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6 6 ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG node, bu hese okens will coninue raveling downhill along he pah, so ha he heigh of he fourh node will never grow o be much larger han (n 3)θ. This process obeys he cu condiion, bu aferwards, here are wo adjacen nodes wih almos 2θ heigh difference. Forunaely, alhough here are now more nodes wih large heighs, he adversary had o pay for his rearrangemen by making he highes queue smaller. In an amorized sense, he siuaion has no become worse. These observaions sugges mainaining heigh bounds for each subse of he nodes, and showing ha hese bounds form an invarian. For convenience, we will wrie b (S) = v S b (v) for any se S V of verices (and similarly for oher quaniies like h (S) and δ (S)). Wih denoing he maximum degree of any verex, we wrie γ = 2 + θ. The key invarian is he following: b (S) n j=n S +1 (H + γ j) for all S V. (2.2) γ H n S Fig An illusraion of Invarian 2.2 Figure 2.1 illusraes he upper bound of his invarian picorially as he sum of he column heighs of he righ-mos S columns. Below, we prove Lemma 2.2, showing ha Inequaliy (2.2) is indeed an invarian over ime for he algorihmsclb θ, agains any adversary respecing he cu condiion. Lemma 2.2. If he adversary respecs he cu condiion, and he invarian (2.2) holds a he beginning of round, hen i holds a he beginning of round + 1. Using his lemma, he proof of Theorem 2.1 is sraighforward. Proof of Theorem 2.1. We prove by inducion ha (2.2) holds a every ime. A ime 1, h 1 (v) H for all v by definiion, so b 1 (S) v S H n j=n S +1 (H + γ j)

7 STABILITY OF LOAD BALANCING ALGORITHMS 7 for all ses S V. The inducion sep from o + 1 follows from Lemma 2.2, and we can apply he resuling guaranee o he singleon ses {v}, yielding a bound of B = H + γ n. Proof of Lemma 2.2. The proof is by conradicion. Assume ha he invarian (2.2) holds a he beginning of round, bu no a he beginning of round + 1. Le S be a se maximizing n Φ(S) := b +1 (S) (H + γ j). j=n S +1 If several ses achieve he maximum value, le S have minimal size among all hese ses. Firs off, noice ha he choice of S guaranees ha eiher all u S have posiive b +1 (u), or hey all have negaive b +1 (u), and hence b +1 (S) = u S b +1(u). Since (2.2) was assumed o hold a he beginning of round, and fails a he beginning of round + 1, we know ha b +1 (S) > b (S). How can he values h (u) for nodes u S change? Adversary sep: Subsiuing he definiions of b and b, we obain ha for any se S, b (S) = h (S) S a +1 = h (S) + δ (S) S a +1 = b (S) + δ (S) S (a +1 a ) b (S) + δ (S) S (a +1 a ) b (S) + e(s). The wo inequaliies hold because of he Triangle Inequaliy and he cu condiion on he adversary. Redisribuion sep: Fix an edge e = (u, v) wih u S and v / S. Because S maximizes Φ and has minimal size, u has imbalance b +1 (u) > H + γ (n S + 1), since oherwise Φ(S u) Φ(S). In paricular, b +1 (u) > γ. Because v was no included in S, is imbalance b +1 (v) mus eiher have sign opposie o he sign of b +1 (S), or have absolue value b +1 (v) H + γ (n S ). In eiher case, b +1 (u) b +1 (v) > γ, and simply subsiuing he definiion of γ, we also obain h +1 (u) h +1 (v) > 2 + θ. During he Redisribuion sep of round, a mos okens can have moved o or from nodes u and v, so heir heighs can have changed by a mos each, and herefore heir previous heigh difference is a leas h (u) h (v) > θ. Figure 2.2 illusraes he gap of a leas γ beween he queue heighs of nodes in S and ouside S. If b +1 (u) 0, hen h (u) h (v) > θ, so he algorihm SCLB θ moves a oken from u o v along e, and no okens from v o u, hereby decreasing b (u) by 1. On he oher hand, if b +1 (u) < 0, hen h (v) h (u) > θ, and a oken mus be moved from v o u along e, increasing he (negaive) imbalance b (u) by 1. Because b +1 (u) > γ = θ + 2, and herefore b (u) > θ +, he sign of he imbalance does no change during he Redisribuion sep, even if okens were moved o or from u, and hence, b (u) decreased by 1 as a resul of edge e. This holds for every edge e e(s), and using he fac ha he average a does no change during he Redisribuion sep, we obain ha b +1 (S) = u S b +1(u) ( u S b (u) ) e(s) = b (S) e(s).

8 8 ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG b (u) +1 γ H+γ(n- S +1) H+γ(n- S )..... Fig The gap of γ beween he queue heighs of nodes in S and ouside S in he case ha b +1 (S) is posiive. S Puing he argumens for he wo seps ogeher, we obain ha b +1 (S) b (S) e(s) b (S). This conradics our assumpion ha b +1 (S) > b (S), and hus complees he proof. Noice ha our bound B = H +γ n is asympoically igh. To see his, consider a simple pah of lengh n. I is cerainly legal for he adversary o inser one oken a node n in every round, and never remove okens. Afer abou θ n2 2 rounds, each node k will conain abou kθ okens, and hence he imbalance of node n is abou θ n 2.3. Capaciies, dynamic neworks, and ime windows. The resul of Theorem 2.1 can be easily exended o he case ha he edges have (ineger) capaciies c e associaed wih hem, and up o c e okens can be sen along e in every round. We assume ha whenever he algorihm decides o send okens from u o v along e, i sends as many as possible, i.e., bounded only by he capaciy c e and he number of okens a u. The cu condiion now requires ha he imbalance creaed by he adversary be resriced by he oal capaciy of he cu. Showing ha he above algorihm is sill sable in his capaciaed version is easy given Theorem 2.1. Corollary 2.3. In he above capaciaed scenario, for any adversary respecing he cu condiion, and any θ 1, he algorihm SCLB θ is sable, i.e., here is a consan B (depending on H, θ, G, and he maximum edge capaciy max e c e ), such ha b (v) B for all nodes v a all imes. Proof. Replace each edge wih capaciy c e by c e parallel edges of capaciy 1. By Theorem 2.1, he algorihm is sable on his new graph, and herefore, i is also sable on he capaciaed graph. Noice ha he consan γ and hence he bound B now depend on he maximum capaciy, and so insead of B = H + 2 n + θn, he new bound becomes B = H + 2 n max e c e + θn, where is he maximum degree of he graph. 2.

10 10 ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG A each ime, for each edge e = (u, v): Choose i o maximize h (i) (u) h (i) (v). If h (i) (u) h (i) (v) + θ, hen send a oken of commodiy i from u o v. If h (i) (v) h (i) (u) + θ, hen send a oken of commodiy i from v o u. In he case of a single commodiy, his algorihm specializes o SCLB θ. The naural analogue of he cu condiion for a single commodiy is o require ha he adversary saisfy δ (i) (S) S (a (i) +1 a(i) ) e(s) (3.1) i for all node ses S V and imes. This would require ha he oal imbalance for se S creaed by he adversary could be balanced along edges leaving S. Unforunaely, Inequaliy (3.1) is oo weak a resricion i allows he adversary o creae paerns of addiion and removal ha canno be balanced by any algorihm, wheher offline or online. A he end of his secion, we show how o use a reducion from he muli-commodiy flow problem o creae such an adversary wih k 3 commodiies. For he special case k = 2, however, he Max-Flow Min-Cu Theorem sill holds, and in fac, we can show ha he cu condiion is sufficien o ensure ha algorihm MCLB θ is sable Sabiliy for k=2. We le H = max v V {h (1) 1 (v) + h(2) 1 (v)} be he maximum heigh of he queues a any node during he sar of he execuion, and he maximum degree of any verex. This ime, we define γ slighly differenly, namely γ = 2 + 2θ. Theorem 3.1. There is a consan B (depending on H, θ and G), such ha for any adversary respecing he cu condiion, MCLB θ ensures b (i) (v) B a all imes, for all verices v, and commodiies i = 1, 2. Proof. A he sar of he execuion, b (1) 1 (S) + b(2) 1 (S) v S H, by definiion of H. The key Lemma 3.2 esablishes ha for all S V, imes, and commodiies i = 1, 2, b (1) (S) + b (2) (S) n j=n S +1 (H + γ j) (3.2) We can hen apply he resul o all singleon ses {v}, proving he heorem. Lemma 3.2. If (3.2) holds a he beginning of round, i holds a he beginning of round + 1. Proof. The proof is by conradicion. Le S be a se (of minimum size in case of ies) maximizing Φ(S) := b (1) (S) + b (2) (S) n j=n S +1 (H + γ j). In he case of wo commodiies, a node migh be included in S because i conribues a lo o he imbalance in one of he commodiies, alhough is conribuion o he oher commodiy migh acually be negaive. To capure he imbalance conribuion of a node o each commodiy, we define he signed imbalance β (i) (v) := sgn(b(i) +1 (S))

11 STABILITY OF LOAD BALANCING ALGORITHMS 11 b (i) (v), β(i) (v) := sgn(b(i) +1 (S)) b(i) (v), for every node v V, and every ime sep. Here, sgn denoes he sign of a erm. Noice ha we use he sign of b (i) +1 (S) a all ime seps in he definiion of β (i) (v) (no he sign of b(i) (S) or of b(i) (S)). We can now rewrie he oal imbalance over he se S a ime + 1 as +1 b (1) +1 (S) + b(2) +1 (S) = (β (1) +1 (u) + β(2) +1 (u)). (3.3) u S Again, we show ha he change in he imbalance for se S canno be posiive, by comparing he increase in he imbalance of S during he Adversary sep wih he decrease during he Redisribuion sep, and hus obain a conradicion. Adversary sep: We know ha for each commodiy i = 1, 2, he imbalance on se S afer he Adversary sep is a mos b (i) (S) b (i) (S) + δ (i) (S) S (a (i) +1 a(i) ) by he Triangle Inequaliy. Now, summing over i = 1, 2 and applying he cu condiion on he adversary yields ha b (1) (S) + b (2) (S) b (1) (S) + b (2) (S) + e(s). Redisribuion sep: Fix a node u S, and an edge e = (u, v) of G wih v / S. As in he proof of Lemma 2.2, we can use he definiion of S (as maximizing Φ and being of minimal size) o obain ha he signed imbalances a nodes u and v saisfy (u) + β(2) +1 (u) > β(1) +1 (v) + β(2) +1 (v) + γ. As u and v can lose and gain a mos okens each during he Redisribuion sep, β (1) +1 β (1) (u) + β (2) (u) > β (1) (v) + β (2) (v) + 2θ. (3.4) In paricular, here mus be a commodiy i such ha β (i) (u) > β (i) (v) + θ, and hus also h (i) (u) h (i) (v) > θ. Hence, MCLB θ moved a oken along edge e during he Redisribuion sep (w.l.o.g., i was a oken of commodiy 1). We wan o show ha his oken acually decreased he signed imbalance of node u. Assume ha i did no. This means ha if he oken moved from u o v, hen sgn(b (1) +1 (S)) is negaive, and if he oken moved from v o u, hen sgn(b (1) +1 (S)) is posiive. In eiher case, he signed imbalance for commodiy 1 a node v mus be higher han a node u, and so β (1) (v) β (1) (u) = b (1) (v) b (1) (u) = h (1) (v) h (1) (u). Because MCLB θ maximizes he difference in is choice of commodiy, we obain ha β (1) (v) β (1) (u) = h (1) (v) h (1) (u) h (2) (v) h (2) (u) = β (2) (v) β (2) (u) β (2) (u) β (2) (v). Rearranging his inequaliy yields ha β (1) (v) + β (2) (v) β (1) (u) + β (2) (u), and hus a conradicion wih Inequaliy (3.4). Therefore, every edge (u, v) wih v / S decreases he signed imbalance of u by 1. Summing over all edges and all nodes u S,

13 STABILITY OF LOAD BALANCING ALGORITHMS 13 The firs inequaliy is simply he Triangle inequaliy, and he second inequaliy holds because he balancing algorihm never exceeds he capaciy of any edge wih any of is oken moves, and herefore boh σ r (i) (u, v) and σ r (i) (v, u) lie beween 0 and c (u,v). Flow Conservaion: For any node v and commodiy i, we can wrie ( ) = = (u,v) E (u,v) E r= 1 r= (u,v) E f (i) (u,v) 1 (σ r (i) (u, v) σ r (i) (v, u)) 1 σ r (i) (u, v) r= (u,v) E 1 = h (i) (v) h(i) (v) δ r (i) (v) r= σ (i) r (v, u) = b (i) (v) + a(i) b(i) (v) a (i) ( ) δ (i) (v) = ( )(D δ (i) (v)). The hird equaliy is rue because h (i) (v) h(i) (v) is exacly he number of okens ha enered v during he ime period [, 1], minus he amoun of okens ha lef v during his period. In he las equaliy, we used ha b (i) (v) = b(i) (v) for all i and v, and ha a (i) r = r D for all imes r. Now, if node v is neiher he source nor he sink for commodiy i, hen δ (i) (v) = D, so flow is conserved. If v is he source of commodiy i, hen δ (i) (v) = D + d i, so he oal flow enering node s i is d i. If v is he sink for commodiy i, hen he oal flow enering node i is d i, because δ (i) (v) = D d i by definiion. Hence, f saisfies flow conservaion and all demands. f conserves flow, saisfies all demands, and does no exceed any edge capaciies, so i is a feasible muli-commodiy flow for he given demands. By combining Lemma 3.3 wih he sabiliy of MCLB θ proved in Theorem 3.1, we obain as a corollary an alernae proof of he wo-commodiy Max-Flow Min-Cu Theorem. I only remains o verify ha he adversary A as defined in Lemma 3.3 indeed respecs he cu-condiion. Corollary 3.4 (2-Commodiy Max-Flow Min-Cu). Le G = (V, E) be a graph wih edge capaciies c e, and wo demand pairs (s 1, 1 ), (s 2, 2 ) wih demands d 1, d 2 such ha for any verex se S V, he oal demand of commodiies i {1, 2} wih exacly one of {s i, i } in S is a mos e e(s) c e. Then, here exiss a feasible wo-commodiy flow sending d i unis of flow from s i o i for i = 1, 2. Proof. Define he adversary A as in Lemma 3.3. To show ha he algorihm MCLB θ is sable agains A, we merely have o verify ha A saisfies he cu condiion. Le S V be arbirary. For convenience, we wrie [u S] := 1 if u S, and 0

18 18 ELLIOT ANSHELEVICH, DAVID KEMPE, AND JON KLEINBERG A. Richa, R. Tarjan, and D. Zuckerman, Tigh analyses of wo local load balancing algorihms, in Proc. 27h ACM Symp. on Theory of Compuing, [17] T. Hu, Muli-commodiy nework flows, Operaions Research, 11 (1963). [18] F. Leighon and S. Rao, Mulicommodiy max-flow min-cu heorems and heir use in designing approximaion algorihms, Journal of he ACM, 46 (1999), pp [19] N. Linial, E. London, and Y. Rabinovich, The geomery of graphs and some of is algorihmic applicaions, Combinaorica, 15 (1995). [20] M. Mizenmacher, On he analysis of randomized load balancing schemes, in Proc. 9h ACM Symp. on Parallel Algorihms and Archiecures, [21] S. Muhukrishnan and R. Rajaraman, An adversarial model for disribued dynamic load balancing, in Proc. 10h ACM Symp. on Parallel Algorihms and Archiecures, [22] D. Peleg and E. Upfal, The oken disribuion problem, SIAM J. on Compuing, 18 (1989). [23] P. Seymour, A shor proof of he wo-commodiy flow heorem, J. of Combinaorial Theory, 26 (1979). [24] B. Shirazi, A. Hurson, and K. Kavi, Scheduling and Load Balancing in Parallel and Disribued Sysems, IEEE Compuer Sociey Press, 1995.

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