AD hoc wireless networks consist of wireless nodes that

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1 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE Topology Control n Ad Hoc Wreless Networks Usng Cooperatve Communcaton Mhaela Carde, Member, IEEE, JeWu,Senor Member, IEEE, and Shuhu Yang Abstract In ths paper, we address the Topology control wth Cooperatve Communcaton (TCC) problem n ad hoc wreless networks. Cooperatve communcaton s a novel model ntroduced recently that allows combnng partal messages to decode a complete message. The objectve of the TCC problem s to obtan a strongly-connected topology wth mnmum total energy consumpton. We show that the TCC problem s NP-complete and desgn two dstrbuted and localzed algorthms to be used by the nodes to set up ther communcaton ranges. Both algorthms can be appled on top of any symmetrc, strongly-connected topology to reduce total power consumpton. The frst algorthm uses a dstrbuted decson process at each node that makes use of only 2-hop neghborhood nformaton. The second algorthm sets up the transmsson ranges of nodes teratvely, over a maxmum of sx steps, usng only 1-hop neghborhood nformaton. We analyze the performance of our approaches through extensve smulaton. Index Terms Ad hoc wreless networks, cooperatve communcaton, energy effcency, topology control. æ 1 INTRODUCTION AD hoc wreless networks consst of wreless nodes that can communcate wth each other n the absence of a fxed nfrastructure. Wreless nodes are battery powered and, therefore, have a lmted operatonal tme. Recently, the optmzaton of the energy utlzaton of wreless nodes has receved sgnfcant attenton [9]. Dfferent technques for power management have been proposed at all layers of the network protocol stack. Power savng technques can generally be classfed nto two categores: by schedulng the wreless nodes to alternate between the actve and sleep mode and by adjustng the transmsson range of wreless nodes. In ths paper, we deal wth the second method. To support peer-to-peer communcaton n ad hoc wreless networks, the network connectvty must be mantaned at any tme. Ths requres that, for each node, there must be a route to reach any other node n the network. Such a network s called strongly connected. In ths paper, we address the problem of assgnng a power level to every node such that the resultng topology s strongly connected and the total energy expendture for achevng the strong connectvty s mnmzed. In order to reduce the energy consumpton, we take advantage of a physcal layer desgn that allows combnng partal sgnals contanng the same nformaton to obtan the complete data. Cooperatve communcaton (CC) models have been ntroduced recently n [11], [15]. By an effectve use of the partal sgnals, a specfc topology can be mantaned wth less transmsson power. In ths paper, we frst present some theoretcal results by showng the NP-completeness of the TCC problem and. The authors are wth the Department of Computer Scence and Engneerng, Florda Atlantc Unversty, 777 Glades Road, Boca Raton, FL E-mal: {mhaela, je}@cse.fau.edu and syang1@fau.edu. Manuscrpt receved 14 Apr. 2005; revsed 23 Sept. 2005; accepted 8 Dec. 2005; publshed onlne 17 Apr For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tmc@computer.org, and reference IEEECS Log Number TMC some relevant bounds. We then propose two dstrbuted and localzed algorthms for the TCC problem that start from a connected topology assumed to be the output of a tradtonal (wthout usng CC) topology control algorthm. One algorthm uses 2-hop neghborhood nformaton where each node tres to reduce the overall energy consumpton wthn ts 2-hop neghborhood wthout losng connectvty under the CC model. The other one s based on a 1-hop neghborhood where each node, startng from a mnmum range, teratvely ncreases ts transmsson range untl all nodes n ts 1-hop neghborhood are connected under the CC model. The ntal strongly connected topology s obtaned as a result of applyng a tradtonal topology control algorthm, such as the dstrbuted MST (DMST) [5] that generates an MST-based topology and the localzed MST (LMST) [13] that generates a pseudo MST-based topology. The rest of ths paper s organzed as follows: In Secton 2, we overvew topology control protocols. Secton 3 descrbes the CC model and the correspondng network model. Also, we ntroduce the TCC problem, prove ts NP-completeness, and show the performance rato between TCC and topology control wthout CC. In Secton 4, we propose a dstrbuted and localzed algorthm that can be appled to any symmetrc, strongly connected topology to reduce the total power consumpton. We contnue wth an teratve approach for settng nodes transmsson ranges n Secton 5. Secton 6 presents the smulaton results for the proposed algorthms, and Secton 7 concludes ths paper. 2 RELATED WORK Topology control has been addressed prevously n lterature n varous settngs. In general, the energy metrc to be optmzed (mnmzed) s the total energy consumpton or the maxmum energy consumpton per node. Sometmes topology control s combned wth other objectves, such as to ncrease the throughput or to meet /06/$20.00 ß 2006 IEEE Publshed by the IEEE CS, CASS, ComSoc, IES, & SPS

2 2 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 some specfc QoS requrements. The strongly connected topology problem wth a mnmum total energy consumpton was frst defned and proved to be NP-complete n [3], where an approxmaton algorthm wth a performance rato of 2 for symmetrc lnks s gven. In general, topology control protocols can be classfed as: 1) centralzed and global versus dstrbuted and localzed and 2) determnstc versus probablstc. The localzed algorthm s a specal dstrbuted algorthm, where the state of a partcular node depends only on states of local neghborhood. That s, such an algorthm has no sequental propagaton of states. Comprehensve surveys of topology control can be found n [14] and [20]. Most protocols are determnstc. The work n [18] s concerned wth the problem of adjustng the node transmsson powers so that the resultant topology s connected or bconnected, whle mnmzng the maxmum power usage per node. Two optmal, centralzed algorthms, CONNECT and BICONN-AUGMENT, have been proposed for statc networks. They are greedy algorthms, smlar to Kruskal s mnmum cost spannng tree algorthm. For ad hoc wreless networks, two dstrbuted heurstcs have been proposed, LINT and LILT. However, they do not guarantee the network connectvty. Among dstrbuted and localzed protocols, L et al. [12] propose a cone-based algorthm for topology control. The goal s to mnmze total energy consumpton whle preservng connectvty. Each node wll transmt wth the mnmum power needed to reach some node n every cone wth degree. They show that a cone degree ¼ 5=6 wll suffce to acheve connectvty. Several optmzed solutons of the basc algorthm are also dscussed as well as a beaconng-based protocol for topology mantenance. L et al. [13] devse another dstrbuted and localzed algorthm (LMST) for topology control startng from a mnmum spannng tree. Each node bulds ts local MST ndependently based on the locaton nformaton of ts 1-hop neghbors and only keeps 1-hop nodes wthn ts local MST as neghbors n the fnal topology. The algorthm produces a connected topology wth a maxmum node degree of 6. An optonal phase s provded where the topology s transformed to one wth bdrectonal lnks. Among probablstc protocols, the work by Sant et al. [19] assumes all nodes operate wth the same transmsson range. The goal s to determne a unform mnmum transmsson range n order to acheve connectvty. They use a probablstc approach to characterze a transmsson range wth lower and upper bounds for the probablty of connectvty. Some varants of the topology control problem have been also proposed by optmzng other objectves. Hou and L n [6] present an analytc model to study the relatonshp between throughput and adjustable transmsson range. The work n [7] puts forward a dstrbuted and localzed algorthm to acheve a relable hgh throughput topology by adjustng node transmsson power. The ssue of mnmzng energy consumpton has not been addressed n these two papers. Ja et al. [8] are concerned wth determnng a network topology that can meet QoS requrements n terms of end-to-end delay and bandwdth. The optmzaton crteron s to mnmze the maxmum power consumpton per node. When the traffc s splttable, an optmal soluton s proposed usng lnear programmng. Our work dffers from these approaches by usng cooperatve communcaton [11], [15]. We explore ths model n mnmzng total power consumpton whle achevng a strongly connected topology. A prelmnary work on topology control wth htchhkng model s presented n [2]. In ths paper [2], we ntroduce the Topology Control wth Htchhkng (TCH) problem and desgn a dstrbuted and localzed algorthm (DTCH) that can be appled on top of any symmetrc, strongly connected topology to reduce total power consumpton. 3 MODEL AND PROBLEM DEFINITION In ths secton, we ntroduce the cooperatve communcaton model and the correspondng network model. Then, we defne the Topology control wth Cooperatve Communcaton (TCC) problem, show ts hardness, and show a performance rato between TCC and topology control wthout cooperatve communcaton. 3.1 Cooperatve Communcaton (CC) Model Recently, a new class of technques, called cooperatve communcaton (CC) (or cooperaton dversty), has been ntroduced [11], [15] to allow sngle antenna devces to take advantage of the benefts of MIMO systems. Transmttng ndependent copes of the sgnal from dfferent locatons results n havng the recever obtan ndependently faded versons of the sgnal, thus reducng the fadng effect through multpath propagaton. In ths communcaton model, each wreless node s assumed to transmt data and to act as a cooperatve agent, relayng data from other users. There are wreless network applcatons proposed n lterature that use the CC model, such as energy effcent broadcastng [1] and constructng a connected domnatng set [21]. CC technques are classfed [11] as amplfy-and-forward, decode-and-forward, and selecton relayng. In the amplfy-andforward verson, a node that receves a nose verson of the sgnal can amplfy and relay ths nosy verson. The recever then combnes the nformaton sent by the sender and relay nodes. In decode-and-forward methods, a relay node must frst decode the sgnal and then retransmt the detected data. Sometmes the detecton of a relay node s unsuccessful and cooperatve communcaton can detrment the data recepton at the recever. One method s to have a node decde f t relays ts partner s data based on the sgnal-tonose rato (SNR) of the receved sgnal. In selecton relayng, a node chooses the strategy wth the best performance. The model consdered n ths paper belongs to the decodeand-forward category, where a node makes the relayng decson based on the SNR of the sgnal receved. Such a model requres each node to have a memory that can store several packet amounts of data and a sgnal processor that can estmate the SNR of each receved packet. Ths model, also referred to n lterature as the htchhkng model n [1], [21], takes advantage of the physcal layer desgn that combnes partal sgnals contanng the same nformaton to obtan complete nformaton. By effectvely usng partal sgnals, a packet can be delvered wth less transmsson power. The concept of combnng partal sgnals usng a maxmal rato combner [16] has been tradtonally used n

3 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 3 the physcal layer desgn of wreless systems to ncrease relablty. Smlarly to the model n [1], we consder that messages are packetzed. A packet contans a preamble, a header, and a payload. A preamble s a sequence of predefned uncoded symbols assgned to facltate tmng acquston, a header contans the error-control coded nformaton sequence about the source/destnaton address and other control flags, and a payload contans the error-control coded message sequence. We assume that the header and the payload of a packet are the outputs of two dfferent channel encoders and that the two channel codes are used by all the nodes n the system. The separaton of a header and a payload n channel codng enables a recever to retreve the nformaton n a header wthout decodng the entre packet. The use of the same channel codes enables a recever to enhance the SNR at the nput to the channel decoder by combnng the payloads of multple packets contanng the same encrypted message. We consder two parameters [1] related wth SNR: p, whch s the threshold needed to successfully decode the packet payload, and acq, whch s the threshold requred for a successful tme acquston. The system s characterzed by acq < p. We note wth k the rato of these two thresholds, k ¼ acq = p. We assume that the threshold to successfully decode a header s less than or equal to the threshold to successful tme acquston acq. A packet receved wth a SNR s: 1) fully receved, f p, 2) partally receved, f acq < p, and 3) unsuccessfully receved, f < acq. Therefore, when a packet s fully or partally receved ( acq ), the header nformaton s successfully decoded. Consder that, when a wreless node transmts a packet, the coverage of a node j that receves the packet wth a SNR per symbol s defned as: c j ¼ 1 for >1, c j ¼ for k< 1, and c j ¼ 0 for 0 < k, where ¼ = p. A channel gan s often modeled as a power of the dstance, resultng n ¼ r =d j ¼ðr=d jþ, where s a communcaton medum dependent parameter, r s the communcaton range of node, and d j s the Eucldean dstance between the nodes and j. For example, consder k ¼ 0:125 and ¼ 2. Let us assume node transmts a packet. For a node j wth r=d j ¼ 1=2, the coverage s 0:25, whereas for the case r=d j ¼ 1=3, the coverage s 0. The basc dea n the CC model s that, f the same packet s partally receved n tmes from dfferent neghbors wth acq < p for ¼ 1::n such that P n ¼1 p, then the packet can be combned by a maxmal rato combner [16] and can be successfully decoded. 3.2 Network Model We consder an ad hoc wreless network wth n nodes equpped wth omndrectonal antennas. The nodes n the network are capable of recevng and combnng partal receved packets n accordance wth the CC model ntroduced n Secton 3.1. We represent the network by a drected graph G ¼ðV;EÞ, where the vertces set V s the set of nodes correspondng to the wreless devces n the network and the set of edges E corresponds to the communcaton lnks between devces. Between any two nodes and j, there wll be an edge j f the transmsson from node s receved by the node j wth a SNR greater than acq. Every node 2 V has an assocated transmsson power level p ¼ r. For each edge j 2 E, the coverage provded by node to node j s defned as c j ¼ 1 for p =d j p and Fg. 1. A cooperatve communcaton example. (a) Intal power consumpton based on MST. (b) Power consumpton wth A as the source. (c) B s the source. (d) C s the source. c j ¼ p =ðd j pþ for acq p =d j < p. The case p =d j < acq s not ncluded snce an edge wll exst only when the SNR of the receved sgnal s at least acq, that s, p =d j acq. In ths paper, we consder the cases when equals 2 and 4 and p ¼ Topology Control wth Cooperatve Communcaton (TCC) In ths secton, we ntroduce the Topology control wth Cooperatve Communcaton (TCC) problem. The fully receved packet s defned as follows: Consderng a transmsson from a node to a node j, node j s partally (fully) covered by f 1 >c j acq (c j ¼ 1). If, upon combnng the packets receved from one or more neghbors, say k neghbors, results n a full coverage of node j,.e., k p k =d kj 1, then the packet s fully receved. We defne strong connectvty under the CC model as follows: For any node s sendng a packet, there should be a path to every other node, that s, the packet should be fully receved by all other nodes n the network. The followng rules apply: 1) s has the full packet and 2) only nodes that fully receved the packet are able to forward t, ncludng s. Each node that has fully receved a packet wll forward t only once. Now, we can formally defne the TCC problem as follows: TCC Defnton. Gven an ad hoc wreless network wth n nodes and usng the CC model, assgn a power level to every node such that: 1) the sum of the power levels n all nodes s mnmzed P n ¼1 p ¼ MIN and 2) the resultant CC-based topology s strongly connected. Fg. 1 presents a smple example of strong connectvty usng the CC model, where acq ¼ 0:2. We assume that the power requred to communcate between two nodes

4 4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 to be the square of the dstance between them. The number on each edge represents the coverage provded by the source node to the destnaton node. In Fg. 1a, a mnmum spannng tree (MST) s formed among the three nodes, where each bdrectonal lnk corresponds to two undrectonal lnks. Each node sets ts power to reach ts furthest neghbor on the MST. For example, node B must set ts power to 4 2 þ 6 2 ¼ 52 to reach node C. The topology s strongly connected f, havng any node as the source of a message, all the other nodes can get ths message drectly or by forwardng. In a model wth CC as n Fgs. 1b, 1c, and 1d, communcaton power of a node can be reduced to partally cover some neghbors as long as several partal messages can be combned for a successful message recept at those nodes. Fgs. 1b, 1c, and 1d show that, startng from each node, all other nodes are fully covered, thus the resultng topology s strongly connected. For example, n Fg. 1b, node A has a power of 18 to fully cover B (3 2 þ 3 2 ¼ 18) and to 31 percent cover C (18=ð7 2 þ 3 2 Þ¼31%). Snce B has receved the complete message, t can forward the message to C, provdng 69 percent coverage wth the power level set to 52 6% ¼ 35:86. Thus, C gets the complete message. Usng the same dea, the two other nodes are fully covered f we select node B or C as the source node. Therefore, the graph s strongly connected usng CC. 3.4 NP-Completeness of the TCC Problem Krouss et al. [10] gave a formal proof of NP-completeness for the general graph verson of the topology control (GTC) problem, wthout usng CC. In order to prove that TCC s NP-complete, we show that TCC belongs to the NP-class and GTC s a specal case of TCC. Theorem 1. The TCC problem s NP-complete. Proof. It s easy to see that TCC belongs to the NP-class. Havng assgned a transmsson power for each node n the network, t can be verfed n polynomal tme whether the resultant topology s strongly connected usng CC and whether the cost of ths assgnment (sum of the powers of each node) s less than a fxed value. Next, we show that GTC s a specal case of TCC. When acq ¼ p, we have no case of partal recepton of sgnals. Thus, the TCC problem reduces to the GTC problem, where a sgnal s ether fully receved or the recepton fals. Hence, the GTC problem s a specal case of the TCC problem for acq ¼ p. Because GTC s NP-complete and s a partcular case of the TCC problem and because TCC belongs to the NP-class, we conclude that TCC s an NP-complete problem. tu 3.5 Performance Rato between GTC and TCC Problems In ths secton, we prove that the optmal soluton of the GTC problem has a performance rato of 1=k wth the optmal soluton of the TCC problem, where k s defned n Secton 3.1. Theorem 2. The performance rato between the optmal soluton of the GTC problem and the optmal soluton of the TCC problem s upper bounded by 1=k. Proof. Let us note the optmal soluton of the GTC problem wth OPT GT C and the optmal soluton of the TCC problem wth OPT TCC. It s clear that OPT TCC OP T GT C snce the soluton set of the TCC problem ncludes that of the GTC problem. Next, we show that OP T GT C 1 k OPTTCC. Let us assume there are n nodes n the network, noted wth 1; 2;...;n. Let us note wth r 1 ;r 2 ;...;r n the node transmsson ranges assocated wth OP T TCC. Then, OP T TCC ¼ r 1 þ r 2 þ...þ r n. For a node, we note wth N TCC the set of nodes partally or totally covered by. Then, 8j 2 N TCC, ð r d j Þ k, where d j s the dstance between nodes and j. Let us consder now the case when each transmsson range s ncreased k 1 tmes. Ths corresponds to a soluton SOL wth node transmsson ranges r 0 1 ;r0 2 ;...;r0 n : SOL ¼ 1 k OPTTCC ¼ðr 1 k 1 Þ þ...þðr n k 1 Þ ¼ r 0 1 þ r 0 2 þ...þ r 0 n : For any node ¼ 1::n and for any node j 2 N TCC,we have ð r0 d j Þ ¼ð rk 1 d j Þ ¼ 1 k ðr d j Þ 1. Therefore, all nodes that were prevously partally covered n the TCC soluton are now fully covered and the strong connectvty s preserved. Therefore, SOL s also a soluton of the GTC problem, wth OPT GTC SOL. Ths results n OPT GT C 1 k OP T TCC. To summarze, we have proved that OP T TCC OPT GT C 1 k OP T TCC ; therefore, OPTGTC OPT TCC 1=k. 4 DISTRIBUTED TOPOLOGY CONTROL USING THE COOPERATIVE COMMUNICATION (DTCC) ALGORITHM In ths secton, we propose the dstrbuted topology control usng the cooperatve communcaton (DTCC) algorthm that can be appled to any symmetrc, strongly connected topology to reduce the total power consumpton. Any node decdes ts fnal power based only on local nformaton from ts 2-hop neghborhood. To be dstrbuted and localzed are mportant characterstcs of an algorthm n ad hoc wreless networks, snce t adapts better to a dynamc and scalable archtecture. 4.1 Basc Ideas In descrbng the algorthm, we use the notatons n Table 1. Each node ndependently locks ts 1-hop neghborhood to perform power adjustment to save energy. We take node as the current node for the example n Fg. 2. All the nodes on the nner dashed crcle ncludng j are s 1-hop neghbors. The nodes on the outer dashed crcle, such as k tu

5 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 5 TABLE 1 DTCC Notatons Fg. 2. Illustraton of 2-hop neghbor set of. and l, are s 2-hop neghbors. The man dea of DTCC s to ncrease s power level to contrbute to the coverage of ts 2-hop neghbors so the range of s 1-hop neghbors can be reduced, and, at the same tme, the overall power consumpton can be reduced. To ensure connectvty, 1-hop neghbors should stll be able to reach drectly. Such a process s the 2-hop power reducton process. In fact, n the 2-hop power reducton process, and ts 1-hop neghbors are nvolved n an atomc acton. To mplement such an atomc acton, two approaches can be used: 1. Back-off scheme. After node has selected a new power level, t backs off a perod of tme nversely proportonal to ts calculated gan. The gan of node represents the maxmum decrease n the total power obtaned by adjustng the power of node to one of the predefned values n P ðþ. Ths wll gve prorty to the nodes wth hgher gan to set up ther fnal power frst. If node receves an update durng ths nterval, then t recomputes ts power level and back-off agan. If the tmer expres wthout any updates, then node consders ths power level as ts fnal power and announces ths power level together wth ts neghbors new power levels to the nodes wthn ts 2-hop neghborhood. 2. Lockng scheme. Node needs to securely lock all of ts neghbors (n addton to ts own lock). Once completes ts power reducton process, t releases ts lock and the locks of ts neghbors and announces the power levels of tself and ts neghbors to the nodes wthn ts 2-hop neghborhood. Unlke the back-off scheme that may exhbt occasonal mscoordnaton, the lockng scheme guarantees that nodes execute the 2-hop power reducton process wthout conflct. However, t s more expensve. 4.2 Detaled DTCC Algorthm The DTCC algorthm starts from a symmetrc (bdrectonal lnks), connected topology G, assumed to be the output of a tradtonal topology control algorthm. Two such algorthms, DMST and LMST, are addressed later n ths secton. Intally, each node sets ts power p to the value p 0 needed to reach ts furthest 1-hop neghbor n G. We assume that each node has all the dstance nformaton wthn ts 2-hop neghborhood and the p j values of all 1-hop neghbors. Note that ths knd of nformaton s usually avalable after the tradtonal topology control algorthm completes. Node mantans p j values for all ts 1-hop neghbors. Whenever p j for a node j changes, node j broadcasts ths change to ts neghbors. The goal of the DTCC algorthm, by startng from an ntal power p 0, s to decde the fnal power assgnment by usng the CC model such as to mnmze the total power. Next, we descrbe the mechansm used by each node n order to decde ts fnal power level. The gan of node s computed n ComputeGanðÞ. The gan g ðpþ s defned as the maxmum decrease n the total power, obtaned by ncreasng node s transmsson power level to p 2 P ðþ, n exchange for a decrease of the power levels of some of the node s neghbors. Ths s because, when the power level of node s ncreased, provdes partal or full coverage to more nodes n the network. For example, f k s a 1-hop neghbor of node j, where j 2 NðÞ (see Fg. 2), then an ncrease n the partal or full coverage of node k may facltate reducton of the power level of node j that can provde less coverage to node k. Each node mantans a varable f ntally set to 0, meanng that ths node has not yet decded ts fnal power level. In order to decde ts fnal power, node computes the gan for varous power levels and selects the power level for whch the gan s maxmum. The power levels n P ðþ are those power levels for whch node could reduce the power level of a neghbor j to d j by provdng the addtonal coverage needed for a full coverage of all the neghbors of j. The process of computng the gan s performed for each power level p 2 P ðþ. Once the gans for all power levels n PðÞ are determned, the node selects the power level that produces a maxmum gan, noted wth p new. If there s no power level p such that g ðpþ > 0, then p wll not change. When node announces ts new power level through BroadcastðÞ, all ts neghbors j wth f j 6¼ 1 wll nvoke ReduceðÞ to decrease ther power levels and broadcast the change as a result of the addtonal coverage provded by node. The pseudocode presented next uses a back-off scheme (see Secton 4.1) n order to mplement the 2-hop power reducton process as an atomc acton. Each node backs-off a tme nversely proportonal to ts calculated gan before decdng ts fnal power. If, durng the back-off nterval, node receves a broadcast from a neghbor j, then node frst updates ts power p and then contnues the back-off scheme. Algorthm DTCC() 1: p p 0 2: f 0 3: whle f ¼ 0 do

6 6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE : compute P ðþ 5: ComputeGan() 6: p new power level for whch gan s maxmum 1 7: start a tmer t g ðp new Þ 8: f broadcast message receved from j before t expres then 9: p Reduceðj; p j ;Þ 10: else 11: p p new 12: f 1 13: end f 14: Broadcastð; p ;f Þ 15: end whle ComputeGan () 1: /*Fnd gan for all power levels n PðÞ*/ 2: for all p 2 P ðþ do 3: for all j 2 NðÞ do 4: p red j Reduce ð; p; jþ 5: end for 6: g ðpþ 7: end for Pj2NðÞ ðp j p red j Þ ðp p Þ Reduce (; p; j) 1: /*Reduce the power of node j on the bass of partal coverage provded by node wth power p*/ 2: f f j ¼ 1 then 3: return p j 4: end f 5: for all k 2 NðjÞ do 6: p j ðkþ ð1 c k Þd jk 7: end for 8: return maxfd j ; max k2nðjþ p j ðkþg 4.3 Propertes The complexty of the DTCC algorthm run by each node s polynomal n the total number of nodes n. The complexty of the ComputeGan() procedure takes OðjP ðþj jj 2 Þ tme, where s the maxmal node degree. Ths s because, for each neghbor j 2 NðÞ, the s coverage on each 2-hop neghbor k 2 NðjÞ needs to be computed. Ths process has to be done for each power level n P ðþ. When jpðþj ¼ OðÞ,t s Oð 3 Þ. Therefore, the complexty of the algorthm DTCC run on each node s Oð 4 Þ wth another loop. Next, we show the correctness of the DTCC algorthm: Theorem 3. The power level assgnment provded by the DTCC algorthm guarantees a strongly connected topology wth the CC model. Proof. Intally, each node s assgned the power level needed to reach the furthest 1-hop neghbor n G. The startng topology G s strongly connected, that s, between any two nodes, there exsts a path. We note that there are two cases when a node s power level may change n the DTCC algorthm: 1) n lne 11, but here the value s ncreased, so ths wll not affect connectvty, and 2) n lne 6 of the procedure Reduce(), when a node s power level may be reduced. Let us assume by contradcton that, after applyng the DTCC algorthm, the strong connectvty s not preserved. Then, there exst two nodes and j such that when the node s sendng a packet, ths packet s not fully receved by j. The nodes and j are connected n G, so there exsts a path 0 ¼, 1 ;...; m ¼ j between and j. We show by nducton that m fully receves the packet sent by 0. Frst, 0 has the full packet. If 0 dd not change ts power or has ncreased the power level, then 1 s fully covered by 0 and, therefore, receves the full packet from 0. Let us consder the case when 0 has reduced ts power level. Then, n conformty wth DTCC, the current power of 0 was updated when one of ts neghbors, say k, has set up ts fnal power. In that case, 0 fully covers k and 0 together wth k fully cover all 0 s neghbors, ncludng 1. So, 1 also fully receves the packet. Applyng the same mechansm, we can show that any node on the path fully receves the packet sent by ts predecessor, even f t s not fully covered by ts predecessor. Thus, node m fully receves the packet, contradctng our ntal assumpton that strong connectvty s not mantaned after runnng DTCC. tu 4.4 Two Specal Cases We have appled the DTCC algorthm on two startng topologes output by two dstrbuted algorthms: DMST (Dstrbuted MST) and LMST (Localzed MST). We note wth DMST the Gallegar s dstrbuted algorthm [5] for constructng an MST and, wth DMST-based DTCC, the DTCC algorthm that starts from a topology G generated by DMST. Also, we note wth LMST the algorthm proposed by L et al. [13] for constructng a pesudo MST and, wth LMST-based DTCC, the DTCC algorthm that starts from a topology G generated by LMST. MST has been consdered before as a reference pont n desgnng topology control mechansms n the general model (wthout CC) because of ts mportant propertes and good performance. MST has the mnmum longest edge among all the spannng trees [4], therefore, f every node has assgned a power level needed to reach the furthest neghbor, then the maxmum power assgned per node s mnmzed for the MST compared wth other spannng trees. Ths property results n maxmzng the tme untl the frst node wll deplete ts power resources. Another property of the MST-based topology n the general case (wthout CC) s that t provdes an approxmaton algorthm wth a performance rato of 2 [10]. Next, we prove that an MST-based topology has a performance rato of 2=k for the TCC problem. An MSTbased topology s a mechansm that bulds an MST over all n nodes n the network and then assgns to any node the power needed to reach the furthest neghbor n the MST. Theorem 4. An MST-based topology s an approxmaton algorthm wth rato bound of 2=k for the TCC problem, where k ¼ acq = p s a constant k 2ð0; 1Š and represents a characterstc of the wreless communcaton medum. Proof. Let us note the optmal soluton of the GTC problem wth OP T GTC, the optmal soluton of the TCC problem wth OP T TCC, and the MST-based soluton wth MST. It s proved n [10] that an MST-based topology has a performance rato of 2 for the GTC problem, therefore,

7 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 7 TABLE 2 ITCC Notatons Fg. 3. Example of DTCC ( acq ¼ 0:01, ¼ 2). (a) DMST and power consumpton. (b) LMST and power consumpton. (c) DMST-based DTCC. (d) LMST-based DTCC. MST 2 OP T GTC. In Theorem 2, we proved that OP T GTC 1 k OP T TCC, therefore, MST 2 k OPTTCC. Snce OPT TCC MST, we obtan that OP T TCC MST 2 k OPTTCC and, thus, the theorem holds. tu Snce DMST-based DTCC starts from an MST-based topology and mproves t, usng the CC advantage, DMSTbased DTCC wll also have a performance rato of 2=k for the TCC problem. As DTCC and LMST are localzed, the resultant LMSTbased DTCC s localzed. However, LMST-based DTCC does not guarantee a performance rato snce LMST s not strctly MST-based topology. We present the smulaton results for LMST-based DTCC n Secton 6. Note that, f the DTCC s appled on LMST, the complexty s Oð1Þ. Ths s because, n LMST, the degree of any node n the resultng topology s bounded by 6 [13]. Therefore, the power level of node, jpðþj, s constant n DTCC. The complexty of DTCC n the general case s OðjPðÞj jnðþj 2 Þ, whch s Oð1Þ here. Fg. 3 shows an example of a sx nodes topology. The number on each node ndcates the power level used by that node n mantanng the topology based on 1) DMST and 2) LMST. We use undrectonal lnks to represent full coverage n both drectons, whereas drectonal lnks wth values less than 1 ndcate the amount of partal coverage. In Fg. 3a, we present a DMST-based topology wthout CC. The power level assgned to each node s the power needed to reach the furthest neghbor n DMST. The total cost s 186. In Fg. 3b, we show the topology obtaned after usng the LMST algorthm [13], wth a total cost of 287. LMST uses a localzed way to generate the MST where every node decdes ts 1-hop neghbors ndependently. Therefore, n a global vew, the resultng topology mght be a graph wth cycles. Fg. 3c shows the topology and power assgnment after runnng the DMST-based DTCC algorthm. We assume acq ¼ 0:01 and ¼ 2. Frst, each node computes ts gan. As node F has the largest gan, t ncreases ts power to 34:56, and, thus, nodes A and C decrease ther power to 1 and 34:23, respectvely. In the second round, node B sets ts power to 4 and node E decreases ts power to 61:94. We obtan a total cost of 160:73 and a 13:59 percent power reducton compared wth the output of the DMST algorthm n Fg. 3a. Strong connectvty s also preserved. For example, node A reduces ts power to 1, whch partally covers ts neghbor D wth 0:04, whle node T provdes the addtonal 0:96 coverage. Thus, a message sent from A s fully receved by F, and then A and F can together cover D. Fg. 3d shows the executon of the LMST-based DTCC algorthm wth a total cost of 206:1 and a reducton Fg. 4. Example of ITCC ( acq ¼ 0:01, ¼ 2). (a) DMST and power consumpton. (b) LMST and power consumpton. (c) DMST-based ITCC. (d) LMST-based ITCC.

8 8 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 Fg. 5. Power consumpton of DTCC wth DMST and LMST ( acq 2f0:0001; 0:1; 0:2g). (a) DMST and DTCC when ¼ 2. (b) LMST and DTCC when ¼ 2. (c) DMST and DTCC when ¼ 4. (d) LMST and DTCC when ¼ 4. rato of 28:19 percent compared wth LMST algorthm n Fg. 3b. 5 INCREMENTAL TOPOLOGY CONTROL USING COOPERATIVE COMMUNICATION (ITCC) ALGORITHM In ths secton, we propose a dstrbuted and localzed algorthm that uses a dfferent approach to set up nodes transmsson power. The Incremental Topology control usng Cooperatve Communcaton (ITCC) algorthm s based on 1-hop neghborhood nformaton. Each node, startng from a mnmum power, teratvely ncreases ts transmsson power untl all the nodes n ts 1-hop neghborhood are fully covered under the CC model. 5.1 Basc Ideas The man algorthm notatons are ntroduced n the Table 2. ITCC algorthm starts from a symmetrc, connected topology G, assumed to be the output of a tradtonal topology control algorthm such as DMST and LMST. Each node computes p max and p mn, the transmsson powers needed to reach the furthest and the closest neghbor n NðÞ, correspondng to G. The fnal power selected by node s a value between p mn and p max. The goal of ths algorthm s to fnd a mnmum transmsson power for node n ;p max Š, such that all the nodes n NðÞ are fully covered ½p mn by node usng CC. In the CC model, f a node v fully receves a message transmtted by a node u (drectly or usng CC), then v wll resend the message once usng ts current power level. The ITCC algorthm adopts an teratve process where each node gradually ncreases ts power (ntally, p mn ). To avod smultaneous updates among neghbors, ether a back-off or a lockng scheme can be used (see Secton 4.1). 5.2 Detaled ITCC Algorthm We assume that each node has the dstance and locaton nformaton for ts 1-hop neghborhood NðÞ, nformaton usually avalable after runnng the tradtonal topology control algorthm. Each node mantans ts current power estmate, p and the p j value for each node j 2 NðÞ. When a node decdes ts fnal power value, t sets f to 1. The goal of the ITCC algorthm s, by startng from an ntal power p mn needed to reach the closest 1-hop neghbor for each node, to teratvely ncrement the power untl all nodes n NðÞ are fully covered usng the CC model. When ths condton s met, node declares ts current power estmate as ts fnal power assgnment. Next, we descrbe the mechansm used by each node to decde ts fnal power level. Each node mantans a varable f whch s ntally set to 0, meanng that ths node has not yet decded ts fnal power level. The algorthm executes n at most jnðþj rounds

9 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 9 Fg. 6. Reduced rato of DTCC wth DMST and LMST ( acq 2f0:0001; 0:1; 0:2g). (a) DMST-based DTCC when ¼ 2. (b) LMST-based DTCC when ¼ 2. (c) DMST-based DTCC when ¼ 4. (d) LMST-based DTCC when ¼ 4. (or teratons). In each round, power level p s mnmally ncremented wth p such that at least one node n NðÞ N 0 ðþ s added to N 0 ðþ. p can easly be computed snce node mantans the dstance and locaton nformaton for all nodes n NðÞ. The algorthm fnshes when NðÞ ¼N 0 ðþ, that s, usng the current power estmate p, node covers all nodes n NðÞ usng the CC model. All broadcast messages sent to advertse new power level updates are sent wth power level p max. If, durng the back-off nterval, a broadcast message s receved from a neghbor n NðÞ, then N 0 ðþ and p are updated before contnung the back-off watng. It mght happen that the value p decreases, but ths s safe snce node dd not advertse the new power level yet. When the tme comes for node to broadcast ts advertsement, t updates ts power level p p þ p and the reachable neghborhood set N 0 ðþ. IfNðÞ ¼N 0 ðþ, then the current power level s the fnal power level of node. The rounds should be desgned to have each node advertse ts new power estmate once. Ideally, the nodes wll send the broadcast wthout colldng wth ther neghbors advertsng. To avod collsons, we could use a 1-hop neghborhood lockng scheme or a back-off mechansm (see Secton 4.1). The pseudocode presented next uses a back-off scheme, where each node backs-off a tme nversely proportonal to ts calculated gan before sendng a broadcast. The gan can be computed, for example, as p max ðp þ p Þ. In ths case, nodes wth a smaller power level wll advertse earler, thus helpng the nodes wth a hgher transmsson power through CC. Ths scheme could help to balance power consumpton. If, durng the back-off tme nterval, node receves an advertsement from a neghbor j 2 NðÞ, then node does frst the update and then contnues the back-off scheme. Algorthm ITCC() 1: p p mn 2: f 0 3: Broadcast(; p ;f ) 4: whle f ¼ 0 do 5: compute p, the mnmum ncremental power needed to cover at least one neghbor n NðÞ N 0 ðþ 6: start tmer t 7: f broadcast message receved from j before t expres then 8: update N 0 ðþ, p 9: f NðÞ ¼N 0 ðþ then 10: f 1 11: Broadcast(; p ;f ) 12: return 13: end f

10 10 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 Fg. 7. Power consumpton of ITCC wth DMST and LMST ( acq 2f0:0001; 0:1; 0:2g). (a) DMST and ITCC when ¼ 2. (b) LMST and ITCC when ¼ 2. (c) DMST and ITCC when ¼ 4. (d) LMST and ITCC when ¼ 4. 14: end f 15: f tmer t expres then 16: p p þ p 17: update N 0 ðþ 18: f NðÞ ¼N 0 ðþ then 19: f 1 20: end f 21: Broadcast(; p ;f ) 22: end f 23: end whle 5.3 Propertes The complexty of the DTCC algorthm run by each node s polynomal n the total number of nodes n. Let us note the maxmal node degree n the graph G, that s, ¼ max ¼1...n jn j. The complexty of DTCC s Oð 4 Þ. Ths s because, for a node, there are at most rounds, the tme to update p s at most 2, and durng the back-off at most neghbor updates can be receved. When a node fnshes executng ITCC algorthm, t decdes ts fnal transmsson range p. Usng ths transmsson range, the algorthm assures that node fully covers all the nodes n NðÞ usng the CC model. The coverage relatonshp s transtve. For any three nodes p, q, and r, fp fully covers q and q fully covers r, then p fully covers r as well. Next, we show the correctness of the ITCC algorthm. Theorem 5. The power level assgnment provded by the ITCC algorthm guarantees a strongly connected topology wth the CC model. Proof. Let us assume by contradcton that the resultng topology s not strongly connected, that s, there exst two nodes and j such that a message sent by node s not fully receved by the node j, usng CC. Note that G s strongly connected; that means there s a path n G from to j, 0 ¼ ; 1 ; 2 ;...; m ¼ j, such that kþ1 2 NðkÞ for any k ¼ 0...m 1. When algorthm ITCC ends, each node fully covers all nodes n NðÞ usng the CC model. Therefore, each node k on the path fully covers the successor node kþ1, for k ¼ 0...m 1. Snce the coverage relatonshp s transtve, t follows that ¼ 0 fully covers j ¼ m usng the CC model. Thus, our assumpton s false and the topology resulted after applyng ITCC algorthm s strongly connected. tu The ITCC algorthm dffers from the DTCC algorthm (see Secton 4) n the followng aspects:. DTCC uses 2-hop neghborhood nformaton, whle ITCC uses 1-hop neghborhood nformaton.. DTCC starts from the power needed to reach the furthest 1-hop neghbor and ncreases ths value n order to reduce the power needed by ts chldren.

11 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 11 Fg. 8. Reduced rato of ITCC wth DMST and LMST ( acq 2f0:0001; 0:1; 0:2g). (a) DMST-based ITCC when ¼ 2. (b) LMST-based ITCC when ¼ 2. (c) DMST-based ITCC when ¼ 4. (d) LMST-based ITCC when ¼ 4. ITCC starts from the power needed to reach the closest 1-hop neghbor and ncreases ths value ncrementally untl ts 1-hop neghborhood s fully covered.. DTCC s executed n one round, whle ITCC executes over at most rounds. 5.4 Two Specal Cases We have appled the ITCC algorthm to two startng topologes, DMST (Dstrbuted MST) and LMST (Localzed MST). Frst, we apply the ITCC algorthm to the topology G generated by DMST and note ths algorthm wth DMSTbased ITCC. Snce DMST-based ITCC starts from a MSTbased topology and mproves t, usng the CC model, DMST-based ITCC has a performance rato of 2=k for the TCC problem (see Theorem 4 n Secton 4.4). Then, we apply the ITCC algorthm to the topology G generated by LMST and name ths algorthm LMST-based ITCC. LMST-based ITCC s a dstrbuted and localzed algorthm snce both LMST and ITCC are dstrbuted and localzed. Another mportant observaton s that the degree of any node n the resultng topology G s bounded by 6 [13]. Therefore, each node has jn j6 and, thus, 6. The complexty of the LMST-based ITCC s therefore Oð1Þ. We use the same example as n Fg. 3 to show how the ITCC algorthm works. Fg. 4a s the ntal power assgnment of DMST-based ITCC. The graph s dsconnected wth ths power assgnment (shown n sold lnes), snce each node can only reach ts closest neghbor. Each node then ncreases ts power untl every neghbor s covered. Fg. 4c s the result. For example, ntally, node F 100 percent covers ts neghbor A and 50 percent covers neghbor C. It then ncreases ts power to 1:6 to 80 percent cover C, because the fully covered neghbor A contrbutes an addtonal 20 percent coverage. The fnal total power obtaned s 180. Fg. 4b s the ntal power assgnment of LMST-based ITCC and Fg. 4d s the resultant power assgnment. The fnal total cost obtaned s 214:11. 6 SIMULATION RESULTS In ths secton, we evaluate the DMST-based DTCC, LMSTbased DTCC, DMST-based ITCC, and LMST-based ITCC algorthms for topologes up to 1,000 nodes. We set up our smulaton n a m 2 area. The nodes are randomly dstrbuted n the feld and reman statonary once deployed. We use both DMST and LMST algorthms n the smulaton to generate the startng topologes and to calculate the ntal power assgnment. Snce a localzed algorthm lacks global nformaton, the topology obtaned when runnng LMST wll be less effcent than DMST, that s, the power consumpton wth LMST wll be greater than

12 12 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 Fg. 9. Reduced rato comparson of DTCC and ITCC wth DMST and LMST ( acq 2f0:0001; 0:1; 0:2g). (a) DMST-based DTCC and ITCC when ¼ 2. (b) LMST-based DTCC and ITCC when ¼ 2. (c) DMST-based DTCC and ITCC when ¼ 4. (d) LMST-based DTCC and ITCC when ¼ 4. that usng DMST. In the smulaton, we consder the followng tunable parameters: 1. The node densty. We change the number of deployed nodes from 100 to 1,000 to check the effect of node densty on the performance. 2. The ndex exponent, whch shows the relaton between dstance and power consumpton. We use the values 2 and The parameter acq, whch depends on actual wreless communcaton. In the smulaton, we use the values 0:0001, 0:1, and 0:2. Fgs. 5a and 5b show power consumpton dependng on the number of nodes, when s 2. Fg. 5a llustrates DMST and DMST-based DTCC and Fg. 5b demonstrates LMST and LMST-based DTCC. We observe that the overall power consumpton can be greatly reduced by usng the DTCC algorthm. The smaller the acq, the better the performance. Power consumed by DMST s less than that consumed by LMST. The node densty does not have much effect on the power consumpton, especally when there are more than 200 nodes. Ths s because, when there are more nodes, the average dstance between nodes s smaller and so s the average communcaton power. Therefore, the overall power consumpton changes slghtly. Fgs. 5c and 5d show the power consumpton dependng on the number of nodes when s 4. We can see that the advantage n power effcency when usng DTCC stll holds. The dfference between power consumpton of these two algorthms s less dstnctve. Fg. 6 shows the reduced rato of the consumed power. Fg. 6a shows DMST-based DTCC for ¼ 2, and Fg. 6c when ¼ 4. Fg. 6b represents LMST-based DTCC for ¼ 2 and Fg. 6d when ¼ 4. We observe that LMST-based DTCC wth an of 2 acheves the hghest reducton n the power consumpton, whch can be up to 18.6 percent, whle DMST-based DTCC wth an of 4 has the least power reducton. Fgs. 7 and 8 are the smulaton results of ITCC. Fg. 7 shows the analyss of power consumpton of DMST-based ITCC, LMST-based ITCC, wth dfferent. We can see that ths fgure s qute the same wth Fg. 5, except that when s 2, the effect of parameter acq s more sgnfcant. Fg. 8 shows the reduced rato of power consumpton n ITCC wth dfferent acq. When s 2, the LMST-based ITCC can save more than 21.5 percent of ts energy. Fg. 9 compares the power reducton rato between DTCC and ITCC. When ¼ 2 and acq s relatvely small (say smaller than 0:1), ITCC outperforms DTCC. Otherwse, DTCC acheves more power reducton than ITCC. In general, DTCC acheves more energy savngs than ITCC snce n DTCC the nodes ncrease ther transmsson range only once wth a large ncrement and, therefore, the

13 CARDEI ET AL.: TOPOLOGY CONTROL IN AD HOC WIRELESS NETWORKS USING COOPERATIVE COMMUNICATION 13 TABLE 3 Reducton Rato of Maxmum Transmsson Power among All Nodes CC contrbuton on ther neghbors s hgher. But, the dfference between these two algorthms s slght. Maxmum energy consumpton among all the nodes s an mportant performance metrc. It shows whether the energy consumpton among all the nodes s balanced or not. Table 3 shows the reducton rato of ITCC and DTCC n maxmum transmsson power taken over all the nodes n the network. We can see that, the greater the parameter, the smaller the rato, and the smaller the acq, the greater the rato. The dfference between DTCC and ITCC s slght, but ITCC has a relatvely greater reducton. The maxmum energy n ITCC s always smaller or equal to the one n the orgnal DMST/LMST topology, whle the maxmum energy n DTCC can be greater than the orgnal one. Ths s because ITCC ncreases the node transmsson range gradually and the upper bound of ts power s to reach ts furthest neghbor. However, n DTCC, a node may ncrease ts power greatly f ths can lead to greater reducton of the power of ts neghbors. Thus, usng ITCC provdes a more balanced energy consumpton per node, resultng n a longer network lfetme. In general, LMST-based DTCC/ ITCC has greater reducton rato than DMST-based ones. Smulaton results can be summarzed as follows: 1. Usng the CC model, the proposed DTCC and ITCC algorthms reduce the nodes energy consumpton n topology control by 7 percent to 21 percent. The LMST-based DTCC or ITCC has greater energy reducton than DMST-based ones. 2. Wth ¼ 2, DTCC and ITCC acheve better performance than wth ¼ 4. The former s around 17 percent and the latter around 9 percent. 3. The energy reducton rato s not senstve to the parameter acq when acq s very small; there s no dfference between 0 and 0:0001 of acq s value. Wth ncreasng values of acq, the energy reducton rato wll reduce slghtly. 4. The energy savngs produced by DTCC and ITCC are comparable wth DTCC producng slghtly better results n general. But, ITCC has a smaller maxmum node power whch s good for balanced energy consumpton. 7 CONCLUSIONS In ths paper, we have addressed the NP-complete problem on Topology Control wth Cooperatve Communcaton (TCC) n ad hoc wreless networks, wth the objectve of mnmzng the total energy consumpton whle obtanng a strongly connected topology. Power control mpacts energy usage n wreless communcaton wth an effect on battery lfetme, whch s a lmted resource n many wreless applcatons. We have proposed two dstrbuted and localzed algorthms that can be appled to any symmetrc, strongly connected topology n order to reduce the total power consumpton. The frst one uses a dstrbuted decson process at each node that makes use of only 2-hop neghborhood nformaton. The second uses the cooperatve communcaton of nodes wthn a 1-hop neghborhood n order to set nodes transmsson ranges teratvely, n at most sx rounds. We have analyzed the performance of our algorthms through smulatons. Our future work s, by startng from the DTCC or ITCC algorthm, to desgn an effcent topology mantenance mechansm that effectvely adapts to a dynamc and moble wreless envronment. ACKNOWLEDGMENTS Ths work was supported n part by US Natonal Scence Foundaton grants CCR , ANI , EIA , CCF , and CNS REFERENCES [1] M. Agarwal, J.H. Cho, L. Gao, and J. Wu, Energy Effcent Broadcast n Wreless Ad Hoc Networks wth Htch-Hkng, Proc. IEEE INFOCOM, [2] M. Carde, J. Wu, and S. Yang, Topology Control n Ad Hoc Wreless Networks wth Htch-Hhkng, Proc. Frst IEEE Int l Conf. Sensor and Ad Hoc Comm. and Networks (SECON 04), Oct [3] W.-T. Chen and N.-F. Huang, The Strongly Connectng Problem on Multhop Packet Rado Networks, IEEE Trans. Comm., vol. 37, no. 3, pp , Mar [4] T.H. Cormen, C.E. Leserson, R.L. Rvest, and C. Sten, Introducton to Algorthms, second ed. McGraw-Hll, [5] R. Gallager, P. Humblet, and P. Spra, A Dstrbuted, Algorthm for Mnmum-Weght Spannng Tree, ACM Trans. Programmng Languages and Systems, [6] T.-C. Hou and V.O.K. L, Transmsson Range Control n Multhop Packet Rado Networks, IEEE Trans. Comm., vol. 34, no. 1, pp , Jan [7] L. Hu, Topology Control for Multhop Packet Rado Networks, IEEE Trans. Comm., vol. 41, no. 10, pp , [8] X. Ja, D. L, and D.-Z. Du, QoS Topology Control n Ad Hoc Wreless Networks, Proc. IEEE INFOCOM, [9] C.E. Jones, K.M. Svalngam, P. Agrawal, and J.C. Chen, A Survey of Energy Effcent Network Protocols for Wreless Networks, Wreless Networks, vol. 7, no. 4, pp , Aug [10] L.M. Krouss, E. Kranaks, D. Krzanc, and A. Pelc, Power Consumpton n Packet Rado Networks, Proc. 14th Ann. Symp. Theoretcal Aspects of Computer Scence (STACS 97), pp , [11] N. Laneman, D. Tse, and G. Wornell, Cooperatve Dversty n Wreless Networks: Effcent Protocols and Outage Behavor, IEEE Trans. Informaton Theory, [12] L. L, J.Y. Halpern, P. Bahl, Y.-M. Wang, and R. Wattenhofer, Analyss of a Cone-Based Dstrbuted Topology Control Algorthm for Wreless Mult-Hop Networks, Proc. ACM Symp. Prncples of Dstrbuted Computng, pp , Aug [13] N. L, J. Hou, and L. Sha, Desgn and Analyss of an MST-Based Topology Control Algorthm, Proc. IEEE Infocom, [14] X.-Y. L, Topology Control n Wreless Ad Hoc Networks, Ad Hoc Networkng, S. Basagn, M. Cont, S. Gordano, and I. Stojmenovc, eds. IEEE Press, [15] A. Nosratna, T.E. Hunter, and A. Hedayat, Cooperatve Communcaton n Wreless Networks, IEEE Comm. Magazne, vol. 42, no. 10, pp , Oct [16] J.G. Proaks, Dgtal Communcatons, fourth ed. McGraw Hll, 2001.

14 14 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 6, JUNE 2006 [17] T.S. Rappaport, Wreless Communcatons, second ed. Prentce-Hall, [18] R. Ramanathan and R. Rosales-Han, Topology Control of Multhop Wreless Networks Usng Transmt Power Adjustment, Proc. IEEE INFOCOM, pp , [19] P. Sant, D.M. Blough, and F. Vansten, A Probablstc, Analyss for the Range Assgnment Problem n Ad Hoc Networks, Proc. ACM Mobhoc, pp , Aug [20] C.-C. Shen and Z. Huang, Topology Control for Ad Hoc Networks: Present Solutons and Open Issues, Handbook of Theoretcal and Algorthmc Aspects of Sensor, Ad Hoc Wreless and Peer-to-Peer Networks, J. Wu, ed. CRC Press, [21] J. Wu, M. Carde, F. Da, and S. Yang, Extended Domnatng Set and Its Applcatons n Ad Hoc Networks Usng Cooperatve Communcaton, IEEE Trans. Parallel and Dstrbuted Systems, accepted for publcaton. Mhaela Carde receved the MS and PhD degrees n computer scence from the Unversty of Mnnesota, Twn Ctes, n 1999 and 2003, respectvely. She s currently an assstant professor n the Department of Computer Scence and Engneerng at Florda Atlantc Unversty. Her research nterests nclude wreless networkng, wreless sensor networks, network protocol and algorthm desgn, and combnatoral optmzaton n wreless networks. She s a member of the IEEE and the ACM. Je Wu s a professor n the Department of Computer Scence and Engneerng at Florda Atlantc Unversty. He has publshed more than 300 papers n varous journal and conference proceedngs. Hs research nterests are n the areas of moble computng, routng protocols, fault-tolerant computng, and nterconnecton networks. Dr. Wu served as a program vce char for the 2000 Internatonal Conference on Parallel Processng (ICPP) and as a program vce char for the 2001 IEEE Internatonal Conference on Dstrbuted Computng Systems (ICDCS). He s a program cochar for the IEEE Frst Internatonal Conference on Moble Ad-Hoc and Sensor Systems (MASS 04). He was a coguest edtor of a specal ssue of Computer on ad hoc networks. He also was an edtor for several specal ssues n the Journal of Parallel and Dstrbutng Computng (JPDC) and the IEEE Transactons on Parallel and Dstrbuted Systems (TPDS). He s the author of the text Dstrbuted System Desgn and s the edtor of the text Handbook on Theoretcal and Algorthmc Aspects of Sensor, Ad Hoc Wreless, and Peer-to-Peer Networks. Currently, Dr. Wu serves as an assocate edtor for the IEEE Transactons on Parallel and Dstrbuted Systems and several other nternatonal journals. He s a recpent of the and Researcher of the Year Award at Florda Atlantc Unversty. He served as an IEEE Computer Socety Dstngushed Vstor and s the charman of the IEEE Techncal Commttee on Dstrbuted Processng (TCDP). He s a member of the ACM and a senor member of the IEEE. Shuhu Yang receved the BS and MS degrees n computer scence n 2000 and 2003, respectvely, from Jangsu Unversty, Zhenjang, and Nanjng Unversty, Nanjng, Chna. She s a PhD canddate n the Department of Computer Scence and Engneerng at Florda Atlantc Unversty. Her current research focuses on the desgn of localzed routng algorthms n the wreless ad hoc and sensor networks.. For more nformaton on ths or any other computng topc, please vst our Dgtal Lbrary at