Distributed and Secure Computation of Convex Programs over a Network of Connected Processors

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1 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY Disribued and Secure Compuaion of Convex Programs over a Newor of Conneced Processors Michael J. Neely Universiy of Souhern California hp://www-rcf.usc.edu/ mjneely Absrac We consider he fundamenal problem of opimizing a convex funcion subjec o a collecion of convex inequaliy consrains and se consrains. An ieraive algorihm is developed ha solves he problem o wihin any desired accuracy using a newor of disribued processors. The newor is assumed o form a conneced graph. Each processing node of he graph aes charge of a porion of he original consrains and solves a correspondingly less complex problem, passing ey values o neighboring nodes. The consrains can be assigned o nodes arbirarily, and individual nodes do no require nowledge of he newor opology or he consrains assigned o oher nodes. Furher, we assume ha each node has a se of privae opimizaion variables ha paricipae in he global opimizaion problem bu are unnown o oher nodes of he graph. This esablishes a general framewor for compuaional load sharing and secure opimizaion over a newor. Index Terms Disribued Compuing, Opimizaion, Privacy, Newor Securiy I. INTRODUCTION We consider he fundamenal problem of finding he minimum value of a muli-variable convex funcion subjec o a se of convex inequaliy consrains. Such convex programs have numerous applicaions, paricularly in he area of daa newors, and a variey of compuaional algorihms exis for solving hem [1][2][3][4][5]. In his paper, we develop a novel echnique for disribuing he compuaion over a conneced graph of newor processors. Each processing node aes charge of a subse of he original problem consrains, and ieraively solves a simplified problem involving only his subse. Problem parameers are updaed a every ieraion based on message passing beween neighboring nodes. Furher, we assume each processing node has a se of privae opimizaion variables ha paricipae in he global opimizaion problem bu mus be ep hidden from oher nodes of he graph. The resuling algorihm yields a soluion ha is arbirarily close o he global opimal soluion, where proximiy o opimaliy is conrolled by a parameer V ha affecs a radeoff in he required compuaion ime. These resuls conribue o he growing field of disribued compuing. This field has received a considerable amoun of aenion in recen years due o he inheren iner-neworing capabiliies of modern compuers and he numerous compuaionally inensive experimens performed by he scienific communiy. Recen heoreical resuls consider soring, ordering, and averaging over graphs [6] [7], and disribued compuaion of marix eigenvalues is considered in [8]. Relaed wor considers implemenaion of parallel algorihms over mesh newors [9] [10] [11] and disribued compuaion wih asynchronous updaes [12]. Algorihms for solving convex programs over muliple parallel processors have been developed previously in [5] [3] using a primal-dual mehodology, and disribued algorihms for solving linear programs on a newor have been recenly considered in [2] in he case when each linear inequaliy consrain involves only local variables. Disribued nonlinear opimizaion for sochasic newor flow problems is considered in [13] [14] [15]. In his paper, we develop disribued soluions o convex opimizaion problems wih any number of variables and wih general consrain ses. Our conribuions are hreefold: Firs, we develop a disribued algorihm ha operaes over any conneced graph of processors. Inequaliy consrains can be assigned o processors arbirarily o ensure an equiable sharing of sysem resources, and individual processors do no require nowledge of he consrains of oher processors. Second, we consider he issue of secure opimizaion, where a global opimum is aained wihou requiring individual processors o reveal heir privae opimizaion variables. Third, we presen our resuls in erms of Lyapunov drif heory, which deviaes significanly from he radiional primal-dual approach o convex opimizaion and simplifies he analysis. The ouline of his paper is as follows. In he nex secion we inroduce he opimizaion problem, and in Secion III we presen he disribued opimizaion algorihm. In Secion IV we inroduce Lyapunov drif heory and prove he performance bounds.

2 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY II. PROBLEM FORMULATION Consider an undireced graph wih K nodes and L lins, where nodes represen processors and lins represen iner-processor communicaion channels. We assume he graph is conneced, so ha here is a pah from any node o any oher node. For each node {1,..., K}, le N represen he se of neighboring nodes, ha is, N consiss of all nodes j such ha here is a lin beween nodes and j. We desire o use his graph of processors o compue he soluion o he following convex opimizaion problem: Problem A: Minimize: K =1 g ( x, p ) Subjec o: f ( x, p ) b for {1,..., K} x Ω Ω 1... Ω K (1) p Θ for {1,..., K} (2) where x is a vecor of public variables in R M for some ineger M, p is a vecor of privae variables in R M for some ineger M for each, g ( ), f ( ) are convex funcions of heir muli-variable argumens (where we define a vecor valued funcion o be convex if each componen funcion is convex), Ω, Ω 1,..., Ω K are convex subses of R M, and Θ 1,..., Θ K are convex subses of R M1,..., R MK, respecively. To preven infinie soluions o he above opimizaion, we assume ha all ses are compac and all funcions are bounded. Furher, we mae he following non-negaiviy assumpions: We assume ha b > 0 and g ( ) 0, f ( ) 0 for all {1,..., K} (where he inequaliies are aen enrywise), and ha he se Ω resrics he public variables x o having non-negaive enries. I is no difficul o show ha general convex opimizaion problems wih bounded funcions over compac ses can be wrien as opimizaions ha conform o he nonnegaiviy assumpions. 1 Noe ha we have defined K ses of inequaliies so ha we can assign each se o a paricular processor. In he case when here are no privae opimizaion variables, he paricular assignmen of inequaliies o processors is arbirary. However, he privae opimizaion variables p and heir consrain ses Θ represen decision variables and consrains ha paricipae in he global opimizaion, bu which mus be ep hidden from all oher processors. Such a problem arises, for example, when he privae variables represen prices or consumpion levels ha a paricular individual does no wish o reveal. We 1 Indeed, all ha is required o modify he general problem o mee he non-negaiviy assumpions is o add a sufficienly large posiive consan o boh sides of every inequaliy and mae an appropriae change of variables. now ransform he opimizaion problem ino a form ha is more conducive o disribued implemenaion. We firs designae node 1 as he roo node of he graph, and form he shores pah ree from all nodes o his roo node 1. Specifically, each node i 2 is assigned a paren node P ar(i) from is se of neighbors, and he sequence of successive parens of a given node i erminaes a he roo node 1 and forms a shores hop pah from node i o node 1. Such rees always exis in conneced graphs, and simple disribued algorihms for consrucing hem are given in [16]. We le Child(i) represen he se of all children nodes of a given node i, ha is, Child(i) is he se of all nodes j such ha P ar(j) = i. Problem B: Minimize: K =1 g ( x, p ) Subjec o: f ( x, p ) b for {1,..., K} (3) x Ω Ω for {1,..., K} p Θ for {1,..., K} x x P ar() for {2,..., K} (4) x x P ar() for {2,..., K} (5) The consrains (4) and (5) imply ha children and parens have he same x values. Because he graph is conneced, his implies ha x 1 = x 2 =... = x K. We hus have he following simple lemma, he proof of which is sraighforward and omied for breviy. Lemma 1: The vecor ( x, p 1,..., p K ) is an opimal soluion o Problem A if and only if ( x,..., x, p 1,..., p K ) is an opimal soluion o Problem B. A. The Inerior Poin Assumpion To faciliae analysis, i is useful o assume ha here exiss a vecor ( x, p 1,..., p K ) saisfying he se consrains (1) and (2), and such ha f ( x, p ) < b for all. Tha is, here exiss a poin ha saisfies all inequaliies wih sric inequaliy (noe ha we do no require he opimal poin o have his propery). We define ɛ max as he maximum value of ɛ such ha here exiss a vecor ( x, p 1,..., p K ) saisfying (1) and (2) and addiionally saisfying f ( x, p ) b ɛ for all (where ɛ is a vecor wih all enries equal o ɛ). I is no difficul o show ha, given he exisence of a posiive value of ɛ max, here mus exis a sequence of vecors x (ɛ), p (ɛ) parameerized by posiive values ɛ ɛ max ha saisfy he se consrains (1) and (2) and such ha: f ( x (ɛ), p (ɛ) ) b ɛ for 0 < ɛ ɛ max (6) while x (ɛ) x, p (ɛ) p as ɛ 0, where x and p represen he opimal soluion vecors for Problem A.

3 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY III. DISTRIBUTED AND SECURE OPTIMIZATION We now presen a disribued algorihm for compuing a soluion ha is arbirarily close o he opimal soluion of Problem B (and hence, Problem A). In paricular, each newor node aes charge of he inequaliy consrains f ( x, p ) b and he se consrains x Ω Ω and p Θ. A sequence of vecor values { x [0], x [1],..., x []} and { p [0],..., p []} is compued over ieraions, he average of which approaches he desired soluion. We noe ha he vecor x [] can be viewed as he esimae of he public variables a node a ime. To define he algorihm, le V > 0 be a conrol parameer ha affecs algorihm performance, and define he sequence δ[] =1/ 1 + for {0, 1,...}. Furhermore, we define violaion sequences U [], Y [], and Z [] as follows: Le U [0] = 0 for {1,..., K}, and le Y [0] = Z [0] = 0 for {2,..., K}. On every ieraion we updae he U [] sequences for each node {1,..., K} as follows: U [ + 1] = max[ U [] b, 0] + f ( x [], p []) (7) Liewise, for nodes {2,..., K} we have: Y [ + 1] = max[ Y [] x [] δ[], 0] + x P ar() [] (8) Z [ + 1] = max[ Z [] x P ar() [] δ[], 0] + x [] (9) where he values of p [] and x [] are compued as defined below, and where δ[] represens a vecor wih all enries equal o δ[]. The U [], Y [], and Z [] vecors are analogous o a sequence of slac variables in a dual soluion o he convex opimizaion problem of ineres, bu can inuiively be viewed as queue baclogs in a sloed queueing sysem wih arrivals and deparures deermined by he conrol decision variables x [], p []. Indeed, our algorihm below is inspired by he sable queue conrol policies of [17] [14] [18] [15]. Specifically, if he U [], Y [], and Z [] queue baclogs are ep bounded, hen i mus be he case ha he ime average inpu rae o each queue is less han or equal o he ime average service rae, so ha inequaliy consrains (3), (4), and (5) are saisfied. We noe ha he posiive δ[] sequence is defined o allow he server rae of he Y [] and Z [] queues of (8) and (9) o be slighly larger han he inpu rae o allow for sabiliy, alhough his margin decreases o zero wih increased ieraions. For he following ieraive algorihm, i is useful o define he vecor H [] for each node 2 as follows: H [] = Y [] Z [] (10) The Ieraive Algorihm: On ieraion, every node 2 ransmis is H [] vecor o is paren. Each node {1,..., K} hen compues x [] and p [] as soluions o he following opimizaion: Minimize: V g ( x, p ) + 2U [] f ( x, p ) ( 2 x H [] ) j Child() H j [] Subjec o: p Θ x Ω Ω where he vecor muliplicaion represens he sandard do produc (i.e., a sum of he producs of each enry of he wo vecors being muliplied). Each node hen ransmis is vecor x [] o all of is children (defined as he se Child()). Noe ha Child() is defined as he empy se if a node has no children. The violaion sequences U [], Y [], Z [] are hen updaed according o (7)-(9). Noe ha each node only requires nowledge of is own consrains and consrain ses, and ha he privae variables p [] are nown only o heir corresponding nodes. I is no difficul o show ha hese privae variables canno be inferred by oher nodes if hese nodes do no have exac nowledge of he individual f ( x, p ) funcions. Indeed, if a paricular node j replaces f j ( x j, p j ) and Θ j by a new funcion f j ( x j, pj 2 ) and a new consrain se 2Θ j, he resuling message passing beween neighbors will be exacly he same, alhough he magniudes of he privae variables p j [] will be doubled. I is ineresing o noe ha he above ieraive algorihm is similar o a classical subgradien search algorihm on he dual opimizaion of Problem B (see, for example, [1]), where he sepsize is normalized o 1 uni and he cos funcion is scaled by he conrol parameer V. However, he algorihm is inspired by minimizing he drif of a quadraic Lyapunov funcion of he violaion sequences U [], Y [], Z [], raher han by he classical primal-dual mehodology. One advanage of he Lyapunov approach is ha i yields a sequence of improving soluion esimaes obained by ime averages of he x [], p [] variables, and does no require a global evaluaion of he cos funcion. Specifically, we define empirical averages of he x [] and p [] sequences as follows: x av [] = 1 1 x [τ], p av [] = 1 1 p [τ] (11) Define ( x, p 1,..., p K ) as he opimal soluion vecor of Problem A, and define g as he corresponding opimal cos. Le G max represen he maximum value of K =1 g ( x, p ) over he feasible vecors saisfying he consrains of Problem A. Furher define F max as

4 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY he maximum value of f ( x, p ) 2 over all {1,..., K} and over all vecors ( x, p ) saisfying x Ω Ω, p Θ. Define B max= max b 2. Theorem 1: (Algorihm Performance) If he opimizaion Problem A saisfies he inerior poin assumpion of Secion II-A, hen for all ieraions, he above ieraive algorihm yields vecors ( x av saisfies: K =1 g ( x av []) g + C V []) wih a cos ha (12) Furher, x av [] Ω Ω and p av [] Θ for all and all, and he following inequaliy consrains are saisfied: f ( x av []) C+V KGmax b + 2ɛ max + ɛmax (13) x av [] xav P ar() [] 2 + (C+V KGmax) +1 2 (14) where C is defined: ( ) C=K 2 max 1 + x max x Ω x Ω x 2 + B max + F max Hence, a value of V can be seleced so ha he vecors x av [] have a resuling cos ha is arbirarily close o he minimizing cos g. The algorihm can be run for a number of ieraions unil he consrains (13) and (14) are arbirarily close o he consrains (3)-(5). The proof of his heorem is provided in he nex secion. A. Discussion Noe ha he above algorihm is inherenly disribued, where each processor communicaes only wih is neighboring processor, as specified by he underlying graph srucure. Indeed, each node mainains is own esimae x [] of he public variables x, and he se consrain x Ω Ω 1... Ω K is enforced locally by resricing each esimae x [] o he local se consrain Ω Ω wihou requiring nowledge of he oher se consrains. The algorihm yields a sequence of soluions x av [] wih improved error bounds a each imesep. This is an imporan feaure for disribued applicaions, and a significan deparure from he primaldual opimizaion resuls of [1] which involve mainaining a running bes soluion for he global problem as compuaions proceed. Evaluaing he bes soluion compued so far is no always possible in disribued seings, as i involves complee nowledge of he global problem and usually requires all processors o share all of heir sae variables and consrain ses wih all oher processors. In our algorihm, each node passes vecors H [] only o neighboring nodes, and his informaion is sufficien o ensure ha local esimaes of he public variables ge progressively closer and closer o saisfying he global consrains. We noe ha he inerior poin assumpion leads o an inequaliy consrain (13) ha converges o (3) lie O(1/). In he case when no inerior poin exiss, he updae equaion (7) can be modified by he sequence δ[] as in (8) and (9). However, his would yield a convergence of O(1/ ). IV. PERFORMANCE ANALYSIS To prove he heorem, we firs presen a fundamenal resul concerning Lyapunov drif. Lyapunov drif heory has been useful in developing sable conrol policies for queueing sysems [17] [19] [20] [21] [14], and he heory has recenly been exended o allow for performance opimizaion of sochasic newors [15] [18]. Here, we consider a deerminisic varian of he Lyapunov resul in [15] applied o he violaion sequences of he previous secion. Specifically, le U[] represen a vecor sequence of non-negaive variables indexed by ime {0, 1, 2,...}. Define he Lyapunov funcion L( U[]) = U[] 2. Here we use he Euclidean norm, so ha U 2 represens he sum of squares of he individual enries of vecor U. Define he single-sep Lyapunov drif as follows: ( U[]) = L( U[ + 1]) L( U[]) A any ime, he U[] vecor can be viewed as he curren sae of a dynamic sysem. Le x[], p[] be vecor sequences represening conrol decision variables ha effec he evoluion of U[]. Le g( x, p) be a nonnegaive cos funcion, assumed o be convex in he composie vecor ( x, p). We assume he cos funcion is bounded and define G max as an upper bound on he maximum cos over all possible vecors x and p. Consider any paricular arge vecors x, p ha yield a desired cos. Theorem 2: (Opimizaion via Lyapunov Drif) If U[0] = 0 and if here are posiive consans V, ɛ, C such ha for all imeslos and all sequences U[] we have: ( U[]) C ɛ U[] + V g( x, p ) V g( x[], p[]) hen for all {1, 2,...} we have: (a) g( x av [], p av []) g( x, p ) + C/V (b) U[] C+V Gmax ɛ + ɛ 2 where x av [] and p av [] are empirical averages of he conrol variables, defined as in (11). Proof: To prove par (a), noe ha he drif condiion of he heorem ogeher wih non-negaiviy of he U[] values imply ha for all : ( U[]) + V g( x[], p[]) C + V g( x, p )

5 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY Summing over all imes τ dividing by, we have: L( U[]) {0,..., 1} and + V 1 1 g( x[τ], p[τ]) C + V g( x, p ) (15) Because g( x, p) is convex in ( x, p), by Jensen s inequaliy we have 1 1 g( x[τ], p[τ]) g( xav [], p av []). Using his bound ogeher wih non-negaiviy of he Lyapunov funcion in he inequaliy (15) yields par (a) of he heorem. To prove (b), noe ha he drif condiion implies: ( U[]) C + V G max ɛ U[] Because he U[] vecors have non-negaive enries, i follows ha he maximum incremen in L( U[]) is C + V G max. Furher, L( U[]) canno increase if ɛ U [] C + V G max. Therefore, we have for all imes : L( U[]) L + C + V G max (16) where he value of L is he maximum value of U 2 subjec o ɛ U C+V G max. The maximum is achieved when all weigh is placed on a single enry of U, and hence L = (C + V G max ) 2 /ɛ 2. I follows from (16) (C+V G max) 2 ha U[] ɛ + C + V G 2 max. I is no A difficul o show ha 2 ɛ + A A 2 ɛ + ɛ 2 for all posiive values A, ɛ, and he resul of par (b) follows. A. Proof of Theorem 1 We now compue he Lyapunov drif associaed wih he ieraive algorihm of he previous secion. Firs, le U[], Y [], and Z[] represen composie vecors consising of concaenaed U [], Y [], and Z [] vecors for {1,..., K}. We define he Lyapunov funcion L( U, Y, Z) = U 2 + Y 2 + Z 2. The one-sep Lyapunov drif is hus: ( U[], Y [], Z[]) =L( U[ + 1], Y [ + 1], Z[ + 1]) L( U[], Y [], Z[])) (17) Consider he updae equaions (7)-(9) for he violaion sequences. Taing he squared norm of boh sides of (7) yields: U [ + 1] 2 U [] 2 + b 2 + F max 2U [] ( b f ( x [], p [])) Liewise, aing he squared norm of boh sides of (8) yields: Y [ + 1] 2 Y [] 2 + x [] + δ[] 2 + x P ar() [] 2 2 Y [] ( x [] + δ[] x P ar() []) Similarly, squaring (9) yields: Z [ + 1] 2 Z [] 2 + x P ar() [] + δ[] 2 + x [] 2 2 Z [] ( x P ar() [] + δ[] x []) Using he abbreviaed noaion o represen he Lyapunov drif defined in (17), i follows ha he drif saisfies: C 2 U [] b f ( x [], p [])) 2 ( Y [] x [] + ) δ[] x P ar() [] 2 ( Z [] x P ar() [] + ) δ[] x [] +V g ( x [], p []) V g ( x [], p []) (18) where C is defined in Theorem 1, and where we have added and subraced he opimizaion meric. By shifing he sums, using he definiion H [] = Y [] Z [], and recalling ha Child() is he se of all nodes j such ha P ar(j) =, we have: C 2 U [] ( b f ( x [], p [])) 2 x [] ( H [] j Child() H j []) 2 δ[] ( Y [] + Z []) +V g ( x [], p []) V g ( x [], p []) (19) The righ hand sides of (18) and (19) are idenical. However, from he laer expression i is clear ha he ieraive algorihm defined in he previous secion is precisely designed o minimize he sum of he second, hird, fourh, and fifh erms on he righ hand side of he above inequaliy (19) over all feasible conrol vecors x, p. Hence, he drif is less han or equal o he resuling righ hand side if a paricular se of feasible conrol vecors are plugged ino he second, hird, fourh, and fifh erms. Now recall from he inerior poin assumpion of Secion II-A ha feasible vecors x (ɛ), p (ɛ) exis and saisfy f ( x (ɛ), p (ɛ) ) b ɛ for all and for all ɛ such ha 0 < ɛ ɛ max. Hence, using he righ hand side as expressed in (18), we have: C 2 U [] ɛ 2 δ[] ( Y [] + Z []) +V g ( x (ɛ), p (ɛ) ) V g ( x [], p []) The above drif expression is in he form specified by he Lyapunov drif heorem (Theorem 2). Hence, we have he following for all imes : g ( x av []) g ( x (ɛ), p (ɛ) ) + C/V (20) U[] C+V KGmax 2ɛ + ɛ (21) Y [] + Z[] C+V KGmax 2δ[] + δ[] (22)

6 DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY where we have used he fac ha δ[] decreases wih, and where he norm of he composie vecors U[] and Y [] + Z[] is given by: U[] = K U [] 2 Y [] + Z[] = =1 K Y [] + Z [] 2 =2 The inequaliies (20)-(22) hold for any value ɛ such ha 0 < ɛ ɛ max. Hence, hey can be opimized separaely by choosing he bes ɛ value. Recall ha x (ɛ) and p (ɛ) converge o he opimal operaing poin as ɛ 0. Hence, aing a limi in (20) as ɛ 0 proves he performance bound (12). Conversely, he bound in (21) is minimized by seing ɛ = ɛ max, proving ha: U[] C + V KG max 2ɛ max + ɛ max (23) Recall ha he U [] values represen queue baclogs (see (7)). I follows ha he accumulaed arrivals o he queue during he firs slos are less han or equal o he maximum possible deparures plus he baclog a ime 1: 1 f ( x [τ], p [τ]) b + U [ 1] Dividing he above inequaliy by and using Jensen s inequaliy o push he ime average summaion inside he convex funcion f ( ) yields: f ( x av []) U b + [ 1] (24) Combining (24) and (23) proves he performance bound (13). Similarly, he performance bound (14) can be proven direcly from (22), using he fac ha 1 δ[τ] 2/ 1/ for 1, as well as he 1 fac ha he accumulaed arrivals o he Y [] queues over he firs slos are bounded by he oal service opporuniies over his inerval plus he baclog a ime 1. This proves Theorem 1. V. CONCLUSIONS We have developed a framewor for disribued and secure compuaion of convex programs using an arbirary conneced graph of processors. Each node of he graph paricipaes in he opimizaion by solving a simplified problem involving boh public and privae variables, and he resuling algorihm mainains privacy while achieving global opimaliy. Our analysis uses a Lyapunov drif echnique ha ransforms he opimizaion ino a corresponding queue conrol sraegy. The echnique is quie general and can be exended o rea sochasic versions of his problem. REFERENCES [1] D. P. Berseas, A. Nedic, and A. E. Ozdaglar. Convex Analysis and Opimizaion. Boson: Ahena Scienific, [2] Y. Baral, J. W. Byers, and D. Raz. Fas, disribued approximaion algorihms for posiive linear programming wih applicaions o flow conrol. Siam Journal of Compuing, vol. 33, no. 6, pp , [3] D. P. Berseas and P. Tseng. Parial proximal minimizaion algorihms for convex programming. Massachuses Insiue of Technology Technical Repor, [4] D. P. Berseas and J. N. Tsisilis. Parallel and Disribued Compuaion: Numerical Mehods. Prenice-Hall, Englewood Cliffs, NJ, [5] M. C. Ferris and O. L. Mangasarian. Parallel consrain disribuion. SIAM Journal on Opimizaion, vol. 1, pp , [6] D. Kempe, A. Dobra, and J. Gehre. Gossip-based compuaion of aggregae informaion. Proceedings of FOCS, [7] J. K. Bordim, K. Naano, and H. Shen. Soring on singlechannel wireless sensor newors. Inernaional Symposium on Parallel Archiecures, Algorihms, and Newors, May [8] D. Kempe and F. McSherry. A decenralized algorihm for specral analysis. Proc. of STOC, [9] M. Singh, V. K. Prasanna, and J. D. P. Rolim. Collaboraive and disribued compuaion in mesh-lie wireless sensor arrays. Personal Wireless Communicaions, Sep [10] R. Miller and Q. F. Sou. Parallel algorihms for regular archiecures: Meshes and pyramids. MIT Press, [11] D. Nassimi and S. Sahni. Bionic sor on mesh-conneced compuers. IEEE Trans. on Compuers, vol. c-27, Jan [12] J. N. Tsisilis, D. P. Berseas, and M. Ahens. Disribued asynchronous deerminisic and sochasic gradien opimizaion algorihms. IEEE Transacions on Auomaic Conrol, Vol. AC- 31, no. 9, Sepember [13] M. J. Neely. Dynamic Power Allocaion and Rouing for Saellie and Wireless Newors wih Time Varying Channels. PhD hesis, Massachuses Insiue of Technology, LIDS, [14] M. J. Neely, E. Modiano, and C. E Rohrs. Dynamic power allocaion and rouing for ime varying wireless newors. IEEE Journal on Seleced Areas in Communicaions, January [15] M. J. Neely, E. Modiano, and C. Li. Fairness and opimal sochasic conrol for heerogeneous newors. Proceedings of IEEE INFOCOM, March [16] D. P. Berseas and R. Gallager. Daa Newors. New Jersey: Prenice-Hall, Inc., [17] L. Tassiulas and A. Ephremides. Sabiliy properies of consrained queueing sysems and scheduling policies for maximum hroughpu in mulihop radio newors. IEEE Transacaions on Auomaic Conrol, Vol. 37, no. 12, Dec [18] M. J. Neely. Energy opimal conrol for ime varying wireless newors. Proceedings of IEEE INFOCOM, March [19] L. Tassiulas and A. Ephremides. Dynamic server allocaion o parallel queues wih randomly varying conneciviy. IEEE Trans. on Inform. Theory, vol. 39, pp , March [20] N. McKeown, V. Ananharam, and J. Walrand. Achieving 100% hroughpu in an inpu-queued swich. Proc. INFOCOM, [21] E. Leonardi, M. Melia, F. Neri, and M. Ajmone Marson. Bounds on average delays and queue size averages and variances in inpu-queued cell-based swiches. Proc. INFOCOM, 2001.