Energy-Optimal Online Algorithms for Broadcasting in Wireless Networks

Transcription

1 Eergy-Optimal Olie Algorithms for Broadcastig i Wireless Networks Shay Kutte, Hirotaka Oo, David Peleg, Kuihiko Sadakae, ad Masafumi Yamashita Abstract The paper cosiders the desig of eergy-efficiet olie protocols for the basic problem of message trasmissio to hosts positioed at ukow distaces i ad-hoc wireless etworks. The paper formulates a umber of variats of this problem ad presets optimally competitive algorithms for those variats. 1 Itroductio 1.1 Backgroud We cosider problems related to the desig of eergyefficiet olie message broadcastig protocols i ad-hoc wireless etworks. Recet developmets i portable wireless devices with limited power resources have led to cosiderable iterest i problems ivolvig the costructio of eergy-efficiet multicast trees i the etwork. Wireless devices ca cotrol their trasmissio power i order to save power cosumptio wheever the distace to the iteded destiatio of the trasmissio is kow. The atteuatio of a sigal with power P s is P r = Ps ds,t), where ds, t) δ is the distace betwee hosts s ad t, ad δ 1 is the distace-power gradiet [3]. A message ca be successfully decoded if P r is o less tha a costat γ. Therefore the trasmissio rage of a host s, amely, the maximum distace to which a message ca be successfully delivered from s, isp s /γ) 1/δ. Power cotrol also has a positive effect o reducig the umber of trasmissio collisios betwee earby seders. The problem studied here cocers a sigle seder which has to trasmit a message to a give collectio of receivers Iformatio Systems Group, Faculty of Idustrial Egieerig ad Maagemet, Techio, Haifa, Israel. kutte@ie.techio.ac.il Departmet of Computer Sciece ad Commuicatio Egieerig, Kyushu Uiversity, Fukuoka, Japa. {oo,sada,mak}@csce.kyushu-u.ac.jp Departmet of Computer Sciece ad Applied Mathematics, The Weizma Istitute of Sciece, Rehovot, Israel. david.peleg@weizma.ac.il i a olie settig, amely, whe the hosts do ot kow each other s locatios. The goal is to specify a protocol for the seder allowig it to directly broadcast the message to the recipiets ad receive ackowledgemets, while miimizig the total trasmissio costs. By direct broadcast we mea that the seder is required to trasmit the message itself to every recipiet, amely, multi-hop delivery is ot allowed. This restrictio may be relevat i situatios whe the battery resources of the receivers is severely limited ad it is desired to miimize their trasmissios, or whe whe the reliability of the hosts is ucertai ad oly direct messages from the source ca be trusted. 1. Cotributios Usig varyig levels of trasmissio power is importat for eergy-efficiet commuicatio. As far as the authors are aware, there has bee o olie algorithms with provable worst-case guaratees for eergy-efficiet broadcastig i ad-hoc wireless etworks. The protocols proposed i this paper are based o computig or estimatig the distaces from the seder host to the receiver hosts i a eergy-efficiet way. The most basic case is that of a sigle seder ad a sigle receiver. The geeric doublig protocol employed by the seder is based o repeatedly trasmittig the messages to icreasigly larger distaces, util reachig the receiver. The behavior of this protocol depeds o the choice of the sequece of distaces, ad the problem is to determie them so as to miimize the overall power cosumptio. If a specific probability distributio may be assumed o the hosts, the algorithm ca be optimized [4]. This paper, however, assumes a olie settig i which o a priori iformatio is give about the distace from the seder to the receiver. Therefore the worst-case sceario should be cosidered. This motivatio leads us to apply a competitive aalysis to the algorithm cf. []). We compare the power cosumptio of a algorithm with that of the optimal ifeasible) offlie algorithm that kows the distace d. We show that the optimal competitive ratio for this problem is 3/+, i.e., there exists a olie algorithm for the problem with this competitive ratio, ad o olie algorithm has 1 Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE

2 smaller competitive ratio. The problem is somewhat similar to the famous cow path olie problem [1], but settig the parameter of the algorithm is ot obvious. Furthermore, we study the geeralizatio of this problem where there is more tha oe receiver. This is a propoer extesio of the cow path problem. For this problem we also propose a competitive olie algorithm ad prove its optimality. Iterestigly, the competitive ratio of the geeralized problem is the same, amely, 3/+. The algorithm ad its aalysis appear i Sectio 4. The rest of the paper is orgaized as follows. I Sectio 1.3 we discuss the model employed i the paper. Sectio formally presets our problems ad algorithms. Sectio 3 establishes the competitive ratio for the sigle receiver case. I Sectio 4 we propose algorithms for multiple receiver case. 1.3 Model t The model cosidered i this paper is the followig. Each host has a uique id ad a global clock, ad they ca schedule their trasmissio time i a collisio-free maer. Messages are always delivered correctly to the destiatio. Below we justify our ratioal for usig this model, ad particularly the assumptios of sychroous commuicatio ad failure-freedom. Let us first discuss our assumptio of a sychroous commuicatio model. Note that this model is a reasoable approximatio uder some atural timig assumptios ad assumig the availability of time-out mechaisms. The problem ca also be cosidered i the alterative asychroous commuicatio model, i which o assumptios are made cocerig the timig ad operatio rates of the participats. However, i this model there is o way to limit the umber of times that collisios occur betwee messages set by receivers. As a result, o olie algorithm ca achieve a costat competitive ratio. I cotrast, i a sychroous model it is possible to use the global clock i order to schedule the hosts i a collisio-free maer as follows. Assume that each host has a uique id. Without loss of geerality assume also that the seder has id 1 ad the receivers have id s to. Suppose that the seder broadcasts a message at time t +1. The each receiver with id i seds a ackowledgmet at time t+i if it receives the message. Because the seder kows which receivers have set ackowledgemets, i the ext roud the seder ca sed the list of id s of receivers that have received the message i the last roud. Each receiver ca ow determie the time to sed a ackowledgmet to advace the completio time by usig oly that iformatio. As our criterio is to miimize the total power cosumptio, ad time efficiecy is igored, this schedulig is sufficiet. Clearly, if the problem requires optimizig both the power cosumptio ad the des Figure 1. Illustratio of the doublig algorithm. Numbers represet trasmissio times, with the trasmissio rage doubled by the seder s at each roud util reachig the receiver t. livery completio) time, the other scheduligs should be cosidered. Similarly, our focus o the failure-free model where messages are always delivered correctly to the destiatios) stems from the fact that i a model allowig arbitrary message loss, the worst case competitive ratio caot be bouded. The developmet of suitable models for studyig fault-tolerat variats of the problem is left for future research. Problems ad algorithms The geeric protocol employed by the seder s i the case of a sigle receiver t is give i the followig procedure. Procedure SedMessaget, msg) 1. i 1; f true. while f 3. do Trasmit msg, p i with power p i. 4. Wait. 5. if received ackowledgmet from t 6. f false; 7. i i +1; The algorithm is illustrated i Figure 1. Clearly, the behavior of ay algorithm for the seder ca be specified as a sequece of icreasig power costs {p i }, for i =1,,..., ad hece its performace depeds o the choice of this sequece. A algorithm A succeeds o the first step J such that p J /γ) 1/δ d. For lack of distace iformatio, the receiver will rely o the iformatio cotaied i the received message ad trasmit its ackowledgemet with the same power. The cost of algorithm A is thus costa) = J i=1 p i + p J where the secod term is Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE

3 for the ackowledgmet. Therefore the problem is to determie this sequece so that the overall power cosumptio is miimized. We assume a olie settig i which o a priori iformatio is give about the distace from s to t, ad a worstcase sceario should be cosidered. We compare the power cosumptio of a algorithm with that of the optimal offlie algorithm that kows the distace d. This offlie algorithm sets p i = γd δ ; the it termiates at the first step ad its eergy cosumptio is miimized, hece its cost is cost =γd δ. Of course this is impossible to achieve olie i the absece of kowledge o the distaces, ad our aim is to reduce the cost of our algorithm as much as possible. The problem is formalized as follows. Problem Broadcast+Ack- BA): There is a seder s ad a receiver t at distace d 1, which is ukow. The seder rus a algorithm A to sed the message. Oce it is delivered, the receiver should retur a ackowledgmet. The goal is to miimize the total power cosumptio for the seder ad the receiver. Competitiveess: The competitive ratio of algorithm A is ρa) =sup{costa)/cost } where the supremum is take over all iputs. The problem studied i this paper is to determie the sequece {p i } that miimizes the competitive ratio. For this problem, we use the followig simple algorithm. Algorithm 1 Doublig Algorithm DA[β]) Ivoke Procedure SedMessage settig p 1 = γ ad p i+1 = βp i for the specified parameter β. Theorem.1 The optimal competitive ratio for problem BA is ρ = 3 +, i.e., for problem BA, there exists a olie algorithm whose competitive ratio is ρ, while there is o olie algorithm whose competitive ratio is smaller tha ρ. The proof is give i Sectio 3. Furthermore, the followig geeralizatio of problem 1 BA) is a propoer extesio of the cow path problem where we cosider the case there is more tha oe receiver. Problem Broadcast+Ack- BA): There are hosts i the regio, icludig a seder s ad 1 receivers r 1,...,r 1 at differet distaces from s. The iput ca be specified as a cofiguratio, d where = 1,..., k ), d =d1,...,d k ), k i=1 i = 1 ad d i <d i+1 for 1 i k. I this cofiguratio, the 1 receivers are orgaized so that there are i receivers at distace d i from s, for 1 i k. The seder kows, but does ot kow the groupigs or the distaces. The seder should broadcast a message to all the receivers ad get ackowledgmets from all of them. The goal is to miimize the total power cosumptio for the seder ad the receivers. For this problem we also propose a olie algorithm of optimal competitive ratio. Theorem. For problem BA, there exists a olie algorithm with competitive ratio 3 +, ad there is o olie algorithm with competitive ratio smaller tha 3 +. The algorithm ad its aalysis appear i Sectio 4. 3 Proof for sigle receiver case I this sectio we prove Theorem.1. First we show a upper boud of the competitive ratio of algorithm DA[β]. Propositio 3.1 The doublig algorithm DA[β] achieves the competitive ratio ββ 1) β 1) for the problem BA. Proof: Let d deote the distace betwee the seder s ad the receiver t. If the algorithm DA[β] termiates at step J, the ecessarily β J 1 /γ) 1/δ <d β J /γ) 1/δ. The cost of DA[β] is J i=0 γβ i + γβ J = γβj+1 1) β 1 + γβ J, while the optimal cost is at least γβ J 1. Hece the competitive ratio is at most ββ 1) β 1). By lettig β =1+ 1, the competitive ratio is at most 3 +. We ow show Theorem.1 which gives a lower boud of the competitive ratio. Proof of Theorem.1: Propositio 3.1 guaratees the existece of a olie algorithm whose competitive ratio is 3 +. Thus, we ow cocetrate o showig that a lower boud o the competitive ratio for problem BA is 3 +. Let us cosider a olie algorithm A which achieves the optimal competitive ratio c 3 +. The output sequece of algorithm A amely, the sequece of trasmissio powers used by the seder) is deoted by x 1,x,... Note that for ay iteger, oe must cosider a sceario where the receiver t is positioed at distace x 1 + ɛ from the seder s, for a arbitrarily small ɛ. O such sceario, the optimal cost is x 1 +ɛ, whereas the olie algorithm icurs a cost of x 1 + x + + x 1 + x + x. Therefore, by the defiitio of the competitive ratio, we have the iequalities x x 1 + x + x cx 1 + ɛ) 1) for ay iteger ad for arbitrarily small ɛ>0. Sice A B + ɛ for every ɛ>0 implies A B, we have from iequality 1) that 1 x i +x cx 1 ) i=1 Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE

4 for ay iteger. These iequalities yield ew ecessary coditios as follows. First, takig ) with =we have x c 1 ) x 1, 3) By 3) ad usig ) with =3, we similarly have cx x 1 + x +x 3 that is, x 3 c 1 x + x +x 3, c 1 1 ) x. c 1 Repeatig this argumet, we get the iequalities with the coefficiets α i defied as where µ i = x i+1 α i x i, 4) α i = c µ i { 1, if i =1, µ i 1 /α i 1 +1, if i. Equatio 5) ca be simplified as 5) α i = c + 1 c α i 1, 6) for ay iteger i. Note that α i > 1 holds for ay iteger i by 4) ad x i x i+1. We ow show that this α i is a Cauchy sequece, amely, α i+1 α i 1. Ideed, α i+1 α i 1 = 1 α i = c c + 1 cαi ) 1 1 c 1 ) + c 1 ) 4c. α i c 4c Recall that c 1 ad c 3 +, by the property of the competitive ratio ad Propositio 4.1, respectively. I this rage, c 1 ) 4c 0, thus αi+1 α i 1, ad α i is a Cauchy sequece. Therefore, α i coverges to some value α, which by 6) must satisfy α = c + 1 c α ad its discrimiat satisfies D = c + 1 4c ) = c 3 ) c 3 + ) 0. 4 Hadlig multiple receivers I this sectio we cosider the case the seder must sed the message to more tha oe receiver. 4.1 Equal distace receivers First we cosider the easier special case where all the receivers are at the same distace from the seder. That distace is ukow to the participats. Problem Uiform-Broadcast+Ack- UBA): There are hosts i the area, of which oe is the seder, s, ad the other 1 are receivers, r 1,...,r 1. The receivers are all at the same distace d from the seder, but the seder does ot kow this distace. The seder should broadcast a message to all the receivers ad get ackowledgmets from all of them. The goal is to miimize the total power cosumptio for the seder ad the receivers. Note that this problem is idetical to problem BA if =. For this problem we claim that the doublig algorithm has optimal competitive ratio. Propositio 4.1 For problem UBA, fixig β =1+ 1, the competitive ratio of Algorithm DA[β] is at most The proof is similar to that of Propositio 3.1. We ow show a asymptotically matchig lower boud. Propositio 4. For problem UBA, there is o algorithm with competitive ratio smaller tha Proof: The proof of Theorem.1 ca be easily exteded to the geeral case. Similar to iequalities 4) ad 5), albeit with differet α, wehave: where x i+1 α 1 = c 1, < α i x i α i = c 1 +1 c α i 1, for ay iteger i. The sequece α i is agai a Cauchy sequece because c 1 ad c hold by the property of the competitive ratio ad Propositio 4., respectively. Thus we have As c 1, the right term i the product is positive. Hece the left term must also be positive, implyig c 3 + ad thus yieldig the desired lower boud o the competitive ratio. α = c 1 +1 c α, which implies c Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE

5 4. Variable distace receivers Next we cosider the geeral Problem BA as stated i Sectio 1, where the 1 receivers are placed at differet distaces from the seder. If the cofiguratio, d is kow to all the hosts, the the solutio is trivial the seder trasmits oce, reachig all the receivers, ad each receiver trasmits its ackowledgemet usig the miimal power required). However, i case the cofiguratio is kow oly to the seder but ot to the other hosts), computig the optimal cost is ot obvious. Nevertheless, it ca be computed as follows. Propositio 4.3 I case the cofiguratio, d is kow to the seder, the optimal cost for problem BA ca be computed i liear time. Proof: Give a cofiguratio, d, we ca restrict the distaces chose by the seder s for message trasmissio to the set {d 1,...,d k }, i.e., the trasmissio powers ca be specified from amog p i = γd δ i for i =1,...,k. The schedule of s s broadcast ca be represeted as a ordered set Y of idices correspodig to the distace to which the seder s trasmits the message. For example, usig the schedule Y = {1, 3, 6}, the seder s trasmits the messages to distaces d 1, d 3 ad d 6. Give a schedule Y, for each 1 i k, the i receivers at distace d i from s will receive the message for the first time o roud mi, Y )=mi{j j i, j Y }, ad thus use power p mi,y ) for the ackowledgemet. Hece for ay schedule Y, the total cost of broadcast icludig ackowledgemets) is costy )= i Y p i + k i=1 ip mi,y ). Subsequetly, the goal of this problem is to fid a schedule Y that miimizes costy ). The computatio is based o a recursive formula for the cost fuctio. Suppose that the seder s trasmits the messages to a l-th distace i the schedule Y for, d. The the cost fuctio is expaded as costy ) = p i + i Y i l + p i + i Y i>l l i p mi,y ) i=1 k i=l+1 i p mi,y ). Sice the left ad right parts i this equatio use disjoit idex sets, we have OPT, d ) = mi l=1,...,k {OPT, d 1,l) ) +OPT, d l+1,k) )}, where, d a,b) deotes a,..., b ), d a,...,d b ) ad OPT, d a,b) ) deotes the optimal eergy cosumptio for, d a,b). Based o these observatios, we ca costruct a dyamic programmig algorithm as follows. Deote the set of idices from i to k by S i = {i, i +1,...,k}. For every i, we look for the schedule Y i which miimizes the eergy cosumptio for, d over all schedules Y S i. Note that Y k = {k} ad Y 1 = Y. Oe ca easily verify that { Yi+1 {i}, if costy Y i = i+1 ) >costy i+1 {i}) otherwise, Y i+1 which yields a dyamic programmig algorithm for solvig static BA i liear time. It is easy to show the followig. Propositio 4.4 For problem BA, there is o algorithm with competitive ratio smaller tha 3 +. Proof: Let us cosider a sceario where receivers are positioed ear the seder ad oe receiver is very far away. The the optimal total cost is domiated by the power required to trasmit the message to the farthest receiver ad for that receiver to trasmit its ackowledgmet. By Propositio 4. for =, 3 + is also a lower boud for BA. Next we cosider a upper boud for the problem BA. We propose the followig algorithm. Defie the parameter β k =1+ 1 k for every k. Algorithm Dyamic Doublig Algorithm) Procedure DDA, msg) 1. p γ;. while >1 3. do Trasmit msg, p with power p. 4. Wait. 5. l #received ackowledgmet packets; 6. l; 7. p β p; Note that each receiver seds exactly oe ackowledgmet for a particular message msg. To prove Theorem., we ow show that Algorithm DDA achieves a competitive ratio of 3 + for problem BA. Proof of Theorem.: A lower boud of 3 + is show i Propositio 4.4. We show that Algorithm DDA has the same competitive ratio. Let p i i =1,,...,l) be a sequece of trasmissio powers used the seder s i the optimal cost algorithm. If a receiver r i receives the message, it seds a ackowledgmet with power p k where k is the miimum idex such that p k γds, r i) δ. Note that because a message from s is delivered to multiple hosts the receivers do ot kow the exact distace to the seder. Therefore the optimal cost is idetical to that of the followig istace: Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE

6 There are i receivers at distace d i for i = 1,,...,lwhere d i satisfies p i = γdδ i. O the other had, the cost of ay olie algorithm for the origial istace is at most the cost for the ew istace, it is eough to show the claim for the latter istace. For this istace, Algorithm DDA first uses β =1+ 1. The the ratio of the cost to sed to hosts with distace d 1 to the optimal cost is at most β /β 1)+ 1β 1+1. From Propositio 4.1 this value is miimized if β =1+ 1 i, ad it is icreasig as β decreases. Therefore its maximum value is achieved if β =1+ 1, ad it is upper bouded by , which is o greater tha Next Algorithm DDA uses β =1+ 1. Because the trasmissio power for the first step is larger tha γ, the maximum ratio is at most , which is agai o greater tha 3 +. For the rest of the executio of the algorithm we always have the maximum ratio smaller tha 3 +. Problem BA correspods to the followig problem: Problem Weight-Broadcast+Ack- WBA): There are oe seder ad k receivers r 1,...,r k. The receivers are at differet ad ukow distaces from the seder, ad r i eeds power i γd δ to sed a message to a host with distace d. The problem is to miimize the total power cosumptio for the seder ad the receivers. Therefore we ca hadle the case of idividual power cosumptio rates i.e., where each host has a multiplicative weight defiig its power cosumptio) i the same framework. The competitive ratio ca be further improved if some iformatio is kow about the distaces of receivers. Propositio 4.5 Cosider problem BA assumig that the groupig vector is kow i advace, while the distace vector d is ukow. I this settig, the competitive ratio of Algorithm DDA is 1+ k + 1 k. Proof: Cosider the sceario where the last k receivers are very far ad all the other receivers are positioed ear the seder. The algorithm will reach all the receivers except for the farthest group ad get their acks i oe step, ad from that poit o, it will behave like Algorithm DA[γ k ]. distace has the optimal competitive ratio 3 +. Our algorithms for broadcastig a message to multiple receivers also have optimal competitive ratio. Iterestigly, the competitive ratio of both problems are the same. We believe that our algorithms ca potetially be made practical for actual wireless etworkig. Ackowledgmets The authors would thak Mr. M. Mesbah Uddi ad Prof. Hiroto Yasuura of Kyushu Uiversity, who suggested the problem. This work is supported i part by the Grat-i-Aid of the Miistry of Educatio, Sciece, Sports ad Culture of Japa. Refereces [1] R. Baeza-Yates, J. Culberso ad G. Rawlis, Searchig i the plae. Iformatio ad Computatio 106:34 5, [] A. Borodi ad R. El-Yaiv, Olie Computatio ad Competitive Aalysis, Cambridge Uiv. Press, [3] G.S. Lauer. Packet radio routig, Chapt. 11, Routig i commuicatio etworks, M. Streestrup ed.), pages Pretice-Hall, [4] R. Mathar ad J. Mattfeldt. Optimal Trasmissio Rages for Mobile Commuicatio i Liear Multihop Packet Radio Networks. Wireless Networks, :39 34, Cocludig remarks Usig multiple levels of trasmissio power is importat for eergy-efficiet ad collisio-free commuicatio. As far as the authors kow, there has bee o olie algorithms with provable worst-case guaratees for eergyefficiet broadcastig i ad-hoc wireless etworks. Our algorithm for sedig a message to a receiver with ukow Proceedigs of the Secod Aual Coferece o Wireless O-demad Network Systems ad Services WONS 05) /05 $ 0.00 IEEE