204 IEEE Iteratioal Coferece o Acoustic, Speech ad Sigal Processig (ICASSP) TOPOOGY OPTIMIZATION FOR ENERGY-EFFICIENT COMMUNICATIONS IN CONSENSUS WIREESS NETWORKS Bejamí Béjar ad Marti Vetterli École Polytechique Fédérale de ausae (EPF) Audiovisual Commuicatios aboratory (CAV) {bejami.bejarharo, marti.vetterli} at epfl.ch ABSTRACT Over the past years there has bee a icreasig iterest i developig distributed computatio methods over wireless etworks. A ew commuicatio paradigm has emerged where distributed algorithms such as cosesus have played a key role i the developmet of such etworks. A special case are wireless sesor etworks (WSN) which have foud applicatio i a large variety of problems such as evirometal moitorig, surveillace, or localizatio, to cite a few. Oe major desig issue i WSNs is eergy efficiecy. Nodes are typically battery-powered devices ad thus, it is critical to make a proper use of the scarce eergy resources. This fact motivates the search for optimal coditios that favor the commuicatio eviromet. It is well kow that the rate at which the iformatio is spread across the etwork depeds o the topology of the etwork ad that fidig the optimal topology is a hard combiatorial problem. However, usig covex optimizatio tools, we propose a method that tries to fid the optimal topology i a cosesus wireless etwork that uses broadcast messages. Our results show that exploitig the broadcast ature of the wireless chael leads to more eergy efficiet cofiguratios tha usig dedicated uicast messages ad that our algorithm performs very close to the optimal solutio. Idex Terms Topology optimizatio, power allocatio, cosesus etworks. INTRODUCTION I etworked systems, a umber of devices ca commuicate ad cooperate i order to perform some global task. Such systems offer a advatage over traditioal cetralized oes i terms of cost, robustess ad scalability, makig them particularly suitable for largescale data aalysis ad moitorig. A clear example of a etworked system that has become very popular over the past years is a Wireless Sesor Network (WSN). WSNs are composed by a large umber of low-cost devices (odes) equipped with a variety of sesors to measure quatities such as temperature, humidity, motio, etc. Such odes have limited computatioal ad power resources ad their purpose is to gather ad retrieve iformatio from the eviromet []. A ew commuicatio paradigm has emerged where smart agets ca take autoomous decisios ad iteract with each other without the supervisio of a cetralized etity. The proliferatio of etworked systems i geeral, ad of WSNs i particular, have motivated the developmet of ew methods ad algorithms for distributed processig. As a example, average cosesus algorithms [2 6] have gaied a lot of popularity i recet years ad have bee widely used for the itegratio of the acquired iformatio across the etwork. Uder mild coectivity coditios, average cosesus algorithms appear as simple ad effective mechaisms for the computatio of global averages from local estimates. Furthermore, they ca also be used as a basic tool for solvig more geeral problems i a distributed fashio [7 9]. I the cotext of WSNs, eergy efficiecy is a major desig issue that should be looked at carefully. It is desirable for such etworks to be autoomous ad capable of workig for log periods of time without battery replacemet. The amout of eergy spet by the etwork is directly related to the cofiguratio of the odes themselves ad how they commuicate. I the case of a cosesus etwork with bidirectioal liks (udirected graph), the rate at which iformatio spreads across the etwork is related to the secod smallest eigevalue of the aplacia matrix, also kow as the algebraic coectivity of the graph. Ituitively, the algebraic coectivity of the graph gives us a measure of how well the etwork is coected (ie., the smaller its value the lower the coectivity of the etwork). For istace, it ca be show that a graph is coected if ad oly if its algebraic coectivity is positive [0]. As a cosequece, there have bee some efforts dedicated to fid the optimal topology/way of mixig iformatio over cosesus etworks i order to maximize the rate of covergece of cosesus algorithms [ 4]. A recet approach looks at the topology optimizatio problem from a eergy-efficiet perspective [5]. The authors propose a coectivity model based o the received power at the odes ad try to fid the optimal power allocatio (ad topology) that yields the least eergy cosumptio i a cosesus etwork. However, the authors i [5] oly cosider the case of uicast messages ad do ot take advatage of the broadcast ature of the wireless chael. Exploitig broadcast commuicatios helps to preserve eergy resources, hece elargig the lifespa of the etwork. I this paper, we cosider the problem of topology optimizatio for eergy-efficiet cosesus-type commuicatios i WSNs usig broadcast messages. We formulate the problem as a biary optimizatio problem o the edge variables ad trasmissio powers. I order to solve the problem we use a relaxatio of the biary variables ad allow them to take values withi the uit iterval. We show that the objective fuctio i the relaxed problem is quasi-covex ad ca be efficietly solved usig a bisectio search over a family of semidefiite feasibility problems. Our formulatio is particularly appealig ad our results easily reproducible usig ay geeral-purpose optimizatio package such as CVX [6]. We will show i the experimetal part that our method is more eergy-efficiet tha usig uicast dedicated messages as i [5]. 978--4799-2893-4/4/$3.00 204 IEEE 522
2. PROBEM FORMUATION Cosider a etworked system such a wireless sesor etwork composed of set of elemets, ad where each ode uses broadcast-type messages to commuicate with its adjacet (-hop) eighbors. et p i, i =,..., be the trasmissio power used by the ith ode. Assume a propagatio model where the received sigal gets atteuated with the travelled distace so that the received power at ode j whe ode i is trasmittig is give by p ij = + p i rij r 0 αij, () where r ij = r ji is the distace betwee odes i ad j, r 0 is a referece distace where full power is received, ad α ij = α ji models the atteuatio characteristics of the chael. I order to allow for a reliable sigal decodig, a proper threshold o a certai Quality of Service measure such as the Bit Error Rate (BER) must be imposed. A requiremet i terms of BER ca be equivaletly expressed as a requiremet i terms of the received sigal power. Therefore, coectivity amog the odes is established based o whether a threshold i the received power over the lik betwee them is met. This meas that the received power i both directios (p ij ad p ji) should be above the miimum required threshold. More formally, let E = {,...,} be the set of all possible edges i the etwork (i.e., = ( )/2), ad let x l {0, }, l E, be a biary variable that takes the value if the lth edge (lik) is active ad 0 otherwise. The, based o our propagatio ad coectivity models, it follows that x lij = pij p th, p ji p th 0 otherwise, (2) where l ij E correspods to the edge that liks odes i ad j, ad where p th represets a threshold o the received power that allows for reliable sigal decodig. It is clear from () ad (2) that fixig the trasmissio powers p i would determie the uderlyig graph topology, ad that give a fixed topology, the miimum required trasmissio powers ca also be determied. Also ote that ot all the combiatios of trasmissio powers p i are allowed sice by (2) we are requirig the graph to be udirected, therefore the trasmissio powers must be related to each other. We are iterested i miimizig the eergy cosumptio over a cosesus etwork that uses broadcast-type messages for commuicatios. I order to aalyze the properties of the etwork let us itroduce some otatio that will facilitate the formulatio of the problem. et a l {, 0, }, l E, be a vector such that a l (i) = ad a l (j) = if odes i ad j are adjacet through lik l, ad a l (k) =0for k = i, j. The aplacia matrix of the graph ca the be expressed as a fuctio of the edges variables x l as (x) = x l a l a T l, (3) l= where x = [x,...,x ] T is a vector cotaiig all edges i the etwork. Recall that the rate of covergece for gossip algorithms is related to the secod smallest eigevalue λ 2() (algebraic coectivity) of the aplacia matrix []. Similarly to [5], a measure of the required eergy to achieve cosesus over a graph usig broadcasttype messages is the proportioal to e T p λ 2((x)), (4) where p =[p,...,p ] T is a vector of all trasmissio powers ad is a vector of all oes. Takig ito accout () ad (2), the relatioship betwee the trasmissio powers p i ad the correspodig variables x lij is give by pi,p x lij = j β ij, (5) 0 otherwise where β ij is give by β ij = β ji = p th + rij r 0 αij. (6) Note that the value of β ij represets the threshold o the trasmissio powers p i ad p j for the ijth lik to be active. Our goal is to fid the topology (ad trasmissio powers) that miimize the eergy cosumptio i (4). Based o (5) we ca ow formulate the eergy miimizatio problem as miimize p,x T p λ 2((x)) x l {0, }, l=,..., p i x lij β ij, i,j=,...,, where the last set of costraits captures the relatioship betwee the edge ad trasmissio power variables. 3. REAXATION AND AGORITHMS Problem (7) is hard to solve due to the biary costrait o the edge variable x. Istead of dealig directly with problem (7) we will resort to a relaxed versio of the problem by allowig the variable x to take values o the uit iterval (i.e., 0 x l ). Also ote that we ca get rid of variable p by expressig it i terms of the variable x usig the relatio p i =maxβ j ijx lij. (8) It is easy to show that, at the optimum of problem (7), the trasmissio power at the odes must satisfy equatio (8). With these cosideratios i mid, we ca express the relaxed versio of (7) as miimize x λ 2((x)) 0 x l, l=,...,. A importat property of problem (9) is that it belogs to the class of quasi-covex optimizatio problems ad thus, ca be solved efficietly usig umerical methods. I order to see this, cosider the objective fuctio i (9), ad let us deote it by f(x). By defiitio, a fuctio g(x) is quasi-covex if the set X = {x g(x) δ} is covex for every δ rage (g(x)). Note oly takes positive values. There- that f(x) = fore, we have that f(x) δ i= max j β ij x lij λ 2 ((x)) i= (7) (9) max β j ijx lij δλ 2((x)) 0, (0) where δ R + is a positive umber. I order to show quasicovexity of f(x) we eed to show that the set of poits defied by (0) is ideed covex. To see this, ote that the left-had-side of the secod iequality i (0) cosists of the additio of two terms. The first term is a covex fuctio i x sice it is 523
the additio of covex fuctios (ote that max j β ijx lij is a covex fuctio i x). The secod term, δλ 2((x)) is also covex i x sice λ 2((x)) is cocave i x ad δ is positive. Cocavity of λ 2((x)) ca be easily show sice it is the poitwise ifimum of a family of covex (actually, liear) fuctios of x (see, [7]), that is λ 2((x)) = if u U ut x l a l a T l u, () where U = {u u T =0, u 2 =}. Now that we have show the quasi-covexity of f(x), a simple method for solvig problem (9) is to use a bisectio search over δ. This ca be easily doe by solvig a family of feasibility problems of the form fid l= x δλ 2((x)) 0, 0 x l, l=,...,. (2) Note that feasibility problem (2) is covex i the optimizatio variable x. For its solutio we will use a semidefiite program (SDP) formulatio of the problem that ca be efficietly solved with geeral-purpose optimizatio packages such as CVX [6]. It is easy to realize that problem (2) ca be equivaletly expressed as the followig SDP feasibility problem fid {x, p,s} i= pi δs 0, l= x la l a T l T si, p i β ijx lij,i,j=,...,, 0 x l, l=,...,, (3) where a iequality of the form A B meas that matrix A B 0 is positive semi-defiite. For completeess, the bisectio procedure to fid the optimal solutio of problem (9) is outlied i Algorithm, where ε is a small umber to decide upo covergece. Note that the ε-optimal value of problem (9) correspods to the output f comig from algorithm (). We have ow a method that allows us to fid a lower boud o the optimal value of the origial problem i (7). Usig the solutio to the relaxed problem (9) provided by the bisectio search i Algorithm, we ca fid a topology by thresholdig the value of x. However, this does ot guaratee that the obtaied topology is a valid oe. A alterative method cosist of sortig the values i x i descedig order ad formig differet topologies by addig Algorithm - Bisectio algorithm : Iput: (, u) ower- ad upper-bouds o the optimal value 2: repeat 3: δ =(u + )/2 4: Get (x, p) by solvig feasibility problem (3) 5: if feasible the 6: (x, p,f ) (x, p,δ) 7: Set ew upper-boud: u = δ 8: else 9: Set ew lower-boud: = δ 0: ed if : util u <ε 2: Output: (x, p,f ) average eergy 22 20 8 6 4 2 0 Uicast Broadcast Relaxatio 8 3 3.5 4 4.5 5 5.5 6 umber of odes Fig. : Average eergy cosumptio usig uicast versus broadcast messages for differet sizes of the etwork ad for a path-loss expoet of α =3.5. oe edge at a time. For each topology, we ca easily check whether it costitutes a valid broadcast topology or ot. If the topology is valid, we ca the compute the objective fuctio ad keep the topology with the smallest objective. Note that such a approach requires the evaluatio of at most ( )/2 topologies ad that there will always be at least oe valid topology (e.g., a fully coected etwork). I practice, however we have observed that the foud topology usig this procedure coicides most of the time with the optimal oe. 4. SIMUATIONS I this sectio we provide some umerical examples to illustrate the validity of the proposed method for optimizig the topology of a wireless cosesus etwork usig broadcast messages. For the propagatio model, we have used a threshold i the miimum received power of p th =0.0, ad a referece distace of d 0 =0.. The path-loss expoet has bee assumed to be the same for all liks i the etwork (i.e., α ij = α, for all i, j). All etworks have bee radomly geerated to uiformly lie withi the uit square (odes coordiates are draw from a uiform distributio over the uit square). I order to illustrate the advatage of broadcast-type of commuicatios as compared to uicast, we have ru a simulatio where we varied the umber of odes from =3up to =6. For these umbers, it is still possible to evaluate all possible topologies ad establish a back to back compariso of our method ad the optimal solutio. For higher umber of odes, however, a exhaustive search becomes impractical sice the umber of possible topologies grows as fast as 2 ( ) 2. I Figure we have displayed the average eergy cosumptio over 00 radom etwork realizatios as a fuctio of the umber of odes, ad for a path-loss expoet α =3.5. The optimal topologies for the uicast ad broadcast cases have bee computed (labelled as Uicast ad Broadcast, respectively) ad displayed i the figure. Also, the eergy cosumptio resultig from the topology foud usig our approach (labelled as Relaxatio) has bee also icluded i the figure. As it ca be appreciated, there is a clear advatage of broadcast commuicatios versus uicast, a effect that becomes eve more proouced as the umber of odes icreases. It is importat to metio that i the uicast case we have 524
umber of edges 5 4 3 2 0 9 8 7 6 5 Uicast Broadcast Relaxatio.5 2 2.5 3 3.5 4 4.5 path loss expoet 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.2 0.4 0.6 0.8 Fig. 2: Average umber of edges as a fuctio of the path-loss expoet for a etwork of =6odes. Fig. 3: A fully coected etwork of =20odes. to cout twice the power put o every lik sice the commuicatio is bidirectioal. Also ote that the approach i [5] is lower bouded by the Uicast curve. Whe comparig the optimal solutio to the broadcast sceario ad the proposed method we ca see that our techique comes very close to the optimal solutio ad, cosequetly, outperforms the uicast sceario ad the approach i [5]. Aother iterestig figure to look at is the umber of edges of the etwork. For a give etwork, the amout of active liks should be larger whe the propagatio chael coditios are favorable. To formalize this ituitio we have performed a simulatio where we have averaged the umber of active liks i a etwork of = 6 odes as a fuctio of the path-loss expoet. The results of the experimet are depicted i Figure 2. As expected, whe the chael coditios are very favorable, the best strategy is to use all liks (i.e., fully coected etwork, 5 edges i our example), whereas the optimal topology approaches a tree (e.g., 5 edges i our example) as commuicatio becomes more expesive. Also i Figure 2 we observe a very good agreemet betwee the optimal topology ad the oe foud by our method. As a illustrative example we have also depicted i Figures 3 ad 4 a fully coected etwork of =20odes ad the resultig topology after optimizig with our method. As it ca be appreciated, there is a sigificat reductio i the umber of liks that should be used i order to miimize the eergy cosumptio. 5. CONCUSIONS We have preseted a approach for reducig the amout of eergy cosumed i a cosesus etwork by fidig the optimal topology (ad power allocatio). Our method takes advatage of broadcasttype of messages i order to reduce the eergy cosumptio as compared to the uicast case of [5]. Numerical experimets show that the developed procedure performs close to the optimal case i the tested scearios. Aother advatage of our formulatio is that it ca be easily implemeted with ay geeral-purpose optimizatio software, makig our results easily reproducible. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 0.2 0.4 0.6 0.8 Fig. 4: Best topology obtaied through our method. 6. ACKNOWEDGEMENTS This work has bee coducted as part of the project HANDiCAMS www.hadicams-fet.eu. The project HANDiCAMS ackowledges the fiacial support of the Future ad Emergig Techologies (FET) programme withi the Seveth Framework Programme for Research of the Europea Commissio, uder FET-Ope grat umber: 323944. 7. REFERENCES [] I. Akyildiz, W. Su, Y. Sakarasubramaiam, ad E. Cayirci, Wireless sesor etworks: a survey, Computer Networks, vol. 38, o. 4, pp. 393 422, 2002. [2] M. H. DeGroot, Reachig a Cosesus, Joural of the America Statistical Associatio,, vol. 69, o. 345, pp. 8 2, March 974. [3] V. Borkar ad P. Varaiya, Asymptotic Agreemet i Dis- 525
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