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1 Architcturs, Languags and Pattrns PH Wlch and AWP Bakkrs (Eds) IOS Prss, Paralll Graph Coloring using JAVA Thomas UMLAND Dutsch Tlkom Brkom GmbH, Goslarr Ufr 5, Brlin, Grmany Abstract In this papr a paralll, piplin orintd vrsion of a wll-known squntial graph coloring huristic is introducd Runtim and spdup rsults of an implmntation in JAVA on a four procssor machin ar prsntd and discussd 1 Introduction Th ida of this papr is to introduc a paralll vrsion of a wll-known squntial graph coloring huristic which could b asily implmntd on a shard mmory multiprocssor systm whn using an appropriat programming languag lik for instanc JAVA [1] Th nxt sction shall hlp to undrstand th squntial algorithm In sction th paralll vrsion is prsntd whil in sction 4 runtim and spdup rsults of a JAVA implmntation running on a four procssor SUN SPARC Workstation ar givn Th squntial rst t algorithm For a graph G = (V; E) with vrtics V = fv 1 ; : : : ; v n g and dgs E V V a function f : V! f1; : : : ; kg, v V 7! f(v) is calld a coloring (of th vrtics) of G if for all pairs of vrtics u; v V, u 6= v, (u; v) E ) f(u) 6= f(v) If w call th valu f(v) th color of th vrtx v V thn coloring th graph G = (V; E) simply mans that vry vrtx of G has to b assignd a color with th rstriction that adjacnt vrtics { i thos connctd by an dg { must gt dirnt colors Th minimal valu of k so that f : V! f1; : : : ; kg is a coloring of th graph G = (V; E) is calld th chromatic numbr of that graph Coloring th vrtics of a graph is a problm ndd to b solvd in a varity of applications g schduling, rgistr allocation, printd circuit tsting tc Unfortunatly, th task of nding a coloring with th minimal numbr of colors is not solvabl in polynomial tim for an arbitrary graph (as known so far) Thrfor on is looking for coloring algorithms, so-calld huristics, which do not always nd an optimal coloring but hav polynomial runtim Th rst t algorithm is a wll-known huristic with polynomial runtim for coloring th vrtics of a graph It rquirs an initial ordring of th vrtics of th input graph Th rst vrtx according to this ordr gts th color 1 whil th othr vrtics ar procssd squntially, assigning ach vrtx th last possibl color which dos not produc a conict with th vrtics alrady colord For a dtaild dscription s for xampl [4, 5] Author's nw addrss sinc 1 March 1998 is: Dutsch Tlkom AG, Entwicklungszntrum Nord, Willy-Brandt-Platz, 815 Brmn, Grmany, ThomasUmland@tlkomd

2 1 T Umland / Paralll Graph Coloring v 1 v v 5 v v v 4 v 6 v 8 Figur 1: First t coloring of a graph with 8 vrtics Figur 1 shows a graph with 8 vrtics for which th algorithm producs a coloring with 5 dirnt colors In this gur ach circl symbolizs a vrtx and ach lin an dg btwn two vrtics Th nam of ach vrtx is writtn outsid and th color producd by th algorithm is writtn insid th circl From th indics of th vrtics th initial ordring can b dducd, i vrtx v i is th i-th vrtx of th ordring 1 Bcaus th algorithm is not too complicatd this xampl shall suc for illustrating its opration Paralllizing squntial rst t Although th rst t algorithm is oftn calld inhrntly squntial bcaus of its strictly ordrd procdur, w will now prsnt a paralll variant which in practic yilds not too bad spdup rsults on a paralll machin 1 Basics To gt an ida how our paralll algorithm works w tak at rst a look at a possibl implmntation of th squntial rst t algorithm Th coloring of a vrtx v i could b implmntd using two main stps: 1 Dtrmin a list of all possibl colors for v i i xclud thos colors alrady usd by vrtics v j, j < i adjacnt to v i This could b implmntd using a boolan array L i { calld th possibility list of vrtx v i { with th proprty L i [k] = FALSE, 9v j with j < i; (v i ; v j ) E and f(v j ) = k This stp can b prformd by a procdur Build(L i ; v j ) which xcluds th color of vrtx v j from th possibility list L i of vrtx v i, Dtrmin th smallst of all possibl colors for vrtx v i, i look for th smallst ntry in L i with L i [k] = TRUE and assign color k to v i W put this stp into a procdur Color(L i ; v i ) which colors vrtx v i in dpndnc on its possibility list L i To color a graph with n vrtics w thrfor hav to xcut th following actions: 1 Obviously th givn graph could b colord using only colors whil rst t producs a coloring with 5 colors { as mntiond arlir th rst t huristic dos not always nd th optimal coloring Among othrs in [4, 5, 6] bounds for th dirnc of th chromatic numbr of a graph and th numbr of colors producd by rst t can b found But this topic is not of intrst in this contxt

3 T Umland / Paralll Graph Coloring 1 Color(L i ; v i ) for i = 1; : : : ; n and Build(L i ; v j ) for i = 1; : : : ; n rsp j = 1; : : : ; i? 1 Of cours ths actions cannot b xcutd all in paralll bcaus thr ar tim dpndncis rsulting from th accss to th possibility lists: 1 For all j < i th action Build(L i ; v j ) must b xcutd bfor Color(L i ; v i ) and Color(L i ; v i ) must b xcutd bfor Build(L j ; v i ) for j > i A rst paralll approach Nxt w distribut ths actions ovr n procssors and gt a paralll algorithm which ovrall nds n? 1 stps To mak th dscription asir w distinguish btwn vn and odd stps: 1 For 1 i n during th odd stps i? 1 th action Color(L i ; v i ) is xcutd on procssor P i ; in paralll to ths actions Build(L i?j ; v j ) ar xcutd on procssors P j for 1 j < i During th vn stps i for 1 i < n th actions Build(L i?j+1 ; v j ) will b xcutd on procssors P j with 1 j i It can b asily vrid that this distribution of th actions ovr th procssors dos not violat th abov tim dpndncis Of cours, this algorithm could also b drivd in a mor formal way from th rcurrncs implid by th tim dpndncis This mthod is dmonstratd g in [] whr paralll algorithms for th matrix multiplication ar obtaind from a st of rcurrncs dscribing th multiplication Howvr, bcaus th dpndncis hr ar quit asy to undrstand w omit this stp Figur shows th distribution of th actions ovr 5 procssors for a graph with 5 vrtics Th actions of ach column in th pictur hav to b xcutd squntially on th corrsponding procssor whil th actions listd in ach row can tak plac in paralll at th spcid tim stp of th algorithm Th arrows in th pictur indicat th points whr th control ovr a possibility lists changs btwn two procssors In this rprsntation w s that th control ovr th possibility lists ows through th procssors in a piplind fashion W also s th typical piplin bhavior at th bginning and at th nd of th xcution whr th piplin has to b lld rsp mptid and thrfor only a fw procssors ar busy A gnralizd paralll approach Th rst paralll approach of sction has th major disadvantag that it rquirs as many procssors as thr ar vrtics in th graph Th gnralizd vrsion prsntd now allows any numbr of procssors P 1 ; : : : ; P N (1 N n) to b usd In this cas vry procssor is rsponsibl for coloring a whol subgraph with n=n vrtics instad of a singl vrtx in th prvious vrsion of th algorithm To simplify th notation w assum that N divids n

4 14 T Umland / Paralll Graph Coloring Stp Procssor 1 Procssor Procssor Procssor 4 Procssor 5 1 Color(L 1 ; v 1 ) Build(L ; v 1 ) Build(L ; v 1 ) Color(L ; v ) 4 Build(L 4 ; v 1 ) Build(L ; v ) 5 Build(L 5 ; v 1 ) Build(L 4 ; v ) Color(L ; v ) 6 Build(L 5 ; v ) Build(L 4 ; v ) 7 Build(L 5 ; v ) Color(L 4 ; v 4 ) 8 Build(L 5 ; v 4 ) 9 Color(L 5 ; v 5 ) Figur : Paralll rst t with 5 vrtics and 5 procssors Th xcution of th gnralizd paralll algorithm is illustratd in gur On can s clarly th partition of th graph into N blocks with n=n vrtics ach Each procssor now has to color all th vrtics of his corrsponding block undr considration of th possibility lists prpard on th prvious procssors This action is don by a procdur again namd Color Th action Build(L i ; V j ) usd in th gnralizd vrsion prforms th xclusion of th colors of all vrtics V j = fv 1+(j?1)n=N ; : : : ; v jn=n g containd in th j-th subgraph from th possibility list L i of vrtx v i which will b colord latr by anothr procssor Again th piplind bhavior of th algorithm which is causd by th ow of control ovr th possibility lists can b sn in th pictur Compard to th rst approach th last phas of th algorithm whr th piplin mptis is not as long in this vrsion If you intrprt th pictur of gur as a stat/tim diagram you rcogniz that only roughly half of th possibl computing rsourcs ar usd by this algorithm { du to th lling and mptying procss of th piplin many procssors ar idl for quit a long tim Thrfor th spdup achivd by an implmntation is not xpctd to signicantly xcd half of th numbr of procssors usd Nvrthlss th rsulting cincy of about 50% is not too bad for this typ of algorithm 4 Runtim rsults of a JAVA implmntation In th gnralizd paralll rst t algorithm of sction w assumd that th algorithm runs on N procssors Nvrthlss that approach is still valid if w tak N as th numbr of concurrnt procsss { so-calld thrads in JAVA { which ar schduld by th oprating systm on possibly lss than N physical procssors Thrfor w can combin any numbr of concurrntly running thrads with any numbr of physical procssors

5 T Umland / Paralll Graph Coloring 15 Procssor 1 Procssor Procssor Procssor 4 Color(L 1 ; v 1 ) Color(L ; v ) Color(L ; v ) Color(L 4 ; v 4 ) Build(L 5 ; V 1 ) Build(L 6 ; V 1 ) Build(L 7 ; V 1 ) Build(L 8 ; V 1 ) Build(L 9 ; V 1 ) Build(L 10 ; V 1 ) Build(L 11 ; V 1 ) Build(L 1 ; V 1 ) Build(L 1 ; V 1 ) Build(L 14 ; V 1 ) Build(L 15 ; V 1 ) Build(L 16 ; V 1 ) Color(L 5 ; v 5 ) Color(L 6 ; v 6 ) Color(L 7 ; v 7 ) Color(L 8 ; v 8 ) Build(L 9 ; V ) Build(L 10 ; V ) Build(L 11 ; V ) Build(L 1 ; V ) Build(L 1 ; V ) Build(L 14 ; V ) Build(L 15 ; V ) Build(L 16 ; V ) Color(L 9 ; v 9 ) Color(L 10 ; v 10 ) Color(L 11 ; v 11 ) Color(L 1 ; v 1 ) Build(L 1 ; V ) Build(L 14 ; V ) Build(L 15 ; V ) Build(L 16 ; V ) Color(L 1 ; v 1 ) Color(L 14 ; v 14 ) Color(L 15 ; v 15 ) Color(L 16 ; v 16 ) Figur : Gnralizd paralll rst t (16 vrtics, 4 procssors) Th rprsntation of th graph as a boolan n n matrix as wll as th possibility lists of th vrtics and th list of th alrady dtrmind colors ar locatd within an objct of a nw class calld Graph Th actions Build and Color which hav to b prformd by ach procss can b implmntd asily in JAVA and ar also intgratd as mthods in th Graph class As th JAVA programming modl allows shard objcts btwn thrads w gnrat only on (shard) instanc of th Graph class which is accssibl by all N concurrntly running thrads Th ow of control ovr th possibility lists { illustratd as arrows in gur { is implmntd by passing tokns from thrad to thrad via objcts of a channl class CHAN which implmnts a dirctd point-to-point connction btwn xactly two thrads This is similar to th way communication taks plac in th programming languag OCCAM [] Th class CHAN provids on mthod for snding and anothr for rciving data ovr th channl in ordr to hid th xplicit synchronization constructs availabl in JAVA Th implmntation of th CHAN class is similar to that usd in JavaPP which provids a lot mor OCCAM and CSP mchanisms via JAVA packags As our JAVA implmntation of th coloring algorithm uss a shard graph objct w do not rally nd to snd data objcts via th channls but instad transmit only a tokn to pass th control ovr a possibility list to th nxt thrad Thrfor our channls ar usd simply as an lgant way to synchronizd two thrads in a rndzvous fashion Th JAVA vrsion usd for this implmntation was th JDK 115 with nativ thrad support i concurrntly running thrads ar schduld ovr th availabl procssors by th oprating systm; a just-in-tim compilr was not availabl Finally th S for dtails

6 16 T Umland / Paralll Graph Coloring Runtim [ms] Paralll graph coloring (graph with 000 vrtics) 4 procssors procssors procssors 4 1 procssor Numbr of concurrnt thrads Figur 4: Runtim of th paralll algorithm masurd on 1{4 procssors implmntation was tstd on a multiprocssor workstation with four procssors of th typ SPARC-40 MHz and 18 MByts of mmory running SOLARIS 51 4 Th kind of graph usd as input has a similar structur to that shown in gur 1 It has 000 vrtics, dgs and rst t yilds a coloring with 1001 colors; th structur and siz of th input graph has only an impact on th absolut xcution tim but not on th spdup bhavior of th algorithm as xprimnts showd Th runtim ndd to color this graph was masurd for a dirnt numbr of concurrntly running thrads N Th schduling of th thrads and th distribution ovr th physical procssors has bn carrid out by th oprating systm At rst th program was allowd to us all four procssors, thn th sam xprimnt has bn rpatd whil disabling on, two and nally thr procssors Th runtim rsults ar shown in gur 4: Each curv 5 corrsponds to a xd numbr of physical procssors and shows th xcution tims for N = 1; : : : ; 40 concurrntly running thrads Th diagram shows that th program indd runs fastr whn using mor than on procssor To masur th bhavior of th paralllization, th spdup for ach curv rlativ to th runtim of on thrad has bn calculatd and is shown in gur 5 Th diagram clarly shows dirnt spdups dpnding on th numbr of procssors usd If w hav only on physical procssor in opration and start mor than on thrad only smi-paralll xcution is possibl; thrfor th tim ndd to administrat and switch th thrads is dominant and slows down th obtainabl spdup This ct can b sn in th lowr-most curv of th diagram Whn using mor physical procssors thy could b utilizd to xcut thrads in paralll and th spdup incrass In this cas th administration ovrhad is only dominant if th numbr of thrads is much gratr than th numbr of procssors As xpctd th spdup is bttr if mor physical 4 Although nowadays this is a quit slow machin, th masurd spdup cts should b indpndnt of th procssors spd and ar xpctd to b th sam on machins with mor up-to-dat procssors 5 Th sparat data points hav only bn connctd to as th idntication of points blonging togthr

7 T Umland / Paralll Graph Coloring 17 Spdup 15 1 Paralll graph coloring (graph with 000 vrtics) 4 procssors procssors procssors 4 1 procssor Numbr of concurrnt thrads Figur 5: Spdup of th paralll algorithm using 1{4 procssors procssors ar availabl In sction was mntiond that th maximum spdup of this algorithm is rstrictd to only about half th numbr of procssors If w tak this fact into account th achivd spdup is quit good 5 Conclusions In this papr w showd how an algorithm which sms to b inhrntly squntial could b convrtd to a paralll on Th rsulting piplin structur could b implmntd asily in JAVA Th runtim and spdup rsults hav bn quit satisfactory For futur work it would b intrsting to bnchmark th implmntation on a machin with mor than four procssors in ordr to tst whthr vn mor procssors could b utilizd or to s whn th impact of th shard mmory architctur prvnts a furthr incras of th spdup Rfrncs [1] Jams Gosling, Bill Joy, and Guy Stl Th Java Languag Spcication Addison-Wsly Publishing Company, Rading, Massachustts, 1996 [] INMOS Limitd Occam Rfrnc Manual Prntic Hall Intrnational, Englwood Clis, 1988 [] H T Kung Th structur of paralll algorithms In M C Yovits, ditor, Advancs in Computrs, volum 19, pags 64{11, Nw York, 1980 Acadmic Prss [4] David W Matula, Gorg Marbl, and Jol D Isaacson Graph coloring algorithms In Ronald C Rad, ditor, Graph Thory and Computing, pags 109{1, Nw York, 197 Acadmic Prss [5] T Umland, Ubr huristisch Vrfahrn zur Losung ds Farbungsproblms, Ph D Thsis, Univrsity of Karlsruh, VDI-Vrlag, Dussldorf, 1996 [6] D J A Wlsh and M B Powll An uppr bound for th chromatic numbr of a graph and its application to timtabling problms Th Computr Journal, 10:85{86, 1967

8 18 T Umland / Paralll Graph Coloring