Minimum Energy Reliable Paths Using Unreliable Wireless Links

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1 Minimum Energy Reliable Path Uing Unreliable Wirele Link Qunfeng Dong Department of Computer Science Univerity of Wiconin-Madion Madion, Wiconin Micah Adler Department of Computer Science Univerity of Maachuett at Amhert Amhert, MA 0003 Suman Banerjee Department of Computer Science Univerity of Wiconin-Madion Madion, Wiconin Archan Mira IBM T J Waton Reearch Center 9 Skyline Drive Hawthorne, NY 0532 archan@u.ibm.com ABSTRACT We addre the problem of energy-efficient reliable wirele communication in the preence of unreliable or loy wirele link layer in multi-hop wirele network. Prior work [] ha provided an optimal energy efficient olution to thi problem for the cae where link layer implement perfect reliability. However, a more common cenario a link layer that i not perfectly reliable, wa left a an open problem. In thi paper we firt preent two centralized algorithm, BAMER and GAMER, that optimally olve the minimum energy reliable communication problem in preence of unreliable link. Subequently we preent a ditributed algorithm,, that approximate the performance of the centralized algorithm and lead to ignificant performance improvement over exiting inglepath or multi-path baed technique. Categorie and Subject Decriptor C.2. [Computer-Communication Network]: Network Architecture and Deign Wirele Communication; C.2.2 [Computer- Communication Network]: Network Protocol Routing Protocol; C.2.4 [Computer Communication Network]: Ditributed Sytem; C.4 [Computer Sytem Organization]: Performance of Sytem General Term Algorithm, Deign, Performance, Reliability, Theory Keyword End-to-End Reliable Communication, Energy Efficiency, Multi-path, Routing, Wirele Network Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. To copy otherwie, to republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. MobiHoc 05, May 25 27, 2005, Urbana-Champaign, Illinoi, USA. Copyright 2005 ACM /05/ $ INTRODUCTION Wirele communication network have been deployed at an increaingly fat rate, and are expected to rehape the way we live in thi phyical world. For example, wirele ad hoc network combined with atellite data network [7] are able to provide global information delivery ervice to uer in remote location that could not be reached by traditional wired network. Meanwhile, advance in hardware technology are contantly proliferating variou wirele communication terminal (e.g. mart phone or PDA) to an exploding uer population. In many cenario, deign of wirele communication protocol are guided by two requirement energy efficiency and reilience to packet loe. Efficiently handling loe in wirele environment, therefore, aume central importance. Even under benign condition, variou factor, like fading, interference, multi-path effect, and colliion, lead to heavy lo rate on wirele link [, 6, 36, 9, 33, 35]. Due to the endto-end reliability requirement of many application, it i neceary to tudy how uch reliability can be guaranteed in an energy efficient way in wirele environment. In thi paper we examine the problem of energy efficient routing of traffic in a multi-hop wirele network that appropriately handle packet loe in the wirele environment. There are two well-known way to achieve end-to-end reliability on multi-hop path. The firt approach employ hop-by-hop retranmiion each link layer hop retranmit lot frame a and when neceary. The econd approach aume that link layer are unreliable and retranmiion are performed end-to-end. It i alo poible to conider a mix of the above a a third approach, where link layer perform a few retranmiion if neceary, but perfect reliability i only guaranteed through end-to-end mechanim. Traditional power aware routing cheme [24, 6] do not take link lo rate into account when computing energy efficient path. By ignoring the impact of uch loe, they implicitly aume that every link i totally reliable. That paradigm i obviouly too optimitic, and retranmiion conume power a well. In order to achieve better energy efficiency in realitic cenario, the right metric hould be the cumulative energy conumption due to all packet tranmiion including retranmiion. Prior work by Banerjee and Mira [] olved the problem of computing energy efficient path for the hop-by-hop retranmiion model only and left optimal approache for the end-to-end

2 cae a an open problem. However, all practical mechanim to achieve perfect end-to-end reliability guarantee rely either on the end-to-end model or on the mixed approach (combination of hopby-hop and end-to-end retranmiion). For example, link layer technologie uch a the 802. MAC protocol [8] typically make a bounded number of retranmiion attempt for a lot or corrupted frame. Further loe can be recovered through end-to-end retranmiion. The following are a ummary of example which underline the importance of energy-efficient olution under the end-toend and the mixed retranmiion model: Link layer technologie uch a IEEE 802. [8] typically implement a limited number of retranmiion, which reult in poible delivery failure over loy link. There are link level technologie that do not provide hop-byhop retranmiion (e.g. TRAMA [20]). Given link layer reliability, packet lo may till happen at network layer due to variou reaon (e.g. congetion in WSN [28]). Node may move, leep, or fail. In uch cae, hop-by-hop reliability cannot be aumed. Note that even if a leeping node can receive packet after waking up, the tranport protocol may have timed out. A long a there i ome link in the multi-hop path that cannot guarantee reliable packet delivery, we will have to rely on TCP-like tranport protocol to initiate end-to-end retranmiion back from the ource. In thi paper, we firt olve the problem of computing minimum energy path for reliable communication in the pure end-to-end retranmiion model where none of the link in a wirele path guarantee any reliability. We next proceed to tudy the more general and realitic mixed retranmiion model where ome link may provide partial reliable delivery while the other may not. For example, even if the link level technology upport hop-by-hop retranmiion, ome link may till be unreliable due to other reaon decribed above. The BAMER and GAMER algorithm are deigned for computing minimum energy path in thee model repectively. The hop-by-hop model and the pure end-to-end model are jut pecial cae of the mixed model. Therefore, our algorithm for the mixed model can be ued to find minimum energy path in any network configuration. For implementation in many practical cenario, we may need a imple and lightweight ditributed protocol. In thi paper, we alo propoe a ditributed routing protocol,, for energy efficient routing in the general mixed retranmiion model. Clearly, can be ued in any network configuration, too. We how that i able to find the minimum energy path in the hopby-hop retranmiion model. Simulation reult demontrate that alo effectively improve energy efficiency over the bet known exiting technique in the general mixed retranmiion model. While the main focu of thi paper i on ingle-path routing, we alo examine the problem of reliability through utilization of multiple redundant path. Prior work ha examined the ue of uch multi-path route in improving throughput or reliability [34, 25] at the cot of generally increaed energy conumption. To illutrate thi apect, we performed imulation-baed comparion of our technique with one of thee prior technique, GRAB [34] and the energy conumption of GRAB to achieve reaonable reliability i order of magnitude larger than that of our cheme. Interetingly, we found that by carefully chooing multi-path route for data delivery it i poible to reduce energy conumption more effectively than the optimal ingle-path route can. In particular, we formally analyze the problem of finding the minimum energy multi-path routing cheme and prove that it i actually NP-hard. To the bet of our knowledge, thi paper i the firt to formally invetigate the potential of multi-path routing on energy conervation. Through extenive imulation, we demontrate that our algorithm can ignificantly improve energy efficiency over bet known exiting technique. Moreover, we carefully examine the effect of a number of network parameter on the performance of our algorithm a well a exiting technique. Thi tudy further enhance our undertanding of energy efficient reliable communication in the preence of loy link. The ret of the paper i organized a follow. Section 2 review previou related work. Our network model and problem formulation are preented in Section 3. In Section 4, we preent two algorithm a well a a ditributed routing protocol for finding minimum energy path in the mixed retranmiion model. In Section 5, we examine multi-path routing a a potential mean of energy conervation in the preence of unreliable link, and formally analyze it complexity. An empirical tudy through extenive imulation of our cheme a well a the bet known current cheme i preented in Section 6. Finally, we conclude the paper in Section RELATED WORK Energy efficient routing ha alway been a central reearch topic in wirele network, both in the paradigm of multicat/broadcat [30, 7, 29, 3, 5, 32, 3, 4, 0, 3] and in the paradigm of unicat [22, 24, 23, 5, 6, 26, 25]. In both paradigm, our objective i to deign a routing cheme uch that the total tranmiion power i minimized. In thi paper, we tudy the paradigm of unicat and refer intereted reader to the literature for more knowledge on energy efficient multicat/broadcat routing. By uing Dijktra hortet path algorithm [8], PAMAS [23] find a minimum cot path where the link cot i et to the tranmiion power. If every link in the path i error free, then a ingle tranmiion over each link can uccefully deliver a packet from the ource to it detination with a minimum energy conumption. Scott and Bamboo [22] tudied the cae where link cot include power conumption on the receiver ide, and propoed to find energy efficient path uing a modified form of the Bellman-Ford algorithm [8]. Some reearcher have conidered power aware routing in an alternative approach. The reidual battery power i ued a a routing metric, in order to achieve a more balanced ditribution of power conumption among all the node o that the lifetime of the whole ytem may be increaed. From our perpective, thee cheme may reult in le energy efficient route. We refer the reader to the literature [24, 5, 6, 26] for detailed information. Unfortunately, none of thee previou paper conidered the loy property of wirele link. Banerjee and Mira [] explored the effect of loy link on energy efficient routing and olved the problem of find minimum energy path in the hop-by-hop retranmiion model. Let w and p denote the tranmiion power and the error rate of a hop-by-hop retranmiion link, repectively. [] propoed the link cot to be, which i actually the expected w p energy conumption of delivering a packet over that link. For the hop-by-hop retranmiion model, it i then traightforward to ue a traditional hortet path algorithm (e.g. Dijktra algorithm) to compute minimum energy path. The ame i, however, not true in the end-to-end retranmi- A imilar metric, ETX, wa propoed by DeCouto et al [9] for computing high throughput path.

3 ion model. Therefore the author in [] only propoed an approximate heuritic that define the link cot to be, where w ( p) l l 2 i ome contant, and ued Dijktra algorithm to compute low-energy path. For implicity, in thi paper we denote Banerjee and Mira algorithm by BMA and denote BMA where l = k by BMA-k. In the end-to-end retranmiion model, packet lo at intermediate link will abort the whole delivery thu far and incur end-to-end retranmiion back from the ource, which mean more tranmiion power i wated than in the hop-by-hop model. Intuitively, l 2 make loy link appear to be even more expenive. BMA-l thu prefer le loy link and reduce the rik of incurring end-to-end retranmiion. While uch a choice i reaonable, clearly it i not optimal. Additionally the more general and realitic mixed retranmiion model i not explored in []. Multi-path routing ha been propoed a a mean of improving reliability a well a throughput. GRAB [34] forward packet along an interleaved meh, and control the width of the meh hence the ucce ratio by aigning an appropriate credit to each packet. We here point out that the multi-path cheme of GRAB harnee the high redundancy and large cale of WSN, and i not appropriate for other network model. In contrat, thi paper conider a more general network model. Moreover, GRAB provide only robut delivery intead of reliable delivery, which mean packet are not guaranteed to be delivered in GRAB. Sriniva and Modiano [25] invetigate the problem of minimum energy node/link dijoint path routing in multi-hop wirele network. Clearly, uch cheme reult in increaed energy conumption, compared with the minimum energy ingle path. Moreover, they do not provide guaranteed delivery, either. Again, none of them explicitly conider link error rate. Tranport protocol (e.g. PSFQ [27]) have alo been propoed to provide reliable communication over unreliable wirele link. Unlike routing protocol, tranport protocol do not pay attention to route election hence are beyond the cope of thi paper. 3. FORMULATION In our network model, each network node i aumed to be equipped with an omnidirectional antenna. A wirele network i modelled a a directed graph G = (V, A), where V i the et of node and A i the et of directed link. Each node i aigned a unique ID i [.. V ] and ha a maximum tranmiion power of P max(i). Each directed link (i, j) ha a non-negative weight W (i, j), which denote the minimum tranmiion power required to maintain a reaonably good quality link from node i to node j. Wirele propagation uffer evere attenuation [4, 9, 2]. Let d ij denote the ditance between node i and node j. If i tranmit with power P t(i), the power of the ignal received by node j i given by P r(j) = Pt(i), c d α ij where α and c are both contant, and uually 2 α 4 [2]. In order to correctly decode the received ignal at the receiver ide, it i required that P r(j) β 0 N 0, where β 0 i the required ignal-to-noie ratio (SNR) and N 0 i the trength of ambient noie. Thu, the weight of link (i, j) i given by W (i, j) = c β 0 N 0 d α ij. Each link (i, j) alo ha an error rate (or lo rate) Er(i, j), which i the probability that a tranmiion over link (i, j) doe not ucceed. If Er(i, j) = 0, link (i, j) i conidered reliable. G contain link (i, j) if and only if W (i, j) P max(i) and Er(i, j) <. The expected number of tranmiion (including retranmiion) of a ucceful delivery over link (i, j) i given by N(i, j) = Er(i, j). Each node i capable of adjuting it tranmiion power according to the outgoing link weight, in order to conerve a much power a poible. Typically, energy efficient routing cheme tend to chooe path compoed of a large number of hort ditance link ince long ditance link are much more power conuming given that α 2. Link failure i preumed to be independent and unpredictable, o the metric i defined to be the expected total energy conumption of a ucceful delivery. By minimum energy path from node u to node v, we refer to a path that ha the minimum expected energy conumption of a ucceful delivery from u to v. Let C min(u, v) denote the expected energy conumption of a ucceful delivery along a minimum energy path from u to v. We refer to the general problem of finding the minimum energy routing cheme in the mixed retranmiion model a the Minimum Energy Reliable Communication Uing End-to-end Retranmiion problem and formally define it a follow. MINIMUM ENERGY RELIABLE COMMUNICATION USING END- TO-END RETRANSMISSIONS INSTANCE Directed graph G = (V, A). Link weight function W : A R + 0. Link error rate function Er : A [0, ). Function U : A {0, } indicate whether a link provide hop-by-hop retranmiion. Specified ource and detination t. Non-negative bound B. QUESTION I there a routing cheme uch that the expected energy conumption of a ucceful delivery from to t i no more than B? 4. SINGLE-PATH MIN-ENERGY ROUTES In thi ection, we preent a number of algorithm to compute minimum energy path for reliable communication over loy link in multi-hop wirele network. We tart by tudying the eemingly impler end-to-end retranmiion model, for which we preent the Baic Algorithm for Minimum Energy Routing (BAMER). Then in Section 4.2, we tudy the more general and realitic mixed retranmiion model. The General Algorithm for Minimum Energy Routing (GAMER) i propoed for that cae. In Section 4.3, we how that an appropriate preproceing tage enable BAMER to olve the ame problem in the mixed model a well. While BAMER and GAMER are both centralized algorithm, typically routing need to be carried out in a ditributed fahion. Toward that end, we propoe the Ditributed Algorithm for Minimum Energy Routing () in Section Baic Algorithm for Minimum Energy Routing (BAMER) We firt preent BAMER and how that it find minimum energy path from to all other node in the end-to-end retranmiion model. Baically, BAMER i a generalized extenion of Dijktra hortet path algorithm [8]. In Dijktra algorithm, only edge weight are conidered. Aume that node u precede v in the path from to v, denoted by P(, v). Let P(, u) denote the part of P(, v) between and u. For any path P(i, j), let C(P(i, j)) denote the energy conumption of uccefully delivering a packet

4 BAMER (G,, T, C) for each node v V (G) do 2 T (v) φ 3 C(v) 4 C() 0 5 S {} 6 u 7 while S V (G) do 8 for each node v V (G) S do 9 if N(u, v)[c(u) + W (u, v)] < C(v) 0 T (v) T (u) {(u, v)} C(v) N(u, v)[c(u) + W (u, v)] 2 u v V (G) S.t. C(v) i minimum 3 S S {u} Table : Peudo code decription of BAMER. i the ource. For any node v, T (v) conit of the link of the computed path from to v, whoe cot i C(v). x (0) (8, 2) y (36) (5, 2) (7, 3) (2, 4) (5, 3) (6, 2) (5, 2) (5, 2) (7, 5) (8, 2) z (32) (7, 3) t (80) Figure : Illutration of BAMER along that path from i to j. In Dijktra algorithm, it i clear that C(P(, v)) = C (P(, u)) + W (u, v). Our algorithm take into account both link weight and link error rate. The key obervation i that C (P(, v)) = N(u, v) [C (P(, u)) + W (u, v)]. Indeed, Dijktra algorithm i a pecial cae of BAMER where Er(u, v) = 0 and N(u, v) =, i.e., link (u, v) i reliable. Baed on thi intuitive undertanding, we preent BAMER in Table. Compared with Dijktra algorithm, the only real difference i line 9 and line. We next proceed to how the optimality of BAMER. LEMMA. Let P(, v) denote a minimum energy path from to v, in which node u i the predeceor of v. The part of P(, v) between and u, P(, u), i a minimum energy path from to u. To prove by contradiction, aume that P(, u) i not a minimum energy path from to u, while another path P (, u) i uch a minimum energy path. We can imply replace P(, u) in P(, v) with P (, u). The reulted new path from to v, denoted by P (, v), will have an expected energy conumption of C ( P (, v) ) = N(u, v) [C ( P (, u) ) + W (u, v) ] < N(u, v) [C (P(, u)) + W (u, v)] = C (P(, v)). Thi contradict the fact that P(, v) i a minimum energy path from to v. LEMMA 2. In BAMER, each time a node v i added to S, link in T (v) form a minimum energy path from to v hence C(v) = C min(, v). We prove Lemma 2 by induction on the order of node being added to S. The bae cae i trivially true. Now aume that Lemma 2 hold for every node already in S, and a node v i then choen to be added to S. Conider any minimum energy path P(, v) from to v. If all previou node in P(, v) have been in S, by Lemma and inductive aumption it i clear from the decription of BAMER that C(v) C (P(, v)) = C min(, v). If at leat one previou node in P(, v) ha not been in S yet, let u denote the firt uch previou node in P(, v) (counting from to v), and let P(, u) denote the prefix part of P(, v) between and u. By Lemma, it i clear from the decription of BAMER that C(u) C (P(, u)). Given that BAMER choe v intead of u, it i the cae that C(v) C(u) C (P(, u)) C (P(, v)) = C min(, v) ince u i a previou node in P (, v). So far we have proved that in either cae, C(v) C min(, v). Clearly, it ha to be the cae that C(v) = C min(, v) Let u be the node already in S that aign C(v) to v in BAMER. By inductive aumption, link in T (u ) form a path from to u whoe expected energy conumption i C(u ). Thu, T (v) = T (u ) {(u, v)} form a path from to v, and the expected energy conumption i C(v) = C min(, v). It follow that link in T (v) form a minimum energy path from to v. COROLLARY. For each node v V (G), BAMER compute a minimum energy path from to v. We illutrate BAMER with the example in Figure. In the example network, each link (u, v) i labelled with the (W (u, v), N(u, v)) pair, and each node u i labelled with it ID and C(u). x i the firt node added to S by BAMER, followed by it ucceor z and y in order. BAMER terminate after chooing t, whoe predeceor i z. The minimum energy path are indicated by the dahed link. The minimum expected energy conumption to deliver a packet from to t i 80. BMA- will chooe the path x y t and the expected energy conumption i 82. Without conidering link lo rate, a naive hortet path algorithm (e.g. Dijktra algorithm) will chooe the path z t, incurring an expected energy conumption of General Algorithm for Minimum Energy Routing (GAMER) In Section 4., we preent the BAMER algorithm for finding minimum energy path in the pure end-to-end retranmiion model where no link guarantee per hop reliability through hop-by-hop retranmiion. Thi i in contrat to prior work (BMA) which olved the problem in the idealized model where each link i perfectly reliable. In realitic cenario, we may have to olve the minimum energy path problem in the more general mixed retranmiion model, where different point-to-point link are implemented

5 GAMER (G,, T, C) for each node v V (G) do 2 T (v) φ 3 C(v) 4 C() 0 5 S {} 6 u 7 while S V (G) do 8 for each node v V (G) S do 9 if (u, v) provide per hop reliability 0 if C(u) + N(u, v)w (u, v) < C(v) T (v) T (u) {(u, v)} 2 C(v) C(u) + N(u, v)w (u, v) 3 ele if N(u, v)[c(u) + W (u, v)] < C(v) 4 T (v) T (u) {(u, v)} 5 C(v) N(u, v)[c(u) + W (u, v)] 6 u v V (G) S uch that C(v) i minimum 7 S S {u} Table 2: Peudo code decription of GAMER. Parameter are the ame a in Table. with different link level technologie, or other factor may make ome link unreliable in the preence of inherently reliable link level technologie, etc. In thi ection, we olve the minimum energy path problem in thi mixed retranmiion model with our General Algorithm for Minimum Energy Path (GAMER). GAMER i a further generalization of BAMER, where each individual link may or may not provide per hop reliability. Again, aume that node u precede v in the path from to v, denoted by P(, v), and let P(, u) denote the part of P(, v) between and u. The additional obervation i that if link (u, v) doe not upport hop-by-hop reliability, C (P(, v)) = C (P(, u)) + N(u, v) W (u, v). Baed on thi intuitive undertanding, we preent GAMER in Table 2. Compared with BAMER, the only difference i line 9 2 handling link providing per hop reliability. We next how the optimality of GAMER. We how that Lemma alo hold for GAMER. The cae where (u, v) doe not provide per hop reliability ha been proved in Section 4.. Now conider the cae where (u, v) doe provide per hop reliability. To prove by contradiction, aume that P(, u) i not a minimum energy path from to u, while another path P (, u) i a minimum energy path from to u. We can replace P(, u) in P(, v) with P (, u). The reulted path P (, v) will have an expected energy conumption of C ( P (, v) ) = C ( P (, u) ) + N(u, v) W (u, v) < C (P(, u)) + N(u, v) W (u, v) = C (P(, v)). Thi contradict the fact that P(, v) i a minimum energy path from to v. Lemma 2 and it proof in Section 4. alo hold for GAMER. Thi i eay to verify and we leave the detail to the reader. COROLLARY 2. For each node v V (G), GAMER compute a minimum energy path from to v. To illutrate how GAMER work, let u return to the example in Figure. Now the link from x to t ha been upgraded to upport hop-by-hop retranmiion hence per hop reliability. Thi doe 3 2 u Figure 2: Solid link are reliable and have a weight of, except that link (v, t) ha a weight of 2. Dahed link are free but have an error rate of 3 and do not provide per hop reliability. 4 not change the behavior of traditional hortet path algorithm and BMA. However, GAMER will find the minimum energy path x t and the expected energy conumption goe down from 80 to BAMER for the mixed retranmiion model Although BAMER i motivated by and deigned for the pure end-to-end retranmiion model, it turn out an appropriate preproceing tage will enable BAMER to olve the ame problem in the mixed retranmiion model. To ee why and how, note that GAMER differ from BAMER only in line 9 2 of Table 2, i.e., the cae where link (u, v) provide per hop reliability. Particularly, the only difference that matter i line 2. Note that the right ide of line 2 can be viewed a [C(u) + N(u, v)w (u, v)]. Compared to the right ide of line in Table, we can ee that link (u, v) can be treated a a reliable link that ha a new weight of W (u, v) = N(u, v) W (u, v) and a new N (u, v) =. Therefore, we can preproce link that provide per hop reliability a i decribed above. Then, applying BAMER on the preproceed network graph i provably correct to compute a minimum energy path from to each node in the network. To illutrate how BAMER work in the mixed retranmiion model, we return to the example in Section 4.2. In the preproceing tage, the point-to-point link (x, t) i marked with (48, ) a a reliable link that ha a weight of 48. BAMER i then executed on the preproceed network graph and correctly find the minimum energy path x t. 4.4 Ditributed Algorithm for Minimum Energy Routing () Both BAMER and GAMER are centralized algorithm. In many application, a routing algorithm ha to be implemented a a ditributed routing protocol in a lightweight fahion. However, deigning a lightweight ditributed protocol that can alway find the minimum energy path i not a trivial tak. To ee the tricky part, conider the examplein Figure 2 where an intermediate node u i forwarding packet from three ource,, 2 and 3, to the detination t. There are three route from u to t. The route u v t i totally reliable and the energy conumption i 3. The route via v 2 i the cheapet and ha an expected energy conumption of 4. However, the expected number of end-to-end retranmiion needed to deliver a packet from u to t via v 2 i 6, becaue of the two free but unreliable link in that path. The route u v b t fall in between them. In particular, it ha an expected energy conumption of 8 and the expected number of end-to-end retranmiion needed to deliver a packet along the route i 4. Now, we turn to examine the implication of thee route. The optimal path from to t i 2 u v t, whoe energy conumption i 5. The optimal path from 2 to t i 2 v v 2 b t

6 (G, Nexthop, R, C) x y x y (2, 0, t) /* initialization */ for each node v V (G) do 2 R(v) 3 C(v) 4 R(u) z 5 C(u) 0 t (, 0, ) z (2, 6, t) t (, 0, ) /* periodic route exchange */ 6 for each round of route exchange do (a) (b) 7 broadcat R and C in a route exchange meage M u x (4, 40, z) y (2, 0, t) x (4, 40, z) y (2, 0, t) 8 for each neighbor v do 9 (8, 80, x) 2 Nexthop(w) v z (2, 6, t) t (, 0, ) z (2, 6, t) t (, 0, ) 5 R(w) M v.r(w) (c) (d) 6 ele R(w) N(u, v)m v.r(w) Figure 3: Illutration of. Each node i labelled with a Table 3: running at individual node. (R(t), C(t), Nexthop(t)) tuple. collect a route exchange meage M v from v 3 C(w) M v.c(w) + M v.r(w)n(u, v)w (u, v) 0 4 for each node w V (G) do if link (u, v) provide per hop reliability if M v.c(w) + M v.r(w)n(u, v)w (u, v) < C(w) u v b t, whoe expected energy conumption i 2. Finally, the optimal path from 3 to t i 3 u v 2 b t, whoe expected energy conumption i 4. It i clear that u need to know about every poible path from itelf to t. In the wort cae, the number of poible path can be exponential in the ize of the network. Given that, we here propoe the Ditributed Algorithm for Minimum Energy Routing (), which i lightweight and achieve reaonably good energy efficiency. Unlike BAMER and GAMER which only compute the one-to-all hortet path from a ingle ource to all other node, compute an energy efficient path from each node to every other node. A peudo code decription of i preented in Table 3. u repreent the local node executing. For any node w V (G), Nexthop(w) denote the next hop node that u ue to forward packet to w. R(w) record the expected number of end-to-end tranmiion (including retranmiion) required to deliver a packet from u to w via Nexthop(w). C(w) record the expected energy conumption to deliver a packet from u to w via Nexthop(w). For each detination w, chooe for u the next hop node that minimize the expected energy conumption of delivering a packet from u to w. In the hop-by-hop retranmiion model, can find the minimum energy path to every detination t V a BMA- doe. Becaue every link provide per hop reliability and no end-to-end retranmiion i needed. Therefore, it only remain to optimize the expected energy conumption from the current forwarding node to the detination. Memory requirement at each node i 2 V, in order to tore R(V ) and C(V ). For any node v V, M v denote the route exchange meage broadcat by node v. M v.c(w) and M v.r(w) denote C(w) and R(w) broadcat by node v, repectively. operate in a periodic round by round fahion. During each round of route exchange, R(V ) and C(V ) are exchanged via route exchange meage. In particular, each node broadcat one meage to it neighbor and collect the meage broadcat by each neighbor. LEMMA 3. For any node w V, whenever v i a downtream node on the path from u to w, it ha to be the cae that C(w) > C v(w). C v(w) denote the value of C(w) at node v then. On one hand, for any node w V, C(w) never grow during the execution of. On the other hand, it i clear from line 3 of Table 3 that whenever Nexthop(w) = v, it mut be the cae that C(w) > C v(w). Recurively applying thi rule conclude the proof of Lemma 3. COROLLARY 3. Route generated by are loop free. Baed on the example in Section 4.2, we illutrate in Figure 3 the round by round execution of in finding an energy efficient path from to t. Although we only illutrate a ingle ource-ink pair here, we point out that actually find uch a path for every ource-ink pair in the network. Doe alway find the optimal path? Unfortunately, the anwer ha to be negative. For the examplar network in Figure 2, the path from to t choen by will be 2 u v b t, whoe expected energy conumption i 6. The reaon i that in, each node keep only the minimum energy path from itelf to every other node. For example, v only know the minimum energy path from itelf to t, namely v b t. Conequently, the optimal path from to t cannot be dicovered. We expect to deign more intelligent algorithm that are reaonably lightweight in future work. 5. MULTI-PATH MIN-ENERGY ROUTES In Section 4, we have propoed and proved BAMER and GAMER for computing the minimum energy path for reliable communication in multi-hop wirele network. Interetingly, we here point out that in ome cae a multi-path routing cheme actually minimize the expected energy conumption. Traditionally, multi-path routing i conidered beneficial for improved throughput and reliability [34, 25]. Intuitively, improved throughput and reliability come at the cot of more energy conumption due to the ue of multiple (not necearily dijoint) path imultaneouly. Therefore, it i not urpriing that reearcher have been deigning ingle-path routing algorithm for energy efficient one-to-one communication, a we do in Section 4. Barrett et al [2] tudied the cae where node may either underetimate or overetimate their ditance to the detination. In the preence of uch noiy routing information, they howed that in ome cae multi-path routing may outperform Dijktra hortet path algorithm in term of the total number tranmiion re-

7 t t b b 2 w w 2 x x 2 y y 2 Figure 4: Wirele Multicat Advantage (WMA) quired to uccefully deliver a packet to it detination. Notice that Dijktra hortet path algorithm doe not take into account link lo rate. We here reveal the intereting and counter-intuition fact that even if perfect routing information i given and link lo rate are taken into account, multi-path routing can till potentially reduce the expected energy conumption of one-to-one communication compared with the optimal ingle-path algorithm, BAMER and GAMER. In addition, we then formally analyze the complexity of computing the minimum energy multi-path routing cheme and prove it to be NP-hard. To the bet of our knowledge, thi paper i the firt to formally tudy exploiting multi-path routing in order to reduce energy conumption in one-to-one communication. With an omnidirectional antenna, a ingle wirele tranmiion by a node can be received by every node within it tranmiion range. Thi property of wirele media i referred to a Wirele Multicat Advantage (WMA) [30]. WMA ha been extenively tudied in energy efficient one-to-many communication, e.g. minimum energy broadcat in wirele network [30, 7, 29, 3, 5, 4, 0]. We how that WMA and the ue of multiple path enable u to reduce energy conumption in one-to-one communication over unreliable link a well. Conider the example in Figure 4, where need to communicate with t. Link coming out of have a lo rate of and a weight of. Link coming out of b and b2 are 2 reliable and free. Conider the multi-path routing cheme where every link participate. A packet ent by cannot be delivered to t if and only if all the three link coming out of fail on thi tranmiion. The probability of a ucceful delivery from to t i thu ( 2 )3 = 7. Therefore, the expected energy conumption of a 8 ucceful delivery i 8. On the other hand, it i clear that the expected energy conumption of any minimum energy ingle path i 7 2. In a multi-path routing cheme, an intermediate node may receive multiple copie of the ame packet from uptream node. Before we can proceed to formally analyze multi-path routing cheme, a problem that ha to be anwered i Should the intermediate node forward every copy of the packet? We believe the correct anwer hould be No. Becaue forwarding the ame packet more than once will incur unneceary additional energy conumption at the intermediate node a well a downtream node, without knowing if that really help at all. We here formally analyze the complexity of finding minimum energy multi-path route and prove that it i NP-hard by reducing from the 3-dimenional matching (3DM) problem, which i known to be NP-hard [2] and formally defined a follow. 3-DIMENSIONAL MATCHING (3DM) INSTANCE Set M = {m, m 2,..., m n} W X Y, where W = {w, w 2,..., w q}, X = {x, x 2,..., x q}, and Y = {y, y 2,..., y q} are dijoint et having the ame number q of element. QUESTION Doe M contain a matching, i.e., a ubet M = m m 2 m 3 m 4 Figure 5: Reduction from 3DM {m, m 2,..., m q} M uch that M = q and no two element of M agree in any coordinate? Given an intance of 3DM, we contruct a graph a hown in Figure 5, where node are ditributed into four layer and edge exit only between node in adjacent layer. The exemplar graph in Figure 5 i contructed from the following intance of 3DM. W = {w, w 2}, X = {x, x 2}, Y = {y, y 2}. M = {m, m 2, m 3, m 4}. m = (w, x 2, y 2), m 2 = (w, x, y ), m 3 = (w 2, x 2, y 2), m 4 = (w, x, y 2). The top layer contain only the detination t. In the econd layer, there are three dijoint group of element node, W = {w, w 2,..., w q}, X = {x, x 2,..., x q}, and Y = {y, y 2,..., y q}, repreenting W, X, and Y, repectively. Each element node i connected to t with an edge whoe weight i 0 and error rate i p = e /3q. In the third layer, there are a et M = {m, m 2,..., m n} of triplet node repreenting the n element of M. Each triplet node i adjacent to the three aociated element node. Edge between element node and triplet node have a weight of and an error rate of 0. The bottom layer contain only the ource node, which i adjacent to all triplet node. Edge between triplet node and have a weight of c = (e )q and an error rate of 0. The tranformation i polynomial, and we here how that M contain a 3-dimenional matching of ize q if and only if the minimum expected energy conumption to deliver a packet from to t i c + q p 3q = e2 q e. () We tart with the only if direction. If M contain a matching of ize q, we can route a packet from to t a follow. tranmit the packet to all the q triplet node contained in the matching. Each triplet node in the matching forward the packet to it aociated element node. Each element node forward the packet to t.

8 The energy required to route the packet from to the 3q element node i determinitically c + q. The probability that at leat one of the element node uccefully deliver the packet to t i p 3q. Thu, the expected energy conumption i given by (). We then prove the more tricky if direction. In particular, we how that the cheme decribed in the proof of the only if direction i the only cheme that can uccefully deliver the packet at an expected energy conumption of (). Firt of all, we point out that any routing cheme can be characterized by the the following two parameter. The number of triplet node that participate to forward packet in thi routing cheme, n 0. n 0 n. The number of element node that participate to forward packet in thi routing cheme, q 0. q 0 min(3q, 3n 0). We prove by contradiction, auming that M doe not contain a matching of ize q. Clearly, there exit the following three cae. If n 0 > q, the expected energy conumption i c + n 0 p q 0 > c + q p q 0 > c + q p 3q. If n 0 = q, it ha to be the cae that q 0 < 3q ince we aume that M doe not contain a matching of ize q. Thu, the expected energy conumption i c + q p > c + q q 0 p. 3q If n 0 < q then q 0 min(3q, 3n 0) = 3n 0. The expected energy conumption i thu c + n 0 p q 0 c + n0 p 3n 0. To conclude our proof by contradiction, it only remain to prove that, Note that for any n 0 < q, c + n 0 p > c + q. (2) 3n 0 p3q c+n 0 c+q and are the value of function p 3n 0 p 3q f(x) = c + x p 3x at x = n 0 and x = q, repectively. In order to prove (2), it uffice to prove that f (x) < 0 for any x q o that f(x) i trictly decreaing in [, q], ince n 0 < q. On one hand, for any x, [ p 3x (c + x)p 3x lnp 3] = (lnp 3 ) 2 (c + x)p 3x < 0. On the other hand, p 3x (c + x)p 3x lnp 3 x=q = (e 3q ) 3q [(e )q + q](e 3q ) 3q ln(e 3q ) 3 = e + >. Therefore, for any x uch that x q, p 3x (c + x)p 3x lnp 3 >. It follow that ( c + x p 3x ) = p 3x + (c + x)p 3x lnp 3 ( p 3x ) 2 < 0. Thi complete the proof of (2). 6. EVALUATION We conduct extenive imulation in our empirical tudy in order to anwer the following quetion. Compared with the bet known current cheme, how effectively can our algorithm conerve energy in a variety of network environment? How network parameter affect the performance of exiting algorithm and our? Such parameter include link error rate, value of α, percentage of link upporting hop-by-hop retranmiion, network ize (i.e., node population), and o on. Before we proceed to preent the imulation reult, we tart by decribing ome technical detail of our imulation. In our imulation, 00 node of the ame tranmiion range are ditributed into a 0 0 quare field uniformly at random. Two node are connected if and only if the ditance between them i no larger than their tranmiion range. For each directed link, it link error rate i choen from [0, MaxLER] uniformly at random, where 0 MaxLER repreent the maximum link error rate. Conequently, link (u, v) and link (v, u) may have different error rate. For each parameter etting, 000 uch trial network are generated. In each trial network, we randomly pick a ource node and a detination node. The average energy conumption of the path computed in all 000 trial network i calculated for individual algorithm, repectively. To evaluate the effectivene of our algorithm in conerving energy, we define normalized energy efficiency () of an algorithm to be the ratio of it average energy conumption to that of BAMER and GAMER, ince BAMER and GAMER are guaranteed to find a minimum energy path. For ingle path routing, we compare our algorithm with the bet known BMA algorithm. For multi-path routing, we compare with GRAB [34], a node/link dijoint path [25] clearly conume more energy than the minimum energy ingle path. GRAB claim to be more efficient and flexible than dijoint path in that it forward packet along an interleaved meh, and control the width of the meh by aigning an appropriate credit to each packet. We firt conduct imulation in the end-to-end retranmiion model to compare the energy efficiency of our algorithm and GRAB, ince GRAB aume the unreliable CSMA MAC. Figure 6 demontrate that the energy conumption of GRAB i typically ome order of magnitude larger, in order to achieve a delivery ratio of 95%. For higher link error rate, thi delivery ratio of 95% i not even achievable. Given thi huge performance gap, we only compare with the bet known ingle path routing cheme, BMA, in the equel. 6. Effect of α and link error rate We firt examine the effect of link error rate and α on the energy efficiency of the algorithm we tudy. To fully undertand the behavior of thee algorithm in the general end-to-end retranmiion model, we here invetigate the cae where none of the link upport hop-by-hop retranmiion. Effect of hop-by-hop retranmiion on energy efficiency will be examined later. We conduct extenive imulation for a number of different value of MaxLER, α, and l, and preent the imulation reult in Figure 7. It i clear from Figure 7 that high link error rate generally emphaize the effectivene of our algorithm. Becaue a higher link error rate mean a higher probability of aborting the end-to-end delivery done thu far and retarting a new end-to-end delivery back from the ource. Conequently, the performance of the relatively

9 BMA- BMA BMA- BMA BMA- BMA MaxLER MaxLER MaxLER Figure 7: Effect of α and link error rate on normalized energy efficiency (). Normalization i with repect to GAMER, which find the optimal path. The figure repreent α = 2, 3, 4, repectively. of GRAB alpha=2 alpha=3 alpha= MaxLER Figure 6: Normalized energy efficiency of GRAB. le intelligent BMA algorithm are more ubjective to link error rate. Large α value demontrate the ame effect. Becaue large α value make hort ditance link even cheaper. Conequently, the algorithm tend to chooe path compoed of more and horter link. The more link a packet ha to go through, the more likely that it delivery may fail and abort at ome intermediate link. Thi mean more energy conumption due to delivery abortion and endto-end retranmiion. Another clear meage from Figure 7 i that reaonably large value of l conitently help BMA achieve better performance. Becaue large l value make loy link appear to be prohibitively expenive to BMA. Conequently, BMA prefer le loy link and that reduce the rik of delivery abortion. We alo conduct imulation for l > 4, but typically that doe not help conerve more energy. For legibility, we only preent imulation reult for l 4. We will ee the reaon underlying thi deciion in later ection. Finally, we point out that perform conitently better than BMA in the end-to-end retranmiion model. 6.2 Effect of hop-by-hop retranmiion We have dicued in Section 6. that large l value help BMA conerve energy by avoiding loy link. Clearly there ha to be a cot to thi trick. For example, conider the hop-by-hop retranmiion model. l = find minimum energy path, while larger value of l may give u le energy efficient path. Intuitively, there hould be ome correlation between the optimal value of l and the percentage of link upporting hop-by-hop retranmiion, which i denoted by UP Grate. We here reveal thi correlation by conducting extenive imulation for a number of different value of l, UP Grate, and MaxLER. We aume a moderate etting of α = 2, which i in favor of BMA algorithm a i hown in Figure 7. Simulation reult preented in Figure 8 lead u to the following concluion. Firt, large l value perform better in the preence of a low UP Grate, while mall l value perform better if a ignificant portion of link upport hop-by-hop retranmiion. Thi obervation i conitent with the fact that BMA- find the optimal olution in the hop-byhop model where all link upport hop-by-hop retranmiion. Second, imulation reult demontrate that l > 4 doe not help BMA. Depending on UP Grate and MaxLER, l = 3 or l = 4 turn out to be the bet choice. Third, by comparing different MaxLER value, we can ee that high link error rate are in favor of large value of l. Thi further verifie our previou undertanding of the reaon why large l value help BMA in the end-to-end retranmiion model: peimitic etimate (i.e., large l value) better help BMA avoid high rik link (i.e., high error rate). Finally, even with the optimal etting of l = 4 and a moderate α = 2, BMA till conume more energy than BAMER and GAMER by up to 43%, and conume more energy than by up to 22%. Hop-by-hop retranmiion conitently help. In fact, we have dicued that i able to find minimum energy path in the hop-by-hop retranmiion model, and thi i verified by the imulation reult in Figure Effect of network ize A we have dicued in Section 6., the more link a packet ha to go through, the more likely that it delivery may abort at ome intermediate link. Since a larger network ize (i.e., node population) lead to longer path, the rik of delivery abortion will go up with network ize. Accordingly, BMA need to be more peimitic on etimating link error rate o that it will further avoid loy link to improve energy efficiency in the preence of increaed network ize. We here preent an empirical invetigation of the correlation between network ize and l, a well a the effect of network ize on the energy efficiency of and BMA. For conitency, we till aume that α = 2. We conduct extenive imulation for a number of different value of network ize, l, and UP Grate. Simulation reult are preented in Figure 9. A i hown in Figure 9, increaed network ize require larger value of l. Meanwhile, increaed network ize alo reult in a lower energy efficiency of BMA. For example, when we have 30 node in the network, l = 3 i the bet performing etting and it

10 Figure 8: Effect of hop-by-hop retranmiion on normalized energy efficiency (). Normalization i with repect to GAMER, which find optimal path. The figure repreent MaxLER = 0., 0.4, 0.7, repectively Figure 9: Normalized energy efficiency variation with change in network ize. The figure repreent 30, 50, 250 node, repectively. conume up to 34% more energy than BAMER and GAMER, and conume up to 28% more energy than. When we have 250 node, l = 5 i generally the bet choice, which conume up to 60% more energy than BAMER and GAMER, and conume up to 35% more energy than. Thi fact draw our attention to an even more challenging problem of BMA: without a priori knowledge of network ize, how hould BMA predetermine it optimal etting of l? A i demontrated by the imulation reult, inappropriate l value can reult in ignificantly lower energy efficiency of BMA, while our algorithm do not have thi problem. For example, if BMA expect the network ize to be 30 while the actual ize i 250, it will conume up to 2.7 time the energy conumption of BAMER and GAMER, and conume up to 2. time that of. 7. CONCLUSIONS In thi paper, we tudy the problem of minimum energy routing for reliable one-to-one communication in the preence of loy link. Banerjee and Mira [] olved the problem in the hop-byhop retranmiion model, where each link i aumed to upport link layer hop-by-hop retranmiion and guarantee reliable delivery. However, link layer retranmiion actually cannot guarantee reliable delivery, due to variou reaon. In the end-to-end retranmiion model where ome link in the communication path i unreliable, we rely on TCP-like tranport protocol to initiate endto-end retranmiion. We firt tudy the pure end-to-end retranmiion model where none of the link guarantee per hop reliability, and then proceed to tudy the more general mixed retranmiion model where ome link may guarantee reliable delivery while the other may not. The BAMER and GAMER algorithm are deigned for computing minimum energy path in both model. The hop-by-hop model and the pure end-to-end model are jut pecial cae of the mixed model, o BAMER and GAMER can be ued to find minimum energy path in any network configuration. For implementation in many practical cenario, we alo propoe a lightweight ditributed routing protocol,, which can be ued for energy efficient routing in any network configuration a well. i able to find minimum energy path in the hop-byhop model, and imulation reult demontrate that alo effectively improve energy efficiency over the bet known exiting technique in the general mixed model. Through extenive imulation, we alo carefully examine the effect of a number of network parameter on the performance of our algorithm a well a exiting technique. Thi empirical tudy further enhance our undertanding of energy efficient reliable communication in the preence of loy link. Traditionally, multi-path routing have been utilized to improve throughput or reliability, poibly at the cot of increaed energy conumption. Our another intereting finding i that, in ome cae multi-path routing may reduce the expected energy conumption in the preence of loy link. We formally analyze the problem of finding the minimum energy multi-path routing cheme and prove that it i actually NP-hard. To the bet of our knowledge, thi paper i the firt to formally invetigate the potential of multi-path routing on energy conervation. 8. ACKNOWLEDGMENTS We thank the hepard of our paper, Stephan J. Eidenbenz, a well a the anonymou reviewer for detailed comment that helped improve the quality of the paper. 9. REFERENCES [] S. Banerjee and A. Mira. Minimum energy path for reliable communication in multi-hop wirele network. In ACM MobiHoc, page 46 56, June 2002.