Localized Minimum Spanning Tree Based Multicast Routing with Energy-Efficient Guaranteed Delivery in Ad Hoc and Sensor Networks

Transcription

1 Localized Minimum Spanning Tree Baed Multicat Routing with Energy-Efficient Guaranteed Delivery in Ad Hoc and Senor Network Hanne Frey Univerity of Paderborn D-3398 Paderborn Françoi Ingelret LCAV, EPFL CH-115 Lauanne, Switzerland David Simplot-Ryl IRCICA/LIFL, Univ. Lille 1 CNRS UMR 822, INRIA Futur, France David.Simplot@lifl.fr Abtract We preent a localized geographic multicat cheme, MSTEAM, baed on the contruction of local minimum panning tree (MST), that require information only on 1- hop neighbor. A meage replication occur when the MST panning the current node and the et of detination ha multiple edge originated at the current node. Detination panned by thee edge are grouped together, and for each of thee ubet the bet neighbor i elected a the next hop. Thi election i baed on a cot over progre metric, where the progre i approximated by ubtracting the weight of the MST over a given neighbor and the ubet of detination to the weight of the MST over the current node and the ubet of detination. Since uch greedy cheme may lead the meage to a void area (i.e., no neighbor providing poitive progre), we propoe a new multicat generalization of the well-known face recovery mechanim. We provide a theoretical analyi proving that MSTEAM i loop-free, and achieve delivery of the multicat meage a long a a path to the detination exit. Our reult demontrate that MSTEAM, outperform the bet exiting localized multicat cheme, and i almot a efficient a a centralized cheme in high denitie. 1 Introduction We conider multi-hop ad hoc and enor network, and localized routing cheme, where node maintain knowledge only about patially nearby node. Thu, unlike in centralized cheme, change in the network require only local meage exchange. Becaue of mall batterie, routing mut be energy-efficient to maximize network lifetime. We epecially conider geographic routing, where each node know it own location, and where the next hop i choen baed on the poition of neighbor with repect to the detination. Almot all cheme are baed on greedy heuritic: the bet neighbor according to an evaluation function i elected. For intance, a olution i to chooe the cloet neighbor to the detination. Obviouly, the meage may be blocked (i.e., no neighbor cloer to the detination than the current node). To guarantee delivery, an additional cheme, face routing [1], i generally ued to ecape from thee cae. In multicat routing, a meage i to be delivered from a ource node to a et of detination. When uing geographic routing to olve the multicat problem, the mot challenging quetion i to decide when the meage hould be replicated into different packet. The bet olution i to route the meage uing a common path among all detination, and then to replicate it at the end of thi path. Of coure, the difficulty lie in determining the bet common path by uing only local information at each hop. Moreover, even with a global knowledge, thi problem i NP-complete. In thi paper, we preent a Minimum Spanning Tree baed Energy Aware Multicat cheme (MSTEAM), a generalization for the multicat cae of the cot over progre framework decribed in [6]. In uch cheme, greedy routing i done by electing at each hop the neighbor v providing the mallet ratio between the cot needed to tranmit the meage to v and the progre toward the detination provided by v. We ue a minimum panning tree (MST) a a backbone to decide when a meage ha to be replicated. In addition, we ue an MST-baed localized next hop election cheme which conider energy conumption of ending a meage to a neighbor v over the progre achieved thank to thi node. To ecape from void area, MSTEAM ue a multicat generalization of the face recovery mechanim. A key apect of our olution i that it highly fit wirele dynamic network ince it i localized. Indeed, forwarding node need to contruct local MST uing only information on their 1-hop neighborhood, which may be obtained thank to beacon meage. MSTEAM i alo well-uited for contrained mobile device, ince an MST may be efficiently computed in time O(n log n). Moreover, MSTEAM i loop-free and alway achieve delivery /8/$25. c 28 IEEE

2 (a) Unit dik graph G = (V, E). (b) Euclidean MST (V ). (c) Gabriel graph of G. Figure 1: A unit dik graph G = (V,E) and two ubgraph. 2 Preliminarie 2.1 Network model A wirele network i a graph G = (V,E), V being the et of vertice (network node) and E V 2 the et of communication link: there exit a pair (u,v) E if u i phyically able to communicate with v. The neighborhood et N(u) of a node u i defined a: N(u) = {v V v u (u,v) E}. The denity d i the average number of neighbor per node. The et of edge E depend on the underlying phyical model. The mot well-known one i the unit dik graph (UDG) model. Given a et of node V and a maximum communication range R, E i defined a: E = {(u,v) V 2 u v uv R}, uv being the Euclidean ditance between u and v. Fig. 1a provide an example of a UDG. For a given multicat tak, the et of detination i denoted a T = {,...,t k }. We aume that node can adjut their tranmitting power, i.e., ending a meage from u to v require the mallet poible power. We aume that node collec-hop neighborhood information by uing beacon. 2.2 MAC layer model When a meage i replicated, the forwarding node end the packet to more than one neighbor. We conider two MAC layer model. In the unicat MAC layer, ending a meage to j next hop neighbor i performed by j independent unicat tranmiion. In the multicat MAC layer, ending a meage i done by only one tranmiion, exploiting the broadcat capabilitie of the wirele media. A detailed invetigation on how a reliable communication i achieved either in the unicat or the multicat MAC layer i beyond the cope of thi work. 2.3 Geometric concept The minimum panning tree (MST) i a well-known graph contruction: a tree (u 1,...,u n ) i an MST if it weight (u 1,...,u n ) i minimal. The weight of the tree denote the um of the weight over all tree edge. In a Euclidean MST, illutrated by Fig. 1b, the weight of an edge i equal to it Euclidean length. Such tree may be efficiently computed in time O(nlog n). Note that we ue throughout the paper the notation (S), which i equivalent to (u 1,...,u n ) for any et S = {u 1,...,u n }. The Steiner tree problem i imilar to the MST one: the goal i to contruct a tree Γ(u 1,...,u n ) with minimal weight, while allowing the inertion of additional intermediate Steiner point, to reduce the weight of the reulting panning tree. Thi problem i NP-complete. A planar graph i a graph in which no edge interect. Gabriel graph contruction (cf. Fig. 1c) i a localized contruction method baed on a geometric concept, introduced in [4]. Starting from a UDG G = (V, E), each edge (u,v) E i removed if there exit a vertex w located inide the circle U(u,v) of diameter uv centered at the midpoint of the egment [uv]. Thi graph i intereting for decentralized network ince the removal algorithm may be applied independently by each node, and doe not require any meage exchange. 3 Related Work One of the firt multicat algorithm for ad hoc network i MIP [11], a variant of the BIP algorithm. In BIP, a tree i built from a ource node by adding at each tep the le expenive node. Both cheme are centralized they cannot be applied in a ditributed network without a large overhead, and MIP tree i compoed of ubtree coming from BIP, which i explicitly built for broadcat. A a reult, routing i not optimal becaue there i not enough path reue. The pioneering work in localized multicat i [7], which decribe the Poition-Baed Multicat (PBM) protocol. Forwarding i performed by determining the neighbor ub-

3 et maximizing a weighted um over two conflicting objective: maximizing the number of next hop and minimizing the remaining ditance to the detination. The weight of the two objective i controlled by a parameter λ [,1]. An early packet replication i achieved by a λ cloe to. Le frequent replication are obtained with λ cloe to 1. PBM require teting each poible ubet and electing the one maximizing the objective function. The complexity of PBM i thu O(2 m ) for m neighbor. To recover from void area, face recovery compute an average point p over the detination and tart traveral of the face interected by the traight line p. When a detination can be handled in greedy mode again, the meage i replicated and the packet i handled in greedy mode again, while the other part continue with face recovery. Even under the Gabriel graph contruction, routing loop may occur with PBM. The protocol decribed in [2] perform meage forwarding along a backbone defined by contructing a panning tree over the ource node and the detination. Three panning tree heuritic are conidered. For each panning tree edge t originated at the tart node, a ingle multicat intance i ent to the end node t. The detination ubet addreed by thi meage conit of thoe detination that are reachable over thi tree edge. Thi i comparable to the replication trategy decribed in thi work. However, the mechanim decribed in [2] differ from MSTEAM in the following way: firt, any meage i forced to follow a backbone edge t until it eventually arrive at the detination. Only when arrived at thi node the meage might be replicated again. Thi i in contrat to MSTEAM where premature meage replication are poible at any node. Second, in contrat to MSTEAM, all detination reachable over t in the backbone will be diconnected whenever node t i not reachable from the ource node. Third, multicat routing decribed in [2] i only concerned with localized contruction of multicat overlay. Routing a meage along an overlay edge uv i done in a centralized way by calculating the hortet path from u to v. In thi work, we decribe a localized metric in order to elect an energy-efficient next hop by uing information about the detination and the neighbor only. Fourth, with global information, recovery from greedy routing failure i not an iue in PBM. A recent localized cheme, Geographic Multicat Routing (GMR), can be found in [8]. The algorithm i baed on the cot over progre framework [6] and, oppoed to PBM, doe not require etting a proper network-dependent parameter λ. In the unicat cae, cot over progre denote the relation between cot produced in the next hop and the progre achieved by thi next hop. The multicat extenion of thi framework minimize the number of elected next hop over the progre achieved by the elected et. Progre i the difference between the um over all individual ditance between the current node and detination, and the um over ditance of each next hop and the detination covered by thi node. A node v cover a detination if thi detination i cloet to v compared to all other next hop. In contrat to [7], thi cheme decribe alo an efficient neighbor et election trategy which reduce the cot from O(2 m ) to O(mk min(m,k) 3 ) in the wort cae, where k i the number of detination and m the number of neighbor. Since thi cheme i greedy, a meage can be blocked. The cheme decribe a face recovery trategy which applie traditional unicat face traveral for each detination. To ave bandwidth, however, face meage traveling the ame face are aggregated into a ingle meage. All the cheme conider hop count a the optimization criteria. When the hardware provide ignal trength adaptation, a ingle tranmiion over a large ditance can be more expenive than many mall-ditance tranmiion. The firt localized cheme conidering thi poibility i decribed in [9]. Thi cheme, GMREE, i an extenion of GMR conidering the cot of the total energy conumption of the next tranmiion. 4 MST baed Multicat In thi ection, we provide a motivation of the goodne of uing Euclidean MST to decide a meage replication, and to determine the bet next hop node for a given et of detination. Unle pecified, MST alway denote Euclidean MST. 4.1 Why MST-baed Backbone? When global information i available, finding the optimal multicat tree i poible, though, being an NPcomplete problem. Our goal i to define a localized heuritic to find a tree with a low cot. Quantifying low in term of a formal analyi i beyond our cope, but we will how empirically that the cot obtained thank to the decribed heuritic doe not ignificantly depart from an efficient centralized olution. The quality of an algorithm depend on two factor: The meage replication trategy, which hould aim at meage forwarding along a cot effective backbone. The next hop election, which hould aim at coteffective meage forwarding along thi backbone. Meage forwarding along an edge of the multicat backbone elect the bet neighbor with repect to the conidered metric (e.g., hop count, Euclidean ditance) and the detination reachable along thi backbone edge. Suppoe being the ource and T = {,...,t k } the detination. Let C(u, v) denote the weight of the hortet weighted path from u to v. Under the unicat MAC

4 t 4 t 4 (a) All detination in the ame et. (b) Two ubet {, } and {, t 4 }. Figure 2: The replication trategy ued by MSTEAM: (,,,,t 4 ) i ued to replicate the meage at node. aumption, a weighted Steiner tree Γ(,,...,t k ), uing C(u, v) a the cot function, define the cot optimal multicat backbone. We do not aume that a node i aware of all network node to compute uch a Steiner tree. Moreover, we do not aume that the cot function C(u,v) i even known to the node. Thu computing a Steiner tree a an optimal backbone i impoible in thi general etting. A an approximation we ue the concept of weighted MST, which may be computed even by contrained device. Since the energy model i unknown, we approximate the routing cot by the implified aumption that uv < uw implie C(u, v) < C(u, w). Then, the weighted MST i equivalent to the Euclidean MST. Under the multicat MAC aumption, energy aving are poible at node where the meage i replicated. At thi point, any et of next hop might produce the ame routing cot. In a mall cale multicat, it might thu be more efficient to perform a ingle direct large broadcat tranmiion intead of many mall tranmiion. In a large cale multicat, however, we expect that the cot aving which are poible at replication node will be outweighed by the cot required to route the meage between thoe replication point. Thu, we ue the ame MST approximation under the multicat MAC aumption. 4.2 Replication trategy Let be the current forwarding node and T i T the et of detination handled by. Node calculate the MST ({} T i ) over itelf and T i. Thi tree provide the backbone to ue to reach all detination in T i from. The meage thu ha to be routed along the edge of thi tree, and mut be replicated at node if multiple path tart at thi node. Actually, each of thee path i repreented by an edge originating at, and pan a ubet of detination. Thee are forming exactly a detination ubet to which ha to end an individual meage copy. Thi trategy i illutrated in Fig. 2, where handle the detination,, and t 4. In Fig. 2a, the MST (,,,,t 4 ) ha only one edge originated at node, o all detination are grouped together. In thi cae, the meage i not replicated and will be routed along (, ). In Fig. 2b, there are two edge originated at node : the firt one pan and, the econd one pan and t 4. The meage i thu replicated into two packet. The firt one i routed along (, ) toward {, }, the econd one i routed along (, ) toward {,t 4 }. 4.3 Energy-efficient metric Greedy routing i done with a generalization of the framework decribed in [6] for the multicat cae. In thi framework, two thing need to be etimated: the cot (in term of energy conumption) of chooing a given neighbor v a the next hop, and the progre toward the detination ubet T i provided by the replication trategy. The neighbor with the mallet cot over progre ratio i choen a the next hop for T i. Since the current forwarding node u conider the MST ({u} T i ) a the backbone to route the meage toward T, ({u} T i ) may be ued a a local etimation of the remaining ditance the meage ha to travel. Conidering a neighbor v, ({u} T i ) ({v} T i ) i an etimation of the progre provided by v toward T i. The greedy cheme aume that the current node u alway elect a candidate node v providing poitive progre. Sending a meage from a node u to it neighbor v require an amount of energy denoted f(u,v). Thu, the cot over progre ratio Q(u,v,T) at node u of a neighbor node v i: Q(u,v,T) = f(u, v) ({u} T) ({v} T). The expreion of Q(u,v,T) i a generalization of the unicat routing metric f(u,v)/( ut vt ) decribed in [6]. In thi connection, u i the current node, v the next hop candidate, and t the detination node. When T = {t}, the expreion Q(u,v,T) reduce to thi formula ince an MST over a pair of node i the traight line connecting them.

5 u 8 u 7 v u (,,, ) u 9 u 1 u2 F u 6 u3 u 4 u 5 (a) Node i concave with repect to {,, }. (b) Traveral of the face interected by the MST edge. Figure 3: Minimum panning tree baed face multicat routing i ued in order to recover from concave node. 4.4 Recovery trategy Suppoe that in Fig. 3a ha to end a meage toward {,, }. Detination are connected over (, ) of (,,, ). However, may not elect any of it neighbor u and v, ince they atify (u,,, ), (v,,, ) > (,,, ) : the meage i blocked. Thi happen whether any of thee detination are reachable from or not. We denote uch node a concave with repect to the detination ubet. Face i a unicat routing cheme that can be ued to handle greedy routing failure for each detination individually. We decribe for the firt time a multicat extenion of face that can handle all detination at once. Similar to unicat face routing, the multicat cheme require a topology control mechanim which tranform the underlying network into a planar graph. Here, we employ the Gabriel graph which require the network to comply to the UDG model. A depicted in Fig. 3b, a planar graph partition the plane into face that can be travered by employing the left/right hand rule; a node end the meage along the edge which i lying next in clockwie/counterclockwie direction of the edge it wa received from. For intance, when tarting at node in Fig. 3b, the face F will be travered along the path u 1 u 2...u 9 when uing the right hand rule. Unicat face recovery ha different variant; We employ the one that tranmit the meage along the equence of face that are interected by the line t connecting the ource to the detination t. When the meage arrive at a node cloer to t than, greedy routing i ued again. With a Gabriel graph, thi mechanim implifie to travering the firt face. The idea of multicat face i a follow. Suppoe that a node ha computed a detination ubet T i for which no better greedy neighbor exit. Let t be the edge connecting to ({} T i ). By uing any of the two rule right or left hand node tart traveral of the face interected by the outgoing MST edge t. Face traveral continue until the meage arrive at a node u atifying ({u} T i ) < ({} T i ). At thi node, the detination ubet T i i handled in greedy mode again. A pecial cae occur when no uch node u i found during face traveral. In thi cae, to avoid loop, the meage i dropped if it i about to be ent again over the firt face traveral edge in the ame direction. Referring to Fig. 3b, edge connect to (,,, ). Since i concave with repect to {,, }, it tart traveral of face F, i.e., the face interected by. Auming the right hand rule, face traveral viit node u 1, u 2, and u 3. Since u 3 i the firt one atifying (u 3,,, ) < (,,, ), it handle the detination ubet {,, } in greedy mode again. 5 The MSTEAM protocol 5.1 Decription Given the detination T = {,...,t k }, the ource firt decide whether a meage replication occur. It thu compute the MST ({} T), and group all detination panned by edge originated at (cf. Sec. 4.2). For each ubet T i T obtained, compute a ubet N i () N() containing all neighbor v N() uch that ({v} T i ) < ({} T i ) (neighbor providing poitive progre toward T i ). If N i (), compute the cot over progre ratio Q(,v,T i ) for each neighbor v N i () (refer to Sec. 4.3). The neighbor providing the bet ratio i the next hop toward T i. If N i () =, then the meage i blocked, and face recovery mut be ued. Node applie the trategy preented in Sec. 4.4 to elect the face node v a the router toward T i. The proce i repeated until all ubet T i have been conidered. In the unicat MAC cae, a packet i ent for each ubet T i, with the et of detination, the elected router and the mode (greedy or face). With face routing, the packet alo contain the firt edge travered by the packet in thi mode, and the weight of the MST at the tarting node ( ({} T i ) in thi example). If multicat MAC i conidered, thi information i aggregated into the ame packet. Thi mean that the latter will contain a lit of all next hop

6 t t t 7 t 9 t 5 t 6 t 7 t 9 t 5 t 6 t 8 t 4 (a) The MST ({} T). t 8 t 4 (b) The multicat tree produced by MSTEAM. Figure 4: A ample run of MSTEAM for a et T of 1 detination and a denity d = 35. and for each of them, the et of detination they erve, the mode to ue and the additional face information. In both cae, the packet i ent uing the minimum energy needed for ucceful tranmiion to the next hop(). When a node u receive a packet, it check whether it i a forwarder. If not, the packet i dropped. If o, it check the routing mode currently ued for the given et of detination T i T. In greedy mode, u repeat the proce followed by. In face mode, it check whether it i cloer to the detination (e.g., ({u} T i ) i le than the weight written in the packet). If o, it handle T i in greedy mode. If not, face recovery i applied once again (refer to Sec. 4.4). Of coure, if u i a detination, it remove itelf from T i. Fig. 4 how a ample run of MSTEAM over a random network. Figure 4a give the MST ({} T), while Fig. 4b provide the multicat tree produced by MSTEAM. The MST panning all detination wa ued at the ource. Since two edge originate at, the meage ha been replicated into two packet at. The firt one wa ent toward t and along (,t ), while the econd one wa ent toward the other node along (, ). Figure 4b how that MSTEAM wa able to follow thee edge in an effective way. One can alo oberve that the replication trategy correctly work by looking at the path followed to reach t 4 and t 9 from the node cloe to t 5. Intead of following (t 5,t 9 ), MSTEAM routed the meage along a common path among t 4 and t 9, and then replicated it at the end of thi path. Regarding the complexity of MSTEAM, a node in greedy mode compute an MST for the replication trategy, which ha a time complexity in O(k log k), k being the number of detination. In the wort cae, all detination are handled eparately. For each of them and for each neighbor, a new MST i computed. In thi cae, the complexity in time of MSTEAM i O(mk 2 log k) for the greedy mode, m being the number of neighbor. Thi complexity may be better etimated ince a MST ha a maximum degree of 6, regardle of k. Since face mode ha a complexity in O(k log k), the complexity of MSTEAM in the wort cae i O(mk log k), which i lower than the complexity of GM- REE (O(mk min(m,k) 3 ), till conidering the wort cae). 5.2 Correctne of MSTEAM We now prove that MSTEAM i loop-free and guarantee delivery. We aume the UDG model with radiu R, and the ue of Gabriel graph to contruct the planar graph ued during face recovery. The proof of the two lemma may be found in [3]. Lemma 1 Let be a node where face recovery for the detination {,...,t k } wa tarted, and t the edge connecting to (,,...,t k ). If can reach at leat one detination, then traveral of face F, interected by t, alway arrive at a node u atifying ({u,,...,t k }) < ({,,...,t k }). Lemma 2 A meage addreed to S = {,...,t k } will either be dropped, replicated, or delivered after a finite number of tep. Theorem 1 The decribed multicat routing cheme MSTEAM i loop-free and provide delivery guarantee. Proof 1 Let be the ource and T the et of detination. In MSTEAM, a meage i either kept or replicated in a forwarding tep, and two meage are never merged. Thu, for each poible ubet S T, at mot one intance of a meage addreing thi et may exit. It follow that the number of poible meage intance i finite. Finally, due to Lemma 2 each intance i handled a finite number of forwarding tep. It follow that the total number of forwarding tep i finite, i.e., no routing loop occur. Let t be an element of T. Suppoe there exit a path from to t, and that t i dropped during routing. A meage might only be dropped when it i handled in face mode. Let u be the node where face traveral wa tarted, and S the ubet of detination handled in face mode. Since the meage i dropped, all node v viited during face traveral atifie ({v} S) ({u} S). Since u wa reached by, and ince can reach t, u can reach at leat one node in S. By Lemma 1, it follow that face traveral will viit a node w atifying ({w} S) < ({u} S), a contradiction.

7 1 1 1 MSTEAM with unicat face recovery Full MSTEAM cheme MSTEAM with uniicat face recovery Full MSTEAM cheme 8 8 Total energy conumption Total energy conumption Average denity (a) Unicat MAC layer Average denity (b) Multicat MAC layer. Figure 5: Performance of the MST baed face recovery mechanim ued in MSTEAM (1 detination node). 6 Performance evaluation 6.1 Simulation etting We ued a home-made imulator with the UDG model. Node have a maximum range R = 25, and a number of them are randomly put on an area of ize to obtain a given denity. The 95% confidence interval i given on each figure. The energy conumption f(u,v) of a node u tranmitting a meage to a node v i: f(u,v) = uv α + c e, α being the path lo contant. We ued α = 4 and c e = 1 8, ued in [9]. We elected GMREE [9] a it i the only other geographic energy-efficient localized multicat cheme. GM- REE ue an underlying ubgraph, and author howed that the Local Shortet Path Tree (LSPT) [1] uing f(u, v) a the cot function i the mot efficient one. We thu retricted our imulation of GMREE to the LSPT. GMREE wa compared in [9] to ESP (Energy-Efficient Shortet Path), the centralized application of Dijktra hortet path tree, uing f(u,v) a the cot function. We found that ESP ha variable performance baed on the maximum communication range: thank to it centralized knowledge, ESP generate efficient route, where the length of each hop i cloe to the optimal, while replicating meage too often. When the communication range i too low, route are not enough efficient to counterbalance the wate coming from replication. We believe that thi problem ha been overlooked in [9]. We thu ued another centralized cheme, uing an approximated weighted Steiner tree [5] a the multicat tree. 6.2 Experimental reult Fig. 5 how the efficiency of the new MST-baed face recovery mechanim. Small denitie were ued, to maximize the number of void area. The new cheme i by far uperior to the baic one: conidering the denity d = 5, energy conumption i divided by approximately 27 in Fig. 5a and 16 in Fig. 5b. Even with d = 1, the ratio i till around 1 in both figure. A the denity increae, the number of void area decreae and face recovery i le ued, o the two cheme become equal. Even in the multicat MAC layer cae, where packet are aggregated before tranmiion, the baic cheme i inferior becaue each detination i handled eparately, o that the protocol quickly ue different path and lead to increaed energy conumption. Fig. 6 how reult at increaing denity. A aforementioned, ESP replicate the meage too early and i le efficient a the denity increae. The Steiner cheme ue a centralized backbone and ha very good reult for all denitie. A the denity increae, the Steiner tree become cloer to the optimal tree, and energy conumption decreae. MSTEAM i able to provide cloe reult: when d = 4, it conume only 6% more energy than the Steiner cheme in Fig. 6a and 8% more in Fig. 6b. MSTEAM i alway better than GMREE: ince face routing i not ued in uch denitie, the two reaon are the replication trategy and the MST-baed progre etimation, which i better than the one ued in GMREE. Thi i true even in Fig. 6b where the multicat wirele advantage i conidered. Overall, thi cae doe not change the relative poition of all cheme, and jut lead to lower energy conumption. We finally give reult at increaing number of detination in Fig. 7 (d = 35). A expected, the energy conumption of all cheme increae with the number of detination, ince more packet and more path are generated. The increae i not fully linear becaue of the common path ued among detination. However, Steiner cheme and MSTEAM are more calable than the other one ince their energy conumption increae more lowly. For intance, when changing the number of detination from 1 to 3, the conumption of MSTEAM in Fig. 7a i multiplied by 1.7 while the one of GMREE i multiplied by 1.9. Thi difference i obviouly viible in both unicat and multicat MAC layer, and i caued by a better path reue trategy.

8 Total energy conumption ESP GMREE on LSPT MSTEAM Steiner Total energy conumption ESP GMREE on LSPT MSTEAM Steiner Average denity (a) Unicat MAC layer Average denity (b) Multicat MAC layer. Figure 6: Performance of all elected cheme at increaing denity (1 detination node). Total energy conumption ESP GMREE on LSPT 3.5 MSTEAM Steiner Total energy conumption ESP GMREE on LSPT 3.5 MSTEAM Steiner Number of detination node (a) Unicat MAC layer Number of detination node (b) Multicat MAC layer. Figure 7: Performance of all elected cheme at increaing number of detination (d = 35). 7 Concluion and future work We have preented MSTEAM, a geographic localized multicat cheme, that ue an MST a an approximation of the optimal backbone. Beide guaranteeing delivery, our protocol i well-uited to ad hoc and enor network becaue it i localized and ha low time complexity. We experimentally demontrated that MSTEAM i very energyefficient, even compared to a centralized cheme, and outperform the bet exiting localized multicat protocol. We plan to improve MSTEAM uing a Steiner tree approximation. We focued on MST becaue it i a reaonable approximation of an optimal tree. Moreover, it computation ha a very low time complexity, while a Steiner tree introduce more complexity. However, under the exponential path lo model ued for our experiment, calculating a Steiner tree i poible and one can expect better energy aving. Of coure, a trade-off i then needed. Reference [1] P. Boe, P. Morin, I. Stojmenović, and J. Urrutia. Routing with guaranteed delivery in ad hoc wirele network. ACM Wirele Network, 7(6), Nov. 21. [2] K. Chen and K. Nahrtedt. Effective location-guided overlay multicat in mobile ad hoc network. International Journal of Wirele and Mobile Computing, 3, 25. Special iue on Group Communication in Ad Hoc Network. [3] H. Frey, F. Ingelret, and D. Simplot-Ryl. Localized minimum panning tree baed multicat routing with energy-efficient guaranteed delivery in ad hoc and enor network. Technical Report RT-337, INRIA, 27. [4] K. Gabriel and R. Sokal. A new tatitical approach to geographic variation analyi. Sytemic Zoology, 18(3), Sept [5] L. Kou, G. Markowky, and L. Berman. A fat algorithm for Steiner tree. Acta Informatica, 15(2), June [6] J. Kuruvila, A. Nayak, and I. Stojmenović. Progre baed localized power and cot aware routing algorithm for ad hoc and enor wirele network. International Journal of Ditributed Senor Network, 2(2), June 26. [7] M. Mauve, H. Füßler, J. Widmer, and T. Lang. Poition-baed multicat routing for mobile ad-hoc network. Technical Report TR-3-4, Department of Computer Science, Univerity of Mannheim, Germany, 23. [8] J. Sanchez, P. Ruiz, X. Liu, and I. Stojmenović. GMR: Geographic multicat routing for wirele enor network. In Proc. of the IEEE Communication Society Conference on Senor, Meh, and Ad Hoc Communication and Network (SECON), Sept. 26. [9] J. Sanchez, P. Ruiz, and I. Stojmenović. Energy efficient geographic multicat routing for enor and actuator network. Computer Communication, 3, 27. Special iue on Senor-Actuator Network. [1] S. Wang, D. Wei, and S. Kuo. SPT-baed topology algorithm for contructing power efficient wirele ad hoc network. In Proc. of the ACM Int. World Wide Web Conference, May 24. [11] J. Wieelthier, G. Nguyen, and A. Ephremide. Energy-efficient broadcat and multicat tree in wirele network. Mobile Network and Application, 7(6), Dec. 22.