Multirate Anypath Routing in Wireless Mesh Networks

Transcription

1 Multrate Anypath Routng n Wreless Mesh Networks Rafael Laufer an Leonar Klenrock Computer Scence Department Unversty of Calforna at Los Angeles August 9, Techncal Report UCLA-CSD-TR arxv:9.6v [cs.ni] 9 Sep Abstract In ths paper, we present a new routng paragm that generalzes opportunstc routng n wreless mesh networks. In multrate anypath routng, each noe uses both a set of next hops an a selecte transmsson rate to reach a estnaton. Usng ths rate, a packet s broacast to the noes n the set an one of them forwars the packet on to the estnaton. To ate, there s no theory capable of jontly optmzng both the set of next hops an the transmsson rate use by each noe. We brge ths gap by ntroucng a polynomal-tme algorthm to ths problem an prove the proof of ts optmalty. The propose algorthm runs n the same runnng tme as regular shortest-path algorthms an s therefore sutable for eployment n lnk-state routng protocols. We conucte experments n a.b testbe network, an our results show that multrate anypath routng performs on average % an up to 6. tmes better than anypath routng wth a fxe rate of Mbps. If the rate s fxe at Mbps nstea, performance mproves by up to one orer of magntue. I. INTRODUCTION The hgh loss rate an ynamc qualty of lnks make routng n wreless mesh networks extremely challengng [], []. Anypath routng has been recently propose as a way to crcumvent these shortcomngs by usng multple next hops for each estnaton [] []. Each packet s broacast to a forwarng set compose of several neghbors, an the packet must be retransmtte only f none of the neghbors n the set receve t. Therefore, whle the lnk to a gven neghbor s own or performng poorly, another nearby neghbor may receve the packet an forwar t on. Ths s n contrast to sngle-path routng where only one neghbor s assgne as the next hop for each estnaton. In ths case, f the lnk to ths neghbor s not performng well, a packet may be lost even though other neghbors may have overhear t. Exstng work on anypath routng has focuse on wreless networks that use a sngle transmsson rate. Ths approach, albet straghtforwar, presents two major rawbacks. Frst, usng a sngle rate over the entre network unerutlzes avalable banwth resources. Some lnks may perform well at a hgher rate, whle others may only work at a lower rate. Seconly an most mportantly, the network may become sconnecte at a hgher bt rate. We prove expermental measurements from a.b testbe whch show that ths phenomenon s not uncommon n practce. The key problem s We use the term anypath rather than opportunstc routng, snce opportunstc routng s an overloae term also use for opportunstc contacts []. that hgher transmsson rates have a shorter rao range, whch reuces network ensty an connectvty. As the bt rate ncreases, lnks becomes losser an the network eventually gets sconnecte. Therefore, n orer to guarantee connectvty, sngle-rate anypath routng must be lmte to low rates. In multrate anypath routng, these problems o not exst; however, we face atonal challenges. Frst, we must fn not only the forwarng set, but also the transmsson rate at each hop that jontly mnmzes ts cost to a estnaton. Seconly, loss probabltes usually ncrease wth hgher transmsson rates, so a hgher bt rate oes not always mprove throughput. Fnally, hgher rates have a shorter rao range an therefore we have a fferent connectvty graph for each rate. Lower rates have more neghbors avalable for ncluson n the forwarng set (.e., more spatal versty) an less hops between noes. Hgher rates have less spatal versty an longer routes. Fnng the optmal operaton pont n ths traeoff s the focus of ths paper. We thus aress the problem of fnng both a forwarng set an a transmsson rate for every noe, such that the overall cost of every noe to a partcular estnaton s mnmze. We call ths the shortest multrate anypath problem. To our knowlege, ths s stll an open problem [], [], [] an we beleve our algorthm s the frst soluton for t. We ntrouce a polynomal-tme algorthm to the shortest multrate anypath problem an present a proof of ts optmalty. Our soluton generalzes Djkstra s algorthm for the multrate anypath case an s applcable to lnk-state routng protocols. One woul expect that the runnng tme of such an algorthm s longer than a shortest-path algorthm. However, we show that t has the same runnng tme as the corresponng shortestpath algorthm, beng sutable for mplementaton at current wreless routers. We also ntrouce a novel routng metrc that generalzes the expecte transmsson tme (ETT) metrc [9] for multrate anypath routng. For the performance evaluaton, we conucte experments n an -noe.b wreless testbe of embee Lnux evces. Our results reveal that the network becomes sconnecte f we fx the transmsson rate at,., or Mpbs. A sngle-rate routng scheme therefore performs poorly n ths case, snce Mbps s the only rate at whch the network s fully connecte. We show that multrate anypath routng mproves the en-to-en expecte transmsson tme by % on average an by up to 6. tmes compare to sngle-rate anypath routng

2 at Mbps, whle stll mantanng network connectvty. The performance s even hgher over the sngle-rate case at Mbps, wth an average gan of a factor of. an a maxmum gan of a factor of.. The remaner of paper s organze as follows. Secton II revews the basc concepts of anypath routng an our network moel. In Secton III, we ntrouce multrate anypath routng an the propose routng metrc. Secton IV presents the multrate anypath algorthm an proves ts optmalty. Secton V reveals our expermental results, showng the benefts of multrate over sngle-rate anypath routng. Secton VI presents the relate work n anypath routng. Fnally, conclusons are presente n Secton VII. leavng each noe. We efne ths unon of paths between two noes as an anypath. In the fgure, the anypath shown n bol s compose by the unon of fferent paths between a source s an a estnaton. Depenng on the choce of each forwarng set, fferent paths are nclue n or exclue from the anypath. At every hop, only a sngle noe of the set forwars the packet on. Consequently, every packet from s traverses only one of the avalable paths to reach. We show a path possbly taken by a packet usng a ashe lne. Succeeng packets, however, may take completely fferent paths; hence the name anypath. The path taken s etermne on-the-fly, epenng on whch noes of the forwarng sets successfully receve the packet at each hop. II. ANYPATH ROUTING In ths secton we revew the theory of anypath routng ntrouce by Zhong et al. [6] an Dubos-Ferrère et al. []. The man contrbutons of the paper are presente later n Sectons III an IV. s A. Overvew In classc wreless network routng, each noe forwars a packet to a sngle next hop. As a result, f the transmsson to that next hop fals, the noe nees to retransmt the packet even though other neghbors may have overhear t. In contrast, n anypath routng, each noe broacasts a packet to multple next hops smultaneously. Therefore, f the transmsson to one neghbor fals, an alternatve neghbor who receve the packet can forwar t on. We efne ths set of multple next hops as the forwarng set an we usually use to represent t throughout the paper. A fferent forwarng set s use to reach each estnaton, n the same way a stnct next hop s use for each estnaton n classc routng. When a packet s broacast to the forwarng set, more than one noe may receve the same packet. To avo unnecessary uplcate forwarng, only one of these noes shoul forwar the packet on. For ths purpose, each noe n the set has a prorty n relayng the receve packet. A noe only forwars a packet f all hgher prorty noes n the set fale to o so. Hgher prortes are assgne to noes wth shorter stances to the estnaton. As a result, f the noe wth the shortest stance n the forwarng set successfully receve the packet, t forwars the packet to the estnaton whle others suppress ther transmsson. Otherwse, the noe wth the secon shortest stance forwars the packet, an so on. A relable anycast scheme [] s necessary to enforce ths relay prorty. We talk more about ths n Secton II-B. The source keeps rebroacastng the packet untl someone n the forwarng set receves t or a threshol s reache. Once a neghbor n the set receves the packet, ths neghbor repeats the same proceure untl the packet s elvere to the estnaton. Snce we now use a set of next hops to forwar packets, every two noes wll be connecte through a mesh compose of the unon of multple paths. Fgure epcts ths scenaro where each noe uses a set of neghbors to forwar packets. The forwarng sets are efne by the multple bol arrows Fgure. An anypath connectng noes s an s shown n bol arrows. The anypath s compose of the unon of paths between the two noes. Every packet sent from s traverses one of these paths to reach, such as the path shown wth a ashe lne. Dfferent packets may traverse fferent paths, epenng on whch noes receve the forware packet at each hop; hence the name anypath. B. System Moel an Assumptons In orer to support the pont-to-multpont lnks use n anypath routng, we moel the wreless mesh network as a hypergraph. A hypergraph G = (V, E) s compose of a set V of vertces or noes an a set E of hypereges or hyperlnks. A hyperlnk s an orere par (, ), where V s a noe an s a nonempty subset of V compose of neghbors of. For each hyperlnk (, ) E, we have a elvery probablty p an a stance. If the set has a sngle element j, then we just use j nstea of n our notaton. In ths case, p j an j enote the lnk elvery probablty an stance, respectvely. The hyperlnk elvery probablty p s efne as the probablty that a packet transmtte from s successfully receve by at least one of the noes n. One woul expect that the recept of a packet at each neghbor s correlate ue to nose an nterference. However, we conucte experments whch suggest that the loss of a packet at fferent recevers occur nepenently n practce, whch s also consstent wth other stues [], []. We show these results n Secton V. Wth the assumpton of nepenent losses, p s p = j ( p j ). () Prevously propose MAC protocols have been esgne to guarantee the relay prorty among the noes n the forwarng set [], [], []. Such protocols can use fferent strateges

3 for ths purpose, such as tme-slotte access, prortze contenton an frame overhearng. Relable anycast s an actve area of research [] an we assume that such mechansm s n place to make sure that the relayng prorty s respecte. The etals of the MAC, however, are abstracte from the routng layer. Practcal routng protocols only ncorporate the elvery probabltes nto the routng metrc n orer to abstract from the MAC etals [9], [] an we take the same approach. The only MAC aspect that s mportant s the effectveness of the relayng noe selecton. As long as the relayng noe s actually the one wth the shortest stance to the estnaton, there shoul be no sgnfcant mpact on the routng performance. C. Anypath Cost We are ntereste n calculatng the anypath cost from a noe to a gven estnaton va a forwarng set. The anypath cost D s efne as D = + D, whch s compose of the hyperlnk cost from to an the remanng-anypath cost D from to the estnaton. The hyperlnk cost epens on the routng metrc use. Most prevous works on anypath routng have aopte the expecte number of anypath transmssons (EATX) as the routng metrc [], [6], []. The EATX s a generalzaton of the unrectonal ETX metrc [], whch s efne as j = /p j. The stance j for ETX represents the expecte number of transmssons necessary for a packet sent by to be successfully receve by j. For EATX, the stance s efne as = /p, whch s the average number of transmssons necessary for at least one noe n to correctly receve the transmtte packet. The remanng-anypath cost D s ntutvely efne as a weghte average of the stances of the noes n the forwarng set as D = j w j D j, wth j w j =, () where the weght w j n () s the probablty of noe j beng the relayng noe. For example, let = {,,..., n} wth stances D D... D n. We refer to the probablty p j smply by p j for convenence. Noe j wll be the relayng noe only when t receves the packet an none of the noes closer to the estnaton receves t, whch happens wth probablty p j ( p j )( p j )...( p ). The weght w j s then efne as j p j k= ( p k ) w j = ( p j ), () j wth the enomnator beng the normalzng constant. As an example, conser the network epcte n Fgure. The stance va n Fgure (a) s calculate as D = + D (/) + (/)(/) = + ( /)( /) ( /)( /) =. +. =.. () One woul expect that ang an extra noe to the forwarng set s always benefcal because t ncreases the number of possble paths a packet can take. However, ths s not always true, as shown n Fgure (b). The anypath stance va = {j} s D = + D =. +.6 = 6.. On one han, usng nstea of reuces the hyperlnk cost, that s,. On the other han, the extra noe ncreases the remanng anypath cost, that s, D D. If the ncrease D D s hgher than the ecrease, ang ths extra noe s not worthy snce the total cost to reach the estnaton ncreases. The ntuton here s that when noe j s the only one n that receve the packet, t s cheaper to retransmt the packet to one of the two noes n an take a shorter path from there than to take the long path va noe j. (a) j 9 (b) j 9 Fgure. An anypath cost calculaton example. The weght of each lnk s the expecte number of transmssons (ETX), whch s the nverse of the lnk elvery probablty. The anypath cost n (a) s lower than the cost n (b). Once the cost of an anypath s efne, t s of nterest to fn the anypath wth the lowest cost to the estnaton, that s, the shortest anypath. Ths s calle the shortest-anypath problem []. Interestngly enough, the shortest anypath wll always have an equal or lower cost than the shortest sngle path. Ths s a rect consequence of the efnton of an anypath as a set of paths. Among all possble anypaths between two noes, we also have the anypath compose only of the path wth the shortest ETX. Therefore, f we are to choose the shortest anypath among all these possbltes, we know for sure that ts cost can never be hgher than the cost of the shortest sngle path. III. MULTIRATE ANYPATH ROUTING Prevous work on anypath routng focuse on a sngle bt rate [] []. Such an assumpton, however, conserably unerutlzes avalable banwth resources. Some hyperlnks may be able to sustan a hgher transmsson rate, whle others may only work at a lower rate. To ate, the problem of how to select the transmsson rate for anypath routng s stll open [], []. We prove a soluton to ths problem an ncorporate the multrate capablty nherent n IEEE. networks nto anypath routng. In ths case, beses selectng a set of next hops to forwar packets, a noe must also select

4 one among multple transmsson rates. For each estnaton, a noe then keeps both a forwarng set an a transmsson rate use to reach ths set. As a result, every two noes wll be connecte through a mesh compose of the unon of multple paths, wth each noe transmttng at a selecte rate. Fgure epcts the scenaro where noes use a selecte bt rate to forwar packets to a set of neghbors. We efne ths unon of paths between two noes, wth each noe usng a potentally fferent bt rate as a multrate anypath. In the fgure, assume that a packet s sent from s to over the multrate anypath. Only one of the avalable paths s traverse epenng on whch noes successfully receve the packet at each hop. We show a path possbly taken by the packet usng a ashe lne. We use fferent ash lengths to represent the fferent transmsson rates use by each noe. A shorter ash represents a shorter tme to sen a packet, hence a hgher transmsson rate. Succeeng packets may take completely fferent paths wth other transmsson rates along ts way. s Fgure. A multrate anypath connectng noes s an s shown n bol arrows. Every packet sent from s traverses a path to reach, such as the path shown wth ashe lnes. Dfferent ash lengths represent the fferent bt rates use by each noe, wth a shorter ash for hgher rates. In orer to support multrate, we must exten the system moel n Secton II-B. Let R be the set of avalable bt rates that noes can use to transmt ther packets. For each hyperlnk (, ) E, we now have a elvery probablty p (r) an a stance (r) assocate wth each transmsson rate r R. In real wreless networks, we usually have fferent elvery probabltes an stances for each transmsson rate, whch justfes ths moel extenson. The EATX metrc escrbe n Secton II-C was orgnally esgne conserng that noes transmt at a sngle bt rate. To account for multple bt rates, we ntrouce the expecte anypath transmsson tme (EATT) metrc. For EATT, the hyperlnk stance (r) for each rate r R s efne as (r) = s p (r) r, () where p (r) s the hyperlnk elvery probablty efne n (), s s the maxmum packet sze, an r s the bt rate. The stance (r) s bascally the tme t takes to transmt a packet of sze s at a bt rate r over a lossy hyperlnk wth elvery probablty p (r). The EATT metrc s a generalzaton of the expecte transmsson tme (ETT) metrc [9] commonly use n sngle-path wreless routng. Note that for each bt rate r R, we have a fferent elvery probablty p (r), whch usually ecreases for hgher rates. Ths behavor mposes a traeoff; a hgher bt rate ecreases the tme of a sngle packet transmsson (.e., s/r ecreases), but t usually ncreases the number of transmssons requre for a packet to be successfully receve (.e., /p (r) ncreases). The remanng-anypath cost D (r) now also epens on the transmsson rate, snce the elvery probabltes change for each rate. Snce both the hyperlnk stance an the remanng anypath cost epen on the bt rate, noe has a fferent anypath cost D (r) = (r) + D(r) for each forwarng set an for each transmsson rate r R. The remanng-anypath cost D (r) for a rate r R s efne as D (r) = j w (r) j D j, wth w (r) j =, (6) j where the weght w (r) j n (6) s the probablty of noe j beng the relayng noe an D j = mn r R D (r) j s the shortest stance from noe j to the estnaton among all rates. We refer to the probablty p (r) j smply by p (r) j for convenence. The weght w (r) j s then efne as w (r) j = p (r) j j k= [ p (r) k ] [ ]. () p (r) j j We aress the problem of fnng both the forwarng set an the transmsson rate that mnmze the overall cost to reach a partcular estnaton. We call ths the shortest multrate anypath problem, whch generalzes the shortestanypath problem [] for the multrate scenaro. Interestngly, the shortest multrate anypath wll always have equal or lower cost than the shortest sngle path. Among all possble multrate anypaths between two noes, we also have the sngle path wth the shortest ETT. As a result, the cost of the shortest multrate anypath can never be hgher than the cost of the shortest path. Lkewse, ue to the same argument, the shortest multrate anypath wll also have equal or lower cost than any shortest anypath usng a sngle transmsson rate. IV. FINDING THE SHORTEST MULTIRATE ANYPATH In ths secton we ntrouce the propose shortest-anypath algorthms. In Secton IV-A, we present the Shortest Anypath Frst (SAF) algorthm use n a sngle-rate network wth the EATX metrc. A smlar sngle-rate algorthm along wth the same optmzaton was also propose by Chachulsk []. We, however, erve ths algorthm nepenently an later n Secton IV-B we ntrouce a generalzaton of ths algorthm for multple rates. Surprsngly, the Shortest Multrate Anypath Frst (SMAF) algorthm has the same runnng tme as a shortest sngle-path algorthm for multrate. We only show the proof of optmalty of the SMAF algorthm, snce by efnton ths mples the optmalty of the SAF algorthm.

5 (a) (b) (c) () (e) (f) (g) (h) Fgure. Executon of the Shortest Anypath Frst (SAF) algorthm from every noe to. The weght of each lnk s the expecte number of transmssons (ETX), whch s the nverse of the lnk elvery probablty. (a) The stuaton just after the ntalzaton. (b) (h) The stuaton after each successve teraton of the algorthm. Part (h) shows the stuaton after the last noe s settle. A. The Sngle-Rate Case We now present the Shortest Anypath Frst algorthm use n the smpler sngle-rate scenaro. Gven a graph G = (V, E), the algorthm calculates the shortest anypaths from all noes to a estnaton. For every noe V we keep an estmate D, whch s an upper-boun on the stance of the shortest anypath from to. In aton, we also keep a forwarng set F for every noe, whch stores the set of noes use as the next hops to reach. Fnally, we keep two ata structures, namely S an Q. The S set stores the set of noes for whch we alreay have a shortest anypath efne. We store each noe V S for whch we stll o not have a shortest anypath n a prorty queue Q keye by ther D values. SHORTEST-ANYPATH-FIRST(G, ) for each noe n V o D F D S 6 Q V whle Q o j EXTRACT-MIN(Q) 9 S S {j} for each ncomng ege (, j) n E o F {j} f D > D j then D + D F Lnes ntalze the state varables D an F an lne sets to zero the stance from noe to tself. Lnes 6 ntalze the S an Q ata structures. Intally, we o not have the shortest anypath from any noe, so S s ntally empty an thus Q contans all the vertces n the graph. As n the shortest-path algorthm, the Shortest Anypath Frst algorthm s compose of V rouns, ctate by the number of elements ntally n Q. At each roun, the EXTRACT-MIN proceure extracts the noe wth the mnmum stance to the estnaton from Q. Let ths noe be j. At ths pont, j s settle an nserte nto S, snce the shortest anypath from j to the estnaton s now known. For each ncomng ege (, j) E, we check f the stance D s larger than the stance D j. If that s the case, then noe j s ae to the forwarng set F an the stance D s upate. Fgure shows the executon of Shortest Anypath Frst algorthm usng the EATX metrc. We see n Fgure (a) the graph rght after the ntalzaton. Fgures (b) (h) show each teraton of the algorthm. At each step, the value nse a noe presents the stance D from that noe to the estnaton an the arrows n bolface present the shortest anypath to. Noes wth two crcles are the settle noes n S. The graph n Fgure (h) shows the result of SAF algorthm rght after settlng the last noe. The runnng tme of the Shortest Anypath Frst algorthm epens on how Q s mplemente. Assumng that we have a Fbonacc heap, the cost of each of the V EXTRACT-MIN operatons n lne takes O(log V ), wth a total of O(V log V ) aggregate tme. The runnng tme to calculate both an D n lne epens on the sze of ; however, f we store atonal state, t can be reuce to a constant tme, as we show n the next paragraph. The for loop of lnes takes O(E) aggregate tme an as a result the total complexty of the algorthm s O(V log V +E), whch s the same complexty of Djkstra s algorthm. To reuce the runnng tme of the calculaton of the noe stance D n lne to O(), we keep two atonal state varables for each noe, namely α an β. In α, we store an n β we store α + j p j D j j F β k= ( p k ), () j F ( p j ). (9) Suppose now that we must upate the forwarng set F to nclue a new noe n, that s, F F {n}. Frst, we upate

6 the state varables α an β to α α + β p n D n β β ( p n ), () an fnally we upate D wth D α. () β B. The Multrate Case We now generalze the SAF algorthm to support multple transmsson rates, ntroucng the Shortest Multrate Anypath Frst (SMAF) algorthm. For each noe V, we now keep a fferent stance estmate D (r) for every rate r R. The estmate D (r) s an upper-boun on the stance of the shortest anypath from to usng transmsson rate r. In aton, we also keep ts corresponng forwarng set F (r), whch stores the set of next hops use for to reach usng r. We use D an F wthout the ncate rates to store the mnmum stance estmate among all rates an ts corresponng forwarng set, respectvely. We also keep a transmsson rate T for every noe, whch stores the optmal rate use to reach. SHORTEST-MULTIRATE-ANYPATH-FIRST(G, ) for each noe n V o D F T NIL for each rate r n R F (r) D 9 S Q V whle Q o j EXTRACT-MIN(Q) S S {j} for each ncomng ege (, j) n E o for each rate r n R 6 o D (r) 6 o F (r) {j} f D (r) > D j then D (r) 9 F (r) (r) f D > D (r) + D(r) then D D (r) F F (r) T r The key ea of the SMAF algorthm s that each noe V has an nepenent stance estmate D (r) for each rate r R an we keep the mnmum of these estmates as the noe stance D. At each roun of the whle loop, the noe wth the mnmum stance from Q s settle. Let ths noe be j. For each ncomng ege (, j) E, we check for every rate r R f the stance D (r) s larger than the stance D j of the noe just settle. If that s the case, then noe j s ae to the forwarng set F (r) of that specfc rate an stance D (r) s upate accorngly. If the new stance D (r) s shorter than the noe stance D, we upate the noe stance D as well as the forwarng set F an transmsson rate T to reflect the new mnmum. The runnng tme of the Shortest Multrate Anypath Frst algorthm also epens on the mplementaton of Q. The ntalzaton n lnes takes O(V R) tme. Assumng that we have a Fbonacc heap, the EXTRACT-MIN operatons n lne take a total of O(V log V ) aggregate tme. We assume that the stance calculaton of (r) an D (r) n lne s optmze to take a constant tme, as shown n Secton IV-A. As a result, the for loop n lnes takes O(ER) aggregate tme. The total runnng tme s therefore O(V log V + (E + V )R), whch s O(V log V + ER) f all noes are able to reach the estnaton. Ths s the same runnng tme of the shortest sngle-path algorthm for multple rates. Compare to the SAF algorthm, the SMAF algorthm allows noes to take avantage of ther multple transmsson rates at the cost of just a small ncrease n the runnng tme. In orer to prove the optmalty of the algorthm, we frst ntrouce fve lemmas that show a few propertes of multrate anypath routng. We use δ (r) as the stance of the shortest multrate anypath from a noe to the estnaton, when transmts at a fxe rate r R. Lkewse, φ (r) represents the corresponng forwarng set use n ths multrate anypath. We use δ wthout the ncate rate to represent the stance of the shortest multrate anypath from to va the optmal forwarng set φ an optmal transmsson rate ρ R. That s, δ = mn r R δ (r), ρ = arg mn r R δ (r), an φ = φ (ρ). We use D as the stance of a partcular multrate anypath from to, but not necessarly the shortest one. The proof for each of these lemmas s avalable n Appenx A. Lemma : For a fxe transmsson rate, let D be the stance of a noe va forwarng set an let D be the stance va forwarng set = {n}, where D n D j for every noe j. We have D D f an only f D D n. We use Lemma for the comparsons n lne of the SAF algorthm an n lne of the SMAF algorthm. By ths lemma, f the stance D va s larger than the stance D n of a neghbor noe n, wth D n D j for all j, then the stance D va = {n} s always smaller than D. That s, t s always benefcal to nclue noe n n the forwarng set n orer to obtan a shorter stance to the estnaton. Lemma : The shortest stance δ of a noe s always larger than or equal to the shortest stance δ j of any noe j n the optmal forwarng set φ. That s, we have δ δ j for all j φ. Lemma guarantees that f a noe uses another noe j n ts optmal forwarng set φ, then stance δ can never be smaller than δ j. Ths s equvalent to the restrcton that all weghts n the graph must be nonnegatve n Djkstra s algorthm.

7 Lemma : For any transmsson rate, f a noe uses a noe n n ts optmal forwarng set φ an δ = δ n, we can safely remove n from φ wthout changng δ. The lnk (, n) s sa to be reunant. By Lemma, f the stances δ = δ n of two noes an n are the same, then the stance δ va forwarng set φ s the same as the stance va forwarng set φ {n}. That s, the stance of noe oes not change f t uses n n ts forwarng set or not. Lemma : If the shortest stances from the neghbors of a noe to a gven estnaton are δ δ... δ n, then φ (r) s always of the form φ (r) = {,,..., k}, for some k {,,..., n}. Accorng to Lemma, the best forwarng set φ (r) for transmsson rate r R s a subset of neghbors wth the shortest stances to the estnaton. That s, gven a set of neghbors wth stances δ δ... δ n, the best forwarng set φ (r) when usng rate r R s always one of {}, {, }, {,, },..., {,,..., n}. As a result, forwarng sets wth gaps between the neghbors, such as {, } or {, }, can never yel the shortest stance to the estnaton. Ths property s the key factor that allows us to reuce the complexty of the propose algorthms from exponental to polynomal tme. For n neghbors, we o not have to test every one of the n possble forwarng sets. Instea, we only nee to check at most n forwarng sets. Lemma : For a gven transmsson rate r R, assume that φ (r) = {,,..., n} wth stances δ δ... δ n. If D j s the stance from noe usng transmsson rate r va forwarng set {,,..., j}, for j n, then we always have D D... Dn = δ (r). Lemma explans another mportant property necessary for the SMAF algorthm to converge. Assumng now that the best forwarng set φ (r) for transmsson rate r R s efne as φ (r) = {,,..., n} wth stances δ δ... δ n, the stance D monotoncally ecreases as we use each of the forwarng sets {}, {, }, {,, },..., {,,..., j}. We now present the proof of optmalty of the algorthm. Theorem : Optmalty of the algorthm. Let G = (V, E) be a weghte, recte, graph an let be the estnaton. After runnng the Shortest Multrate Anypath Frst algorthm on G, we have D = δ for all noes V. Proof: Ths proof s smlar to the proof of Djkstra s algorthm [6]. We show that for each noe s V, we have D s = δ s at the tme s s ae to S. For the purpose of contracton, let s be the frst noe ae to S for whch D s δ s. We must have s because s the frst noe ae to S an D = δ = at that tme. ust before ang s to S, we also have that S s not empty, snce s an S must contan at least. We assume that there must be a multrate anypath from s to, otherwse D s = δ s =, whch contracts our ntal assumpton that D s δ s. If there s at least one multrate anypath, there s a shortest multrate anypath α from s to. Let us conser a cut (V S, S) of α, such that we have s V S an S, as shown n Fgure. Let the set be compose of noes n V S that have an outgong lnk to a noe n S. Lkewse, let the set K be compose of noes n S that have an ncomng lnk from a noe n V S. s V S Fgure. The shortest multrate anypath α from s to. Set S must be nonempty before noe s s nserte nto t, snce t must contan at least. We conser a cut (V S, S) of α, such that we have s V S an S. Noes s an are stnct but we may have no hyperlnks between s an, such that = {s}, an also between K an, such that K = {}. Wthout loss of generalty, assume that noe has the shortest stance to among all noes n V S. That s, δ δ j for all j V S. We clam that every ege leavng noe must necessarly cross the cut (V S, S). Thus, for every ege (, j) leavng noe, we must have j S. To prove ths clam, let us assume that noe has an ege (, j) to another noe j V S. By Lemma, we know that n ths case we must have δ δ j. However, snce we assume that noe has the shortest stance n V S, then δ δ j an such an ege coul only exst f δ = δ j. By Lemma, we know that f δ = δ j then the lnk (, j) s reunant an we can safely remove t from the multrate anypath wthout changng ts stance. As a result, for every ege (, j) we must have j S. Fgure shows ths stuaton where noe only has lnks to noes n S. Atonally, we clam that the noes n S were settle n ascenng orer of stance. That s, f δ j < δ k then noe j was settle before noe k. Snce noe has the shortest stance to among all noes n V S, settlng s before mples that s s settle out of orer. For the purpose of contracton, let s be the frst noe settle out of orer. Ths s an assumpton whch s nepenent from the ntal assumpton that D s δ s. We now clam that D = δ at the tme s s nserte nto S. To prove ths clam, notce that K S. Snce s s the frst noe for whch D s δ s when t s ae to S, then we must have D k = δ k, for every k K. Let φ K be the forwarng set use n the shortest multrate anypath from to usng the optmal transmsson rate ρ R. By Lemma, φ s compose of the neghbors of wth the shortest stances to. Assume that φ = {,,..., j} wth δ δ... δ j. Snce s s the frst out-of-orer noe, we know that the noes n S were settle n orer. Therefore, noe was settle before noe, whch was settle before noe, an so on. At the tme noe s settle, the forwarng set F (ρ) s ntalze to = {}. When noe s settle, there s no nee to check F (ρ) the forwarng set {}. By Lemma, ths forwarng set s never optmal so we just check the set {, }. By Lemma, usng {, } always proves a shorter stance than usng K S

8 .. Mbps. Mbps. Mbps. Mbps Delvery probablty.6.. (a) Lnk n rank orer (b) Fgure 6. (a) The locaton of the noes n the testbe, arrange n an approxmate x9 gr. (b) The elvery probabltes of the testbe lnks for each transmsson rate. The ata ponts for each curve are place n orer from largest to smallest (.e., n rank orer). As the rate ncreases, less lnks are avalable an thus path versty ecreases. just {}. The forwarng set s then upate to F (ρ) = {, }. The same proceure s repeate for each settle noe, untl we = φ = {,,..., j}. At ths tme, we also have D (ρ) = δ, whch trggers the upate D = D (ρ) = δ, F = F (ρ) = φ, an T = ρ. Once D s equal to the shortest stance δ, t oes not change anymore an we have D = δ at the tme s s nserte nto S. We can now prove the theorem wth two contractons. Snce noe occurs after noe s n the shortest multrate anypath to, by Lemma we have δ δ s. In aton, we must also have δ s D s because D s s never smaller than δ s. Snce both an s are n V S an noe s was chosen as the one wth the mnmum stance from Q, then we must have D s D an δ δ s D s D. From our prevous clam, we know that D = δ an therefore D = δ δ s D s D, from whch we have fnally have F (ρ) D = δ = δ s = D s. () As a result, s s not settle out of orer snce has the shortest stance n V S an δ s = δ. From ths we conclue that the noes n S are settle n ascenng orer of stance. Atonally, we also have D s = δ s at the tme s s ae to S, whch contracts our ntal choce of s. We conclue therefore that for each noe s V we have D s = δ s at the tme s s ae to S. V. EXPERIMENTAL RESULTS We evaluate the propose multrate algorthm usng an -noe.b noor testbe. Each noe s a Stargate mcroserver [] equppe wth an Intel -MHz Xscale PXA processor, 6 MB of SDRAM, MB of Flash, an an SMC ElteConnect SMCW-B PCMCIA.b wreless network car usng the Prsm chpset. Ths car has a maxmum transmsson power of mw an t efaults to a propretary power control algorthm. The noes of the testbe are strbute over the celng of the Center for Embee Networke Sensng (CENS) at UCLA. The noes are locate n an approxmate x9 gr an roughly ten meters apart from each other. Fgure 6(a) epcts the locaton of the noes n the testbe. Each noe s equppe wth a -B omn-rectonal rubber uck antenna for the wreless communcaton. In orer to emulate a wreless mesh network wth multple hops, we use a -B SA-XX attenuator between the wreless nterface an ts antenna. The attenuator weakens the sgnal urng both the transmsson an the recepton of a frame, emulatng a large stance between noes. For Mbps, we have paths of up to hops between each par of noes, wth. hops on average. For Mbps, we have a longer transmsson range, whch reuces the maxmum path length to hops, wth an average of. hops between each par of noes. We use the testbe to measure the elvery probablty of each lnk at fferent transmsson rates. For that purpose, each noe broacasts one thousan -byte packets an later on we collect the number of receve packets at neghbor noes. We repeat ths process for,,., an Mbps to have a lnk estmate for each transmsson rate. We use the Clck toolkt [] an a mofe verson of the MORE software package [] for the ata collecton. Our mplementaton s capable of senng an recevng raw. frames by usng the wreless network nterface n montor moe. We mofe the HostAP Prsm rver [9] for Lnux n orer to allow not only. frame overhearng but also frame njecton whle n montor moe. In aton, we extene the HostAP rver to enable t to control the transmsson rate of each. frame sent. The Clck toolkt tags each frame wth a selecte transmsson rate an ths nformaton s then passe along to the rver. For each frame, our mofcaton reas the nformaton tagge by Clck an notfes the wreless nterface frmware about the specfe transmsson rate. Fgure 6(b) shows the strbuton of the elvery probablty of each lnk n the testbe at fferent.b transmsson rates. Every noe par contrbutes wth two lnks n the graph,

9 one for each recton. Lnks of each rate are place n orer from largest to smallest (.e., n rank orer). The ponts of each curve are sorte separately an, therefore, the elvery probabltes of a gven x-value are not necessarly from the same lnk. In wreless mesh networks, hgher transmsson rates usually have shorter rao ranges an therefore a lower network ensty. We can see ths behavor n Fgure 6(b). As the transmsson rate ncreases, we can see that we have less lnks avalable an therefore less path versty between noes. For nstance, as shown by the ashe horzontal lne, the number of lnks wth a elvery probablty hgher than % s at Mbps, 9 at Mbps, 9 at. Mbps, an only at Mbps. Wth less paths avalable at hgher rates, we have an nterestng traeoff for multrate anypath routng. Wth a lower transmsson rate, we have more path versty an a shorter number of hops to traverse, but also a lower throughput. On the other han, a hgher rate results n a hgher throughput, but also n less path versty an a larger number of hops. Our algorthm explores ths traeoff an selects the optmal transmsson rate an forwarng set for every noe. Fg. shows the results of an experment we conucte to test the nepenence of recevers. In our experment, a noe broacasts, ata frames at Mbps to four neghbors an each frame has bytes. The x-axs represents the 6 possble set of recevers for the frame (.e., set correspons to the frame beng lost by all neghbors an set correspons to every neghbor correctly recevng the frame). The y- axs represents the probablty of each set. The observe hstogram s rectly erve from the ata. The nepenent hstogram s erve by assumng that the loss probablty at each recever s nepenent of each other, so t s calculate smply by multplyng the respectve probabltes of each nvual recever. We can see that both functons are pretty close ncatng that the elvery probabltes of each recever are loosely correlate. Ths experment was repeate for other noes n the testbe an a smlar behavor was observe. Our result are also consstent wth other stues [], []. Probablty Set of recevers Observe Inepenent Fgure. (a) Dstrbuton of frame receptons at four neghbors. For four neghbors, we have = 6 subsets an each one represents a fferent set of neghbors who correctly receve the frame. The shortest multrate anypath always has an equal or lower cost than the shortest sngle-rate anypath. Otherwse, we woul have a contracton snce we can fn another multrate anypath (.e., the sngle-rate anypath) wth a shorter stance to the estnaton. It s mportant, however, to quantfy how much better multrate anypath routng s over sngle-rate anypath. For ths purpose, we calculate the gan of multrate over sngle-rate anypath. We efne the gan of a gven par of noes as the rato between the sngle-rate anypath stance an the multrate anypath stance between these two noes. Ths metrc reflects how many tmes the en-to-en transmsson tme s larger when usng sngle-rate as oppose to multrate anypath routng. Fgure (a) shows the strbuton of ths gan for every par of noes n the network. Each curve represents the gan over sngle-rate anypath routng at a fxe rate. We see that the en-to-en transmsson tme wth multrate anypath routng s at least % an up to. tmes shorter than wth snglerate anypath routng at Mbps, wth an average gan of.. For hgher rates, we also see an nterestng behavor epcte by the vertcal lnes. These lnes ncate that several noe pars have an nfnte gan. The nfnte gan occurs because these noes can not talk to each other at that partcular rate ue to the poor lnk qualty; the network therefore becomes sconnecte. We have (.6%) noe pars that can not reach each other at both an. Mbps an (.%) noe pars out of reach at Mbps. For the network to be connecte, we must then ether use a lower rate (e.g., Mbps) for the whole network at the cost of a lower throughput or use multrate anypath routng. For Mbps, f we remove the noe pars wth nfnte gans, we have a gan of at least 9% an up to.6, wth an average of.. For. Mbps, we have a gan up to., wth an average of %. Fnally, for Mbps, we have a gan up to 6., wth an average of %. Fgure (b) shows the reason why multrate always performs better than sngle-rate anypath routng. In ths graph, we show the strbuton of the optmal transmsson rates selecte by each noe to reach every other noe. We can see that the optmal transmsson rates are not concentrate at a sngle rate, but rather strbute among over several possbltes. We have.% of noe pars usng Mbps, % usng. Mbps, an % usng Mbps as the optmal rate. Interestngly enough, no noe par selecte Mbps as the optmal rate snce t was more benefcal to use another rate nstea. If these rates were concentrate at a partcular rate, then multrate an sngle-rate anypath routng woul have the same cost. Ths assumpton, however, oes not hol n practce an therefore multrate anypath routng always has a hgher performance, sometmes manyfol hgher as shown n Fgure (a), than sngle-rate anypath routng. VI. RELATED WORK Most of the work n anypath routng focuses on usng a sngle transmsson rate. The followng works are all snglerate anypath routng schemes.

10 . Gan of multrate anypath routng 6. Mbps. Mbps. Mbps. Mbps Probablty.... Noe par n rank orer (a).... Transmsson rate (Mbps) (b) Fgure. Results of the SMAF algorthm for the wreless testbe. (a) Gan of multrate over sngle-rate anypath routng. For each noe par, we ncate n the y-axs how many tmes multrate anypath routng s better than sngle-rate anypath. (b) Hstogram of the transmsson rate chosen by each noe. Optmal transmsson rates are not concentrate at any partcular rate, ncatng that a sngle-rate algorthm can not perform as well as a multrate algorthm. Zorz an Rao [] use a combnaton of opportunstc an geographc routng n a wreless sensor network. The authors assume that sensor noes are aware of ther locatons an ths nformaton s use for routng. The forwarng set of a gven noe s compose of the neghbors whch are physcally closer to the estnaton. Packets are broacast an neghbors n the set forwar the packet respectng the relay prorty explane n Secton II. As an avantage, ths routng proceure oes not nee any sort of route ssemnaton over the network. Usng just the physcal stance as the routng metrc, however, may not be the best approach snce t oes not take lnk qualty nto account. We ntrouce the EATT routng metrc that takes not only the lnk qualty but also the multple transmsson rates nto account urng route calculaton. Ye et al. [] present another sngle-rate opportunstc routng protocol for sensor networks. The key ea s that each packet carres a cret whch s ntally set by the source an s reuce as the packet traverses the network. Each noe also mantans a cost for forwarng a packet from tself to the estnaton an noes closer to the estnaton have smaller costs. Packets are sent n broacast an a neghbor noe forwars a receve packet only f the cret n the packet s hgh enough. ust before forwarng the packet, ts cret s reuce accorng to the noe cost; therefore, more crets are consume as the packet moves away from the shortest path. A mesh aroun the shortest path s then create on-the-fly for each packet. Yuan et al. [] use a smlar ea for wreless mesh networks. Although packet elvery s mprove, ths routng scheme ncreases overhea snce t s base on a controlle floong mechansm. Therefore, robustness comes at the cost of uplcate packets. In our proposal, a packet s forware by a sngle neghbor n the forwarng set an a MAC mechansm, such as the one propose by an an Das [], s n place to guarantee that no uplcate packets occur n the network. Bswas an Morrs [] esgne an mplemente ExOR, an opportunstc routng protocol for wreless mesh networks. ExOR follows the same guelnes of sngle-rate anypath routng explane n Secton II. Bascally, a noe forwars a batch of packets an each neghbor n the forwarng set wats ts turn to transmt the receve packets. The authors mplement a MAC scheulng scheme to enforce the relay prorty n the forwarng set. As a result, a noe only forwars a packet f all hgher prorty noes fale to o so. The authors show that opportunstc routng ncreases throughput by a factor of two to four compare to sngle-path routng. Our results go beyon an show that an even better performance can be acheve wth multrate anypath routng. Atonally, n our esgn, each packet s route nepenently wthout storng any per-batch state at ntermeate routers. Chachulsk et al. [] ntrouce MORE, a routng protocol whch uses both opportunstc routng an network cong to ncrease the network en-to-en throughput. Upon the recept of a new packet, a noe encoes t wth prevously receve packets an then broacasts the coe packet. Results show that MORE allows a hgher throughput than ExOR an snglepath routng. Network cong, however, requres routers to store prevous packets n orer to coe them wth future packets, ang sgnfcant storage an processng overhea to the forwarng process. Furthermore, the authors only focus on opportunstc routng wth a sngle transmsson rate. Our results ncate that performance coul be further mprove wth multrate anypath routng. An analyss of multrate anypath routng an network cong s also an open problem an an nterestng topc for future work. Beses usng a sngle bt rate, the above-mentone systems also o not have a systematc approach for the selectng the forwarng set for a gven estnaton. The selecton s commonly base on the heurstc that f a neghbor has a smaller ETX stance to the estnaton, then t shoul be n the forwarng set. However, the ETX s a sngle-path metrc an o not represent correctly the noe s true stance when

11 usng anypath routng. Zhong et al. [6] was the frst to propose the expecte anypath number of transmssons (EATX) metrc escrbe n Secton II, whch was also use n [], []. The authors propose an algorthm for forwarng set selecton n [], but ths algorthm may not reach an optmal soluton epenng on the orer that neghbors are teste. Dubos-Ferrère et al. [] ntrouce a shortest anypath algorthm capable of fnng optmal forwarng sets. The authors generalze the well-known Bellman-For algorthm for anypath routng an prove ts optmalty. Performance tests n a wreless sensor network show that anypath routng sgnfcantly reuces the requre number of transmssons from a noe to the snk. Chachulsk [] presents a generalzaton of Djkstra s algorthm for anypath routng that s very smlar to the one we nepenently erve n Secton IV-A, but the author oes not prove any proof of optmalty. Both of these algorthms, however, are esgne for networks usng a sngle transmsson rate. Instea, our algorthm n Secton IV-B generalzes anypath routng for multple rates, gvng noes the ablty to choose both the best rate an the best forwarng set to a partcular estnaton. We also prove the proof of optmalty for our algorthm. As a result, the optmalty of the sngle-rate algorthm n [] s also prove snce ths s a partcular case of our algorthm. More recently, multple transmsson rates have been aresse n opportunstc routng. Raunovc et al. [] presents an optmzaton framework to erve routng, scheulng, an rate aaptaton schemes. Zeng et al. [] presents a lnearprogrammng formulaton to optmze the en-to-en throughput of opportunstc routng, conserng multple rates an transmsson conflct graphs. However, n both cases the problem beng solve s NP-har. Heurstcs are then apple to fn a soluton, whch s not necessarly optmal. VII. CONCLUSIONS In ths paper we ntrouce multrate anypath routng, a new routng paragm for wreless mesh networks. We prove a soluton to ntegratng opportunstc routng an multple transmsson rates. The avalable rate versty mposes several new challenges to routng, snce rao range an elvery probabltes change wth the transmsson rate. Gven a network topology an a estnaton, we want to fn both a forwarng set an a transmsson rate for every noe, such that ther stance to the estnaton s mnmze. We pose ths as the shortest multrate anypath problem. Fnng the rate an forwarng set that jontly optmze the stance from a noe to a gven estnaton s consere an open problem. To solve t, we ntrouce the EATT routng metrc as well as the Shortest Multrate Anypath Frst (SMAF) algorthm an presente a proof of ts optmalty. Our algorthm has the same complexty as Djkstra s algorthm for multrate sngle-path routng, beng easy to mplement n lnk-state routng protocols. We conucte experments n a -noe.b testbe to evaluate the performance of multrate over sngle-rate anypath routng. Our man fnngs are: () when the network uses a sngle bt rate, t may become sconnecte snce some lnks may not work at the selecte rate; () multrate outperforms -Mbps anypath routng by % on average an up to a factor of 6. whle stll mantanng full connectvty; () multrate also outperforms -Mbps anypath routng by a factor of. on average an up to a factor of.; () the strbuton of the optmal transmsson rates are not concentrate at any partcular rate, corroboratng the assumpton that hyperlnks n sngle-rate anypath routng usually o not transmt at ther optmal rates. ACKNOWLEDGMENTS Ths work was one n part whle the frst author was vstng the Ecole Polytechnque Féérale e Lausanne (EPFL). We woul lke to thank Henr Dubos-Ferrère an Martn Vetterl for hostng the frst author at EPFL an ntroucng anypath routng to hm. We thank Deborah Estrn for her help an scussons over the years an for the CENS testbe. We also thank Ee Kohler, Fan Ye, an Lxa Zhang for nsghtful comments on an early raft. We are grateful to Martn Lukac for hs help wth the testbe. Ths work was supporte by the U.S. Natonal Scence Founaton uner Grants NBD-96 an CCF-. Any opnons, fnngs, an conclusons or recommenatons expresse n ths materal are those of the authors an o not necessarly reflect the vews of the Natonal Scence Founaton. REFERENCES [] D. Aguayo,. Bcket, S. Bswas, G. u, an R. Morrs, Lnk-level Measurements from an.b Mesh Network, n Proceengs of the ACM SIGCOMM Conference, Portlan, OR, USA, Aug., pp.. [] M. Campsta, P. Esposto, I. Moraes, L. H. Costa, O. C. Duarte, D. Passos, C. V. e Albuquerque, D. C. Saae, an M. Rubnsten, Routng Metrcs an Protocols for Wreless Mesh Networks, IEEE Network, vol., no., pp. 6, an.-feb.. [] A. Seth, D. Kroeker, M. Zahara, S. Guo, an S. Keshav, Low-cost Communcaton for Rural Internet Kosks usng Mechancal Backhaul, n Proceengs of the ACM Mobcom 6 Conference, Los Angeles, CA, USA, Sep. 6, pp.. [] S. Chachulsk, Trang Structure for Ranomness n Wreless Opportunstc Routng, Master s thess, Massachusetts Insttute of Technology, Cambrge, MA, USA, May. [] S. Bswas an R. Morrs, ExOR: Opportunstc Mult-Hop Routng for Wreless Networks, n Proceengs of the ACM SIGCOMM Conference, Phlaelpha, PA, USA, Aug., pp.. [6] Z. Zhong,. Wang, S. Nelakut, an G.-H. Lu, On Selecton of Canates for Opportunstc AnyPath Forwarng, ACM SIGMOBILE Moble Computng an Communcatons Revew, vol., no., pp., Oct. 6. [] H. Dubos-Ferrere, M. Grossglauser, an M. Vetterl, Least-Cost Opportunstc Routng, n Proceengs of the Allerton Conference, Montcello, IL, USA, Sep.. [] K. Zeng, W. Lou, an H. Zha, On En-to-En Throughput of Opportunstc Routng n Multrate an Multhop Wreless Networks, n Proceengs of the IEEE Infocom, Phoenx, AZ, USA, Apr., pp. 6. [9] R. Draves,. Pahye, an B. Zll, Routng n Mult-Rao, Mult-Hop Wreless Mesh Networks, n Proceengs of the ACM MobCom Conference, Phlaelpha, PA, USA, Sep., pp.. [] S. an an S. R. Das, Explotng Path Dversty n the Lnk Layer n Wreless A Hoc Networks, A Hoc Networks, vol. 6, no., pp., ul.. [] C. Res, R. Mahajan, M. Rorg, D. Wetherall, an. Zahorjan, Measurement-Base Moels of Delvery an Interference n Statc Wreless Networks, n Proceengs of the ACM SIGCOMM 6 Conference, Psa, Italy, Sep. 6, pp. 6.