOCOM,2002.[5] Alzoubi, K., Wan, P. J., Frieder, O. New distributed algorithmfor connected dominating set in wireless Ad Hoc networks. InProc. 35thHawaii Int’1 Conf, 2000,3881-3887.[6] Wu, J. and Li, H. On calculating connected dominating set forefficient routing in ad hoc wireless networks. In Proc. the 3rdACM Int’l workshop Disc. Algor. and Methods for MobileComputing and Commun., 1999,7-14.[7] Butenko, S., Cheng, X., Oliveira, C.A.S, and Pardalos, P. M. Anew heuristic for the minimum connected dominating setproblem on ad hoc wireless networks. Cooperative Controland Optimization, 2004,61-73.[8] Min, M., Huang, C. X., Huang, S. C.-H., Wu, W., Du, H. andJia, X. Improving construction for connected dominating setwith Steiner tree in wireless sensor networks. GlobalOptimization, 2004.[9] Wu, J., Dai, F., Gao, M., and Stojmenovic, I. On CalculatingPower-Aware Connected Dominating Sets for EfficientRouting in Ad Hoc Wireless Networks. In Proc. IEEE Int’lICPP, 2001,346-356.[10] Acharya, T. and Roy, R. Distributed Algorithm for PowerAware Minimum Connected Dominating set for Routing inWireless Ad Hoc Networks. In Proc. ICPPWorkshops, ,2005,387-394.[11] Wu, W., Du, H., Jia, X., Li, Y., Huang, C.-H., Du, D-Z.Minimum connected dominating sets and maximalindependent sets in unit disk graphs. technical report 04-047,Department of Computer Science and Engineering, Universityof Minnesota, 2004.802
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