Near-Perfect Load Balancing by Randomized Rounding

Transcription

1 Near-Perfec Load Balancng by Randomzed Rondng Tobas Fredrch Inernaonal Comper Scence Inse 947 Cener S, Se Berkeley, CA, USA Thomas Saerwald Inernaonal Comper Scence Inse 947 Cener S, Se Berkeley, CA, USA ABSTRACT We consder and analyze a new algorhm for balancng ndsble loads on a dsrbed nework wh n processors The am s mnmzng he dscrepancy beween he maxmm and mnmm load In eery me-sep pared processors balance her load as eenly as possble The drecon of he excess oken s chosen accordng o a randomzed rondng of he parcpang loads We proe ha n comparson o he correspondng model of Raban, Snclar, and Wanka 998 wh arbrary rondngs, he randomzaon yelds an mproemen of roghly a sqare roo of he acheed dscrepancy n he same nmber of me-seps on all graphs For he mporan case of expanders we can een achee a consan dscrepancy n Olog nlog log n 3 ronds Ths s opmal p o log log n- facors whle he bes preos algorhms n hs seng eher reqre Ωlog n me or can only achee a logarhmc dscrepancy Ths resl also demonsraes ha wh randomzed rondng he dfference beween dscree and connos load balancng anshes almos compleely Caegores and Sbec Descrpors: F [Theory of Compaon: Analyss of Algorhms and oblem Complexy General Terms: Algorhms, Theory INTRODUCTION Consder an applcaon rnnng on a parallel or dsrbed nework conssng of n processors conneced n an arbrary opology Each processor has nally a collecon of obs whch we call okens The goal of load-balancng s o reallocae he okens by ransmng hem along edges so ha each processor has nearly he same amon of load The problem has manfold applcaons n ob schedlng, rong, adape mesh paronng, fne elemen compaons, and n smlaons of physcal phenomena Permsson o make dgal or hard copes of all or par of hs work for personal or classroom se s graned who fee proded ha copes are no made or dsrbed for prof or commercal adanage and ha copes bear hs noce and he fll caon on he frs page To copy oherwse, o repblsh, o pos on serers or o redsrbe o lss, reqres pror specfc permsson and/or a fee STOC 09, May 3 Jne, 009, Behesda, Maryland, USA Copyrgh 009 ACM /09/05 $500 There are aros models for load balancng A common smplfyng assmpon s ha he okens are dsble Ths dealzed process s ery well ndersood [8 Howeer, he dsbly assmpon s nald for many applcaons [ I has been shown ha he deaon can be qe sgnfcan [, and he qeson of a precse qanae relaonshp beween he dscree and he dealzed process has been posed by seeral ahors [0,, 5, 8, Exsng models also dffer n he assmpons regardng commncaon n he nderlyng nework Some models resrc he nmber of okens send across a lnk a a me [, 0, 6 On he oher hand, he dffson model allows load o be moed from each processor o all s neghbors n parallel n each me sep [8, 8, As poned o by Ghosh and Mhkrshnan [, s more effcen o send a sream of many okens o one neghbor han o send one oken o each neghbor Ths moaes s o sdy he balancng crc model [3 where each erex ransfers an arbrary nmber of okens o exacly one neghbor n each rond [5,, 9 The load-balancng process of a balancng crc s goerned by a seqence of no necessarly perfec machngs ogeher wh an orenaon of each edge In each rond, wo pared erces balance her loads as eenly as possble If hs s no possble, he excess oken s send n he drecon of he edge For an arbrary graph s no clear how o choose a good seqence of machngs ha balance he load qckly Ths s dfferen for hghly srcred graphs There s for example a canoncal choce of machngs for he hypercbe ha ses n rond all edges across dmenson mod log n [ In general sch machng seqences wh a fxed perod d are called perodc balancng crcs noe ha ypcally d s of order he maxmm degree of G Le x 0 R n be he nal load ecor of he nework The am of load-balancng algorhms s o redce he nal dscrepancy K = max x 0 x 0 whn a ceran nmber of seps For hs, Raban, Snclar and Wanka [9 nrodced he so-called local dergence ha prodes an pper bond on he deaon beween he dealzed and he dscree model oer all me seps They [9, Corollary 5 showed n he perodc balancng crcs model ha whn Od logkn/ seps he dscrepancy can be redced o Od log n/ λ, where λ s he egenale gap of he balancng marx Noe ha he same edge may hae dfferen orenaon oer me The precse model s descrbed n Secon

2 Graph class Ronds Dscrepancy Orenaon Reference General Graph Balancng Crc Consan-Degree Expander Balancng Crc d-dm Tors Balancng Crc Ωmax{damG, log n/log } arb oposon 3 OdlogKn/ λ Odlog n/ λ arb [9, Corollary 5 OdlogKn/ λ O d log n/ λ rand Corollary 45 OdlogKn/ λ Od log log n/ λ rand Corollary 47 OlogKn Olog n arb [9, Corollary 5 OlogKn Olog log n rand oposon 48 OlogKnlog log n 3 O rand Corollary 5 OlogKn n /d Od n /d arb [9, Theorem 8 OlogKn n /d O d n /d log n rand Corollary 49 On /d logkn + d n +/d sd [9, Corollary 0 General Graph OlogKn/ λ Ologn/ λ arb [9 Random machng OlogKn/ λ O logn/ λ log n rand Corollary 54 Expander OlogKn Ologn arb [9 Random machng OlogKnlog log n 3 O rand Theorem 5 Table Smmary and comparson of or new pper and lower bonds on he dscrepancy for dfferen graphs Arb, sand, and rand refer o an arbrary, sandard and randomzed orenaon of all edges, respecely K s he nal dscrepancy of he load ecor In he balancng crc model, λ s he second larges egenale of he rond marx P = d k= Pk In case of random machngs, λ s he second larges egenale of he sandard dffson marx Q cf Secon n absole ale Or resls When balancng an odd nmber of ndsble okens, s crcal o decde n whch drecon o send he excess oken The resls of Mhkrshnan, Ghosh and Schlz [8 and mos resls of Raban e al [9 hold regardless of he orenaon of he machng edges We obsere n Secon 3 ha n hs case on eery graph n log n me-seps he dscrepancy canno be redced below Olog n/ log log n and ha n me-seps are necessary for consan dscrepancy Ths s raher nsasfyng as he dealzed process reqres only a logarhmc nmber of ronds o reach a consan dscrepancy on many mporan graphs lke expanders, complee graphs and hypercbes In order o redce he deaon beween he dealzed and he dscree process we follow a sggeson of Raban e al [9 and dsrbe he excess oken n a more balanced manner, ha s, n a random drecon Inely s clear ha hs dscree randomzed model shold be closer o he dealzed process Howeer, s srprsng ha hs small dfference n he model resls n a sch a as redcon of he dscrepancy In more deal, or resls ha are smmarzed n Table are as follows For general graphs we redce he dscrepancy o } d log log n O mn n O d logkn seps whp cf Corollary 45 and 47 The analogos resl of [9, { d log n, Corollary 5 only achees a dscrepancy of O d log n he same nmber of ronds Whle hs ges a qadrac mproemen for many graph, he mproemen for consandegree expanders from Olog n o Olog log n s een exponenal Ineresngly, or proof also reeals ha he deaon beween he dscree and dealzed model s Olog log n for consan-degree expanders, whch we proe o be gh on any consan-degree expander cf Theorem 40 For he d-dmensonal ors graph we achee a dscrepancy of O d n /d log n cf Corollary 49 n a nmber of ronds where [9, Theorem 8 only achees Od n /d whp As mgh be hard o defne canoncal machngs for nonsrcred graphs, s poplar o se random machngs Raban e al [9 dd no consder random machngs, b a sraghforward adapaon of her echnqes yelds hs bond n nsead, see eg [4, 5, 0, We proe resls ha hold for a large class of randomly generaed machngs ncldng he models of [5, 0, For arbrary graphs we proe a bond of O log n log n afer O logkn ronds whp cf Corollary 54 If / λ s he domnan erm, we ge agan a qadrac mproemen oer he model wh arbrary drecons of he machng edges For he mporan case of an expander where / λ s consan, we do a separae analyss and show ha he dscrepancy ges down o O n nearly opmal me OlogKnlog log n 3 whp cf Theorem 5 Ths resl can also be exended o consan-degree expanders n he balancng crc model wh approprae deermnsc machngs cf Corollary 5 Oerall, or resls demonsrae ha wh an approprae randomzaon he gap beween he dscree and dealzed model decreases sgnfcanly For he mporan class of expander graphs he dfference dsappears almos compleely On hese graphs s neresng o look a he me-dscrepancy rade-off, whch can be measred as he prodc of me and he acheed dscrepancy All preos rade-offs had a me-dscrepancy prodc of Ωlog n + logkn [, 8, 0, 8, 9, 6 whle we achee a prodc OlogKnlog log n 3 whch s ery close o he naral lower bond of ΩlogKn Relaed work Balancng crcs were nrodced by Aspnes, Herlhy and Sha [3 They consrced a seqence of Θlog n machngs ha achee a dscrepancy of one for all nps for a specfc orenaon of he edges Ths resl was mproed by Klgerman and Plaxon [3, 4 who consrced for he same problem a seqence of only Θlog n machngs Noe ha n conras o he model we hae descrbed before, he orenaon of all edges ms be fxed and here s no resrcon on he se of machng edges hs can be ewed as a balancng crc on complee graphs Recenly, a specal balancng crc called block nework was examned nder he assmpon ha edges are orened nformly a random [ In [7 he ahors showed ha he cascade of wo block neworks ges a dscrepancy of

3 7 n log n seps Howeer, he analyss of [7 s raher alored for hs specal nework, whle or resls apply o all graphs Anoher ery srprsng relaonshp beween a dscree process and s dealzed connos conerpar appears for so-called deermnsc random walks Cooper and Spencer [7 show a remarkable smlary beween he expecaon of a random walk he dealzed process and a deermnsc analoge where nsead of dsrbng okens randomly, each erex seres s neghbors n a fxed order If an almos arbrary dsrbon of oken s placed on he erces of an nfne grd Z d and does a smlaneos walk n he deermnsc random walk model, hen a all mes and on each erex, he nmber of okens deaes from he expeced nmber he sandard random walk wold hae goen here, by a mos a consan Rondng he nmber of okens x + x / shared a me by wo pared erces and a random can be seen as dependen randomzed rondng of half-negral nmbers Ths ery general approach of randomzed rondng [0 p or down wh probably dependng on he fraconal par s a sandard mehod for approxmang he solon of a dscree problem by rondng he solon of an dealzed connos problem Organzaon of he paper In Secon we ge a more formal descrpon of or loadbalancng model Secon 3 presens he basc mehod and proes some general resls In Secon 4 we nrodce he local p-dergence Ψ p o bond he dfference beween he dscree and dealzed process n he perodc seng In Secon 5 we sdy random machngs and ge n Secon 5 resls for arbrary graphs In Secon 5 we show ha we can also achee a dscrepancy of O on any expander a he cos of an log log n 3 facor n he rnme PRELIMINARIES AND DEFINI- TIONS Or Model and Noaons Le G = V, E be a graph wh erces V = [n, edges E V, mn-degree δ = mn V deg, and max-degree = max V deg For any erex V and neger k N, we defne B k := { V : ds, k} All logarhms are o he base By whp wh hgh probably we refer o an een ha holds wh probably a leas n c for some consan c > The erae load-balancng process s goerned by a seqence of no necessarly perfec machngs M, M, Eery machng corresponds o a dobly sochasc commncaon marx P wh P n M, P = / f and are mached = f s no mached n M, and P = 0 oherwse The specal case of perodc balancng crcs was nrodced by Aspnes e al [3 In he langage of he model descrbed aboe hey refer o a seqence of commncaon marces P where wo marces P and P are dencal f mod d We sar wh an arbrary load ecor ξ 0 R n + In rond, f wo erces, are mached, hey balance her loads as closely as possble For dsble okens he process s s a Marko chan and can be descrbed by ξ = ξ P In he dscree process wh ndsble okens, we hae o decde where o send he excess oken f he sm of he okens of wo mached erces s odd Mos of he resls [9 hold for an arbrary orenaon of all edges Howeer, for her resls abo perfec load balancng, hey reqre he so-called sandard orenaon where each edge {, } wh < s orened owards, ha s an excess oken s sen o In spe of he hge bas, hs orenaon s parclarly sefl for he redcon o sorng neworks [3, 4, 9 As a drawback, he sandard orenaon reqres global conssency Therefore s nely appealng o consder a random orenaon of all edges {, } Tha way, he load s dsrbed more eenly n expecaon Moreoer, a random orenaon can be comped locally and offers falolerance agans crashes or replacemen of he commncaon lnks [ The precse model s as follows Wheneer he erces and perform a balancng operaon n rond, he erex wh < flps an nbased con a sep and accordng o he ocome ges eher x +x / or x +x / okens The remanng nmber of okens s sen o Noe ha hs corresponds o a randomzed rondng [0 of he dealzed process whch sends x +x / o boh erces To descrbe he drecon of he excess oken n me sep, we se Φ, o specfy a edge {, } M where s send s + f he excess oken s send o and f s send o In hs seng, he sandard orenaon of Raban e al [9 corresponds o Φ, = for < and Φ, = oherwse The randomzed rondng mples ha for each Φ, {, } M, Φ, s chosen nformly and ndependenly a random from {, } We wll refer o a machng edge {, } M wh < shorly as [ : elmnares Followng he approach by [9 of relang he dscree and dealzed process, we se x 0 = ξ 0, and herefore ξ = ξ 0 P k k= Noe ha each machng marx P = P k s a symmerc, dobly sochasc marx ha sasfes P = P Hence all egenales of P k are real and non-negae and so are he ones of k= Pk Noe ha he dealzed process s well-ndersood, n parclar f all P k s are he same We defne aerage load o be x = n = x0 /n We resae he followng wo wellknown resls Lemma eg [8, Lemma For any me sep 0, ξ x ξ 0 x λ P Theorem eg [9, Theorem In he dealzed process, he nmber of ronds for acheng a dscrepancy of l for an nal load ecor ξ 0 wh dscrepancy K s bonded aboe by O log Kn P l

4 Fnally, we defne Q := I L wh L beng he Laplacan marx of G Noe ha Q s a dffson-marx and cor- + responds o a naral random walk wh loops ha moes o each neghbor wh he same probably We call a graph wh /δ = O an expander graph, f / λ Q s bonded aboe by a consan Unless oherwse saed, we do no reqre an expander o be of consan-degree 3 Lower Bonds on Arbrary Rondng By relaely smple argmens we show ha who randomly chosen drecons for he excess okens, he dscrepancy can sll be large oposon 3 Le G be any graph and le M, M,, M T a seqence of T machngs Then here s an orenaon of each machng edge and an nal load ecor x 0 sch ha he dscrepancy of x T s a leas max{damg, log n } log T Hence, for T = Olog n he bes dscrepancy we can ge wh an arbrary orenaon s Ω log n Noe ha hs log log n resl maches he resl of [9, Corollary 5 for consandegree expanders p o a facor of log log n assmng ha he nal dscrepancy K s polynomally bonded To ge down o a consan dscrepancy for an arbrary np and orenaon of he machng edges, een Ωn ronds are necessary In sharp conras, or general resl achees for any expander a consan dscrepancy n only Olog nlog log n 3 ronds Wh he same consrcon as n oposon 3 we oban he followng resl Corollary 4 There s a graph G sch ha for any seqence of machngs M, M,, M T wh he sandard orenaon, here s an nal load ecor x 0 sch ha he dscrepancy of x T s a leas max{damg, log n log T } 3 THE BASIC METHOD We hae seen ha f he Φ, are arbrary or all, we canno hope for a dscrepancy ha s sgnfcanly less han logarhmc Tha s why we ncorporae he dea of randomzed rondng ha can be descrbed as follows We hae x = x P + e 3 for e s he excess load allocaed as a resl of rondng p and down More precsely, for all and [n, e s gen by e = { x + x Φ, 0 oherwse f {, } M, where we defne o be one f s { odd and zero oherwse Wh } M := {, } M x + x =, hs s he same as e = [: M x + x Φ, = [: M Φ, wh denong he -h n-dmensonal row n ecor Unwndng eqaon 3 yelds x = x 0 P k + k= By eqaons 4 and, x ξ = = l= x ξ = l= l= e l l= el k=l+ Pk [: M l Φ l, [: M l k=l+ P k 4 k=l+ Pk Φ l, w[l,, 5 wh w [l,, := k=l+ Pk, k=l+ Pk, To bond he smmand or more generally, a sbse of he smmand of eqaon 5 we need he followng lemma Ths lemma allows s o assme ha all machng edges recee an odd nmber of okens and herefore we may deal wh a sm of ndependen random arables A less general, b smlar lemma has been shown n [ Lemma 3 For any rple of me seps T le W := l= [: M l Φ l, w[l,,, and W := l= [: Ml Φl, w[l,, Then for any δ 0, [ W δ [ W δ We wll also freqenly se he followng basc lemma Lemma 3 For any pars of seps l, wh l T and any erex, we hae [: Ml w[l,, k=l+ Pk We sae wo resls for general seqences of machng marces ha are sed laer on Theorem 33 For any graph G and machngs M, M,, M T fx wo me seps, wh T and any erex Then, [ l= [: M l Φ l, w[l,, 8 k= P k log n n 4 Theorem 33 roghly saes ha he rondng errors ha hae small mpac small norm conrbe only lle o x ξ For he specal case =, he heorem aboe drecly ges s he followng general resl Corollary 34 For any graph G, V and sep, he deaon beween he dscree and dealzed process afer ronds s a mos 8log n whp 4 PERIODIC BALANCING CIR- CUITS 4 Upper Bonds We now consder he perodc balancng crc model In hs case, wo machng marces P and P are dencal f mod d Hence, makes sense o defne he rond marx P = d k= Pk

5 For bondng he deaon beween he dscree and dealzed process, we sae he followng defnon ha generalzes he one of [9 o arbrary p-norms Defnon 4 The local p-dergence s defned by Ψ p = max P k p /p V, = [: M l k=l+ k=l+ P k, Noe ha Ψ P Ψ P, b also Ψ P Ψ P, snce k=l+ Pk s a sochasc marx Raban e al [9 expressed he deaon beween he dscree and dealzed process n erms of Ψ P and showed ha redcng he dscrepancy o OΨ P for any nal ecor wh dscrepancy K can be acheed whn O d logkn seps 3 P Theorem 4 For any me sep 0, he maxmm deaon beween he dscree and dealzed process a sep s a mos OΨ P log n whp If he nal dscrepancy K s polynomal n n, he maxmm deaon oer all me seps s a mos OΨ P log n whp Usng hs and Theorem, we mmedaely ge Corollary 43 We achee a dscrepancy of O Ψ P log n for any ecor wh dscrepancy K whn O d logkn seps whp P Le s now bond Ψ P n erms of he second larges egenale d log n P Theorem 44 For any rond marx P, Ψ P = d O P Compared o he resl Ψ P = O of [9, Theorem 4, or bond on Ψ P s mch smaller, becase he l -conergence of P s faser han he l -conergence A drec applcaon ges he followng resl Corollary 45 We achee a dscrepancy of d O log n afer O d logkn P seps whp P By a more sble analyss, we ge a mch beer resl for small ales of λ P and d Agan we frs consder he deaon beween he dscree and connos process Theorem 46 For any me sep, he maxmm deaon beween he dscree and dealzed model a sep s a mos O whp If he nal dscrepancy K s dlog log n P polynomal n n, he maxmm deaon oer all me seps s a mos O whp d log log n P oof We proceed smlarly as n Theorem 4, b here we spl he sm and bond he crcal smmand drecly n erms of λ = λ P Le be an arbrary, b fxed me sep x ξ = l= [: M l Φ l, w[l, = 8d log log n l= [: M l + 8d log log n l= +, Φ l, w[l,, [: M l Φ l, w[l,, 3 Noe ha n conras o Raban e al [9 we con he facor of d o make or resls more comparable o he ones n Secon 5 8d log log n, snce for eery l I remans o bond The second erm can be bonded rally by [: M l he frs erm Noe ha for any l, w [l,, [: M l Range [: M l = [: M l [ Φ l, w[l,, w [l,, k=l+ Pk λ l k=l+ Pk, k=l+ Pk, d by Lemmas 3 and and herefore 8d log log n [ l= [: M l Range Φ l, w[l,, log log n 8d l= λ l d 8 log log n = O d λ λ = O d log 4 n λ Applyng Lemma 3 and Hoeffdng s bond we oban, [ 8 log log n λ l= [: M l Φ l, w[l,, δ exp exp 8log log n l= δ / O Choosng δ = Θ d log 4 n λ d P δ [: M l Range [ Φ l, w[l,,, we ge ha he deaon s a 8d log log n d log log n mos δ + = O a me wh probably n For he second clam, assme ha he nal dscrepancy sasfes K n C for a consan C By Theorem, afer = O log n ronds he dscrepancy of he dealzed process s less han Moreoer, by he argmen aboe and sng a non bond, we fnd ha he maxmm deaon p o sep s a mos O dlog log n wh probably n Now a sep, he dscrepancy of he dscree process s a mos he deaon o he dealzed process pls dlog log n one, whch s O Snce he dscrepancy n he dscree process may no ncrease, and he dscrepancy of he dealzed process s also non-ncreasng and a mos, he maxmm deaon s O dlog log n also for all me seps larger han We wll see n Theorem 40 ha he maxmm deaon s ndeed Θlog log n for consan-degree expanders Before we proe hs, we sae he followng mplcaon of Theorem 46, Corollary 47 We achee a dscrepancy of O afer O seps whp dlog log n P dlogkn P

6 oposon 48 For any consan-degree expander G here s a rond-marx P conssng of a mos d+ machngs sch ha /P = Θ Hence, we can achee a dscrepancy of Olog log n n me OlogKn, where K s he nal dscrepancy Hence for he mporan class of consan-degree expanders, we achee a dscrepancy of Olog log n n opmal me We now apply or resls o d-dmensonal ors graphs [9, Theorem 8 showed Ψ P = Θd n /d Applyng Corollary 43 and sng he fac ha Ψ P Ψ P ges he followng Corollary 49 Consder he d-dmensonal ors wh consan d For any load ecor wh nal dscrepancy K, we achee afer Od n /d logkn ronds a dscrepancy of O dn /d log n whp 4 Lower Bonds We now proe he followng lower bond ha maches Theorem 46 Theorem 40 Le G be an arbrary d-reglar expander graph wh d = O Then here s an nal load ecor wh dscrepancy K = Θlog log n sch ha he maxmm deaon beween he dscree and dealzed process s a leas Ωlog log n whp oof Choose a sbse S V sch ha wo erces n S hae a dsance of a leas 4c log log n o each oher, where c s a large consan o be deermned laer Snce for eery erex V, B c log log n = Olog n, we can fnd sch a sbse S of sze a leas Ωn/log n cf [9 Defne he load ecor x 0 = max{0, c log log n ds, S}, where c c s a small consan ha s specfed laer Clearly, he nal dscrepancy eqals c log log n We sar by examnng he dealzed process Noe ha f we rn he dealzed process for less han c log log n ronds, he load balancng processes whn B c log log ns and B c log log ns for s, s S, s s are ndependen Hence o compe ξ c log log n whn B c log log ns for a fxed s S, we may also replace ξ 0 by a ecor y 0 whch concdes wh x 0 whn B c log log ns, b s 0 elsewhere Now he nal qadrac error can be bonded aboe by y 0 y c /4 log log n =0 = Olog n δ c 4 log log n n + n n By Lemma, we defne c > 0 o be he consan sch ha afer c log log n ronds, y y Hence by Lemma, we hae for = c log log n, c O ha y y = O Snce y = O, we hae for all B c / log log ns, ξ = O Le s now consder he dscree process Snce each wo dfferen erces s, s S hae a dsance of a leas 4c log log n o each oher, he balancng processes whn B c log log ns and B c log log ns are ndependen drng he frs c log log n seps Now he probably ha n B c log log ns,s S all machng edges are orened owards s drng he frs c log log n ronds s a leas B c log log ns c log log n n, f c s sffcenly small chosen Snce we hae a leas Ωn/ log n ndependen eens, here s a leas one s S sch ha all machng edges n B c log log ns are orened owards s drng he frs c log log n seps whp In hs case, ξ c log log n s x c log log n s c log log n O, and he clam follows The same proof echnqe n Theorem 40 also works for ors graphs Theorem 4 For he d-dmensonal ors graph wh d = O, here s an nal load ecor wh dscrepancy K = Θpolylogn sch ha he maxmm deaon beween he dscree and dealzed process s Ωpolylogn whp Whle he acal lower bond may ery well be polynomal n n, hs lower bond ogeher wh Theorem 46 sll demonsraes an exponenal gap beween ors graphs and consan-degree expanders 5 RANDOM MATCHINGS For ceran graphs, mgh be non-ral o consrc explc machng marces sch ha her egenale gap or local p-dergence can be bonded Also, for many graphs s no obos whch machng seqences are o be consdered good Ths moaes s o examne seqences of random machngs We choose ndependenly for each P a random machng by a local algorhm There are many sch algorhms aalable, eg he LR algorhm of Ghosh and Mhkrshnan [ or he dsrbed synchronos algorhm of Boyd, Ghosh, abhakar and Shah [5 Defnon 5 We call a seqence of machngs M, [T, a random machng f There s a consan α > 0 sch ha [{, } M α/ for all [T and {, } E The random decsons whn one mesep do no parwse correlae negaely, ha s, [{, } M {, } M [{, } M for all [T and {, } {, } = All random decsons beween dfferen me-seps are ndependen Throgho hs secon, we wll denoe by λ he second larges egenale of Q n absole ale Or am s o sae bonds n erms of he explc egenale λ Q nsead of he egenales of he machng marces whch are random arables

7 x x + x x + x x + x x + x x + x x + x x + x x + a x x b x x c x < x d x < x Fgre Illsraon of he for cases of Defnon 56 wh {, } M 5 Coarse balancng and elmnares Le s frs consder he dealzed process Consder an arbrary sep Le ξ be any n-dmensonal load ecor and le ξ + = ξ P +, where P represens he random machng marx of rond The followng basc lemma assers ha he qadrac error decreases exponenally n λ = λ Q, f α s a consan greaer han zero Lemma 5 from [ Consder a randomly generaed machng wh he propery ha each {, } E s chosen wh probably a leas α wh 0 < α Then, [ E ξ x ξ x α 3 λq ξ x Usng condonal expecaons and Marko s neqaly, we oban he followng heorem cf [5 for a smlar saemen Theorem 53 For any nal ecor ξ 0 and any me sep, [ ξ x α / 3 λq ξ 0 x α / 3 λq Usng he heorem aboe and Corollary 34 we mmedaely oban Corollary 54 The dscrepancy afer O logkn log n ronds s a mos O log n whp For expanders, where / λ s a consan, hs resl s raher weak, as wll be shown by a more noled analyss below To hs end we sae he followng basc lemma ha relaes he shor-erm decrease of ξ o he egenale gap Lemma 55 Consder a randomly generaed machng wh he propery ha each {, } E s chosen wh probably a leas α wh 0 < α Then f ξ =, [ ξ α 9 ξ λq + n + α 3 λq 5 Near-Perfec Balancng In hs secon, we address he problem of near-perfec load balancng By hs we mean a sae where he dscrepancy has been redced o a consan ale Noe ha [9 gae resls for perfec balancng where he dscrepancy s redced o Howeer, s no oo dffcl o see ha wh a random orenaon of all edges, a dscrepancy of can only be acheed n Ωn ronds for any graph and any machngs cf [7 Therefore, we confne orseles wh acheng a consan dscrepancy, where he consan s a leas We wll see ha on expanders for whch /δ = O, we can achee a consan dscrepancy n Ologknlog log n 3 seps To descrbe how packages of okens moe hrogh he nework oer me, we defne canoncal pahs Ths allows s o fx a ceran package rong hrogh he nework and analyze he conrbon o he load deaon only on hs pah Ths fxed-ew of he w pahs s crcal for he analyss, as aods he necessy of a non bond oer all possble pahs Defnon 56 The seqence, +,, s called he canoncal pah of from me o f for all mes wh < he followng holds: If s nmached n M, hen + = Oherwse, le V be sch ha {, } M Then, f x f x f x f x x x < x < x and Φ, = hen + =, and Φ, = hen + =, and Φ, = hen + =, and Φ, = hen + = We wll denoe sch a canoncal pah from o ha sars a as P [, Obsere ha he endpon dp of he pah depends on he randomly chosen mached edges, her randomly chosen orenaons and he load ecor a sep Also noe ha f wo eqal loads are balanced, he canoncal pah says where s ff he randomly chosen orenaon of he edge pons pwards We sae wo more facs abo canoncal pahs Fac 57 Gen he load ecor x a sep, a erex and all chosen machng edges beween sep and, a pah P [, performs a smple random walk on G, e, wheneer here s an adacen machng edge, he pah swches along hs machng edge wh probably /, and oherwse says a he crren erex Fac 58 Two canoncal pahs P [, =,,, κ and P [, =,,, κ do no nersec on a erex on he same me, ha s, for all [κ The saemens and proofs of hs secon se dfferen consans whch are defned as follows Defnon 59 Or proofs se he followng consans

8 C := α/0 5 γ 5 > 0, C := 50 C ln exp C /50 8 > 0, C 3 := α 34 λq 0,, C 4 := / + α λq 0,, 54 C 5 := C C 4/ > 0, C 6 := max {0, 4 ln 8 ln exp C } 5 8 C 5 C 0, C 7 := C ln C /3 / 4 + > 0, C /3 4 C 8 := 8/log C 4 > 0, C 9 := 3 C 7 + C 6 + C + + /4 C 8 > 0, ϑ := C 7 + C + C 6 + > For a canoncal pah P, le dp he desnaon of he pah, ha s he erex a sep of P I s easy o erfy ha a canoncal pah P [, ss n expecaon a leas C 6 + mes sx dfferen erces n a row The followng defnon wll be mporan n he remander Defnon 50 For a canoncal pah P [,, we defne for, Y =, f mod 6 = 0 and he canoncal pah ss sx dfferen erces whn he me-neral [, + 5, and Y = 0 oherwse For a fxed me sep, an arbrary me sep, and a erex, we se Λ := k= P k, where s he -h n-ecor Defnon 5 For eery canoncal pah P = P [, wh desnaon dp, we defne he followng hree eens, TransP := BalP := SccessP := C l= =l+ { C C 6 Y C l, Λ dp C4 + n Φ l, w[l,, l= [: M l }, dp C 7 + C + C 6 Le s descrbe he nons behnd hese defnons TransP denoes he een ha he canoncal pah ss approxmaely he correc nmber of mes sx dfferen erces BalP means ha l -norm of he weghs of he random arables represenng he rondng errors a rond ha conrbe o x s exponenally decreasng n dp Fnally, SccessP denoes he een ha he conrbon of he rondng errors close o x dp s small Or sraegy s frs o proe ha TransP holds wh consan probably and hen proe he mplcaons TransP BalP SccessP Lemma 5 For eery canoncal pah P, [TransP 7 8 Lemma 53 For eery canoncal pah P, [BalP TransP 7 8 Lemma 54 For arbrary, wh <, log n, le P = P [, be a canoncal pah Then, [SccessP BalP / For he proof of or man resl, we nex nrodce he followng poenal fncon Then we proe ha f he eens SccessP and BalP hold, he mproemen of he poenal along a fxed canoncal pah s sgnfcan Defnon 55 We defne he poenal n rond as Υ = n = Υ, where Υ = x x ϑ f x x ϑ, and Υ = 0 oherwse Moreoer, we defne he mproemen of he poenal by := Υ Υ + and he local change of a erex by := Υ Υ + For smplcy, we may assme ha x = 0 when dealng wh Υ n he followng, whch can be sfed as follows Clearly, for any nal load ecor x 0 here s a nmber γ Z sch ha he aerage of x 0 = x 0 + γ s beween 0 and, where denoes he all-ones ecor By defnon of or load balancng algorhm, x = x for any sep, n parclar, he dscrepancy of x and x are he same We frher obsere he followng properes of Υ Lemma 56 For any graph, any nal load ecor x 0, and me N, Υ N, Υ + Υ, If {, } M, hen If ϑ x + x + ϑ 4 + and {, } M, hen x x Υ = 0 mples ha x has dscrepancy a mos ϑ + The defnon of he mproemen of he poenal s narally exended o canoncal pahs as follows Le P = P [, be a canoncal pah and for each [,, le be he erex a sep sed by P Defne { 0 f s nmached a sep, P := + f s mached wh a sep of lengh ϑ, hen P = Lemma 57 Consder a canoncal pah P [, C 8 log log n If x ϑ +, x = P 4 x ϑ C 9 log log n dp oof Fx a erex and consder he canoncal pah P [, We proceed by a case dsncon on x Frs, sppose ha ϑ + x 3ϑ + /4 C 8 log log n + ϑ By x ϑ and Defnon 50 here s a leas one dp sep on he pah P on whch x s mached wh anoher erex whose load s a leas wo okens closer o x = 0 and x ϑ + Hence n hs case by Lemma 56, P x ϑ = 3ϑ + /4 C 8 log log n For he second case, le x 3ϑ + /4 C 8 log log n + ϑ For smplcy, we wll

9 assme n he followng calclaons ha each erex on P [, s mached a sep wh anoher erex when P s locaed on P = x x 4 + ϑ x x + ± ϑ 4 x x + x x + / ϑ + /4 x x + / ϑ + /4 as x x + N The sm n he preos expresson s mnmzed when all x x + are he same Therefore, P x ϑ C 8 log log n ϑ + /4 C 8 log log n ϑ + /4 x ϑ 4C 8 log log n x ϑ ϑ + /4 4C 8 log log n x ϑ ϑ + /4 4C 8 log log n x ϑ, 3 C8 log log n where he las neqaly holds snce x 3ϑ + /4 C 8 log log n + ϑ mples ha x ϑ ϑ + /4 4C 8 log log n The nex lemma follows drecly by Theorem 33 and a non bond oer all n erces and log n me seps Lemma 58 Le E be he een ha for all erces [n and all meseps < = Olog n wh k= + Pk log n 4 holds ha l= [: M l Φ l, wl, Then, [E n Inely, he lemma says ha he rondng errors from sep o for he load of a erex a sep neer case a large deaon, proded ha he load s well dsrbed a he neghborhood of drng he seps and The nex lemma s a smple applcaon of Theorem 53 Lemma 59 Le E denoe he een ha he dscrepancy of he dealzed process s afer C logkn/ λ seps, where C s a sffcenly large consan Then, [E n 4 Bascally, we now proe ha he een SccessP mples ha he erex w = dp a he end of a canoncal pah P deaes from he aerage by only a consan Then we se hs fac ogeher wh Lemma 57 o show ha he poenal along each P wh SccessP decreases Lemma 50 Le ΩlogKn/ Assme ha E and E hold and consder a canoncal pah P = P [,, 8 log log n = + log C 4 wh desnaon dp = w sch ha SccessP and BalP holds Then, x w x C 7 + C 6 + C + Now we are ready o proe he man resl of hs secon Theorem 5 For expanders wh a consan degreerao /δ, he dscrepancy of any nal load ecor of dscrepancy K s redced o ϑ + whn OlogKnlog log n 3 seps 8 log log n oof Choose = + log C 4 Moreoer assme ha we know Υ Then by lneary of expecaons, he fac ha wo canoncal pahs are erex-dson and Lemmas 54, 58, 59, and 50, E [Υ Υ E E = [ E P [, E E [n [E E [SccessP BalP E E [n [ E P [, SccessP BalP E E 4 4n x ϑ C 9 log log n [n 8C 9 log log n Υ [ I follows from Corollary 54 ha E Υ n whp, f = Θ logkn As shown aboe for eery par of meseps, = + 8 log log n log C 4, [ E Υ Υ 8C 9 log log n Ierang hs argmen yelds E [Υ +8C 9 log log n C 8 log log n 6 ln n 8C9 log log n 6ln n Υ 8C 9 log log n n 6 n = n 4

10 Fnally, sng he negraly of Υ we arre a [ x +8C 9 log log n C 8 log log n 6 ln n x > ϑ + [ Υ +8C 9 log log n C 8 log log n 6 ln n > 0 [ E Υ +8C 9 log log n C 8 log log n 6 ln n n 4 Or echnqes also apply o consan-degree expanders n he balancng crc model Corollary 5 Le G be a consan-degree expander and consder he balancng crc model wh a rond marx P sasfyng λ P = O Then we can achee a dscrepancy of O n O logkn P log log n3 ronds whp 6 CONCLUSIONS We presen he frs analyss of a naral local load balancng algorhm ha drecs he excess okens a random I s shown ha on many mporan graphs hs smple randomzaon mproes oer s deermnsc conerpar [9 by a leas a qadrac facor for many graphs For he mporan case of expanders, we show ha he load s balanced almos perfecly whn OlogKnlog log n 3 ronds Ths resl s opmal p o a facor of Olog log n 3 whle all preos approaches were only opmal p o a facor of Olog n Neerheless, an neresng open qeson s o close he gap beween or pper bond of OlogKnlog log n 3 and he ral lower bond of ΩlogKn Acknowledgmens We are hankfl o Rober Elsässer, Alsar Snclar and Danny Vlenchk for helpfl dscssons Ths work was sppored by posdocoral fellowshps from he German Academc Exchange Serce DAAD References [ W Aello, B Awerbch, B M Maggs, and S Rao Approxmae load balancng on dynamc and asynchronos neworks In 5h Annal ACM Symposm on Theory of Compng STOC 93, pages 63 64, 993 [ H Arnd Load balancng: Dmenson exchange on prodc graphs In 8h Inernaonal Parallel and Dsrbed ocessng Symposm IPDPS 04 IEEE Comper Socey, 004 [3 J Aspnes, M Herlhy, and N Sha Conng neworks J ACM, 45:00 048, 994 [4 P Berenbrnk, T Fredezky, and Z H A new analycal mehod for parallel, dffson-ype load balancng J Parallel Dsrb Comp, 69:54 6, 009 [5 S Boyd, A Ghosh, B abhakar, and D Shah Randomzed gossp algorhms IEEE Transacons on Informaon Theory and IEEE/ACM Transacons on Neworkng, 56: , 006 [6 F Cedo, A Cores, A Rpoll, M A Senar, and E Lqe The conergence of realsc dsrbed loadbalancng algorhms Theor Comp Sys, 44:609 68, 007 [7 J Cooper and J Spencer Smlang a random walk wh consan error Comb obab Comp, 5:85 8, 006 [8 R Elsässer, B Monen, and S Schamberger Dsrbng n sze workload packages n heerogenos neworks J Graph Algorhms & Applcaons, 0:5 68, 006 [9 U Fege, D Peleg, P Raghaan, and E Upfal Randomzed broadcas n neworks Random Srcres & Algorhms, 4: , 990 [0 B Ghosh, F T Leghon, B M Maggs, S Mhkrshnan, C G Plaxon, R Raaraman, A W Rcha, R E Taran, and D Zckerman Tgh analyses of wo local load balancng algorhms SIAM J Comp, 9:9 64, 999 [ B Ghosh and S Mhkrshnan Dynamc load balancng by random machngs Jornal of Comper and Sysem Scences, 533: , 996 [ M Herlhy and S Trhapra Randomzed smoohng neworks J Parallel and Dsrbed Compng, 665:66 63, 006 [3 M Klgerman Small-deph conng neworks and relaed opcs PhD hess, Deparmen of Mahemacs, Massachses Inse of Technology, Sepember 994 [4 M Klgerman and C Plaxon Small-deph conng neworks In 4h Annal ACM Symposm on Theory of Compng STOC 9, pages 47 48, 99 [5 L Loász and P Wnkler Mxng of random walks and oher dffsons on a graph Sreys n combnaorcs, pages 9 54, 995 [6 R Lülng and B Monen A dynamc dsrbed load balancng algorhm wh proable good performance In 5h ACM Symposm on Parallel Algorhms and Archecres SPAA 93, pages 64 7, 993 [7 M Maroncolas and T Saerwald The mpac of randomzaon n smoohng neworks In 7h Annal ACM ncples of Dsrbed Compng PODC 08, pages , 008 [8 S Mhkrshnan, B Ghosh, and M H Schlz Frs- and second-order dffse mehods for rapd, coarse, dsrbed load balancng Theory Comp Sys, 34:33 354, 998 [9 Y Raban, A Snclar, and R Wanka Local dergence of Marko chans and he analyss of erae load balancng schemes In 39h Annal IEEE Symposm on Fondaons of Comper Scence FOCS 98, pages , 998 [0 P Raghaan and C Thompson Randomzed rondng: a echnqe for proably good algorhms and algorhmc proofs Combnaorca, 7: , 987 [ R Sbramanan and I D Scherson An analyss of dffse load-balancng In 6h ACM Symposm on Parallel Algorhms and Archecres SPAA 94, pages 0 5, 994