A parallelisable multi-level banded diffusion scheme for computing balanced partitions with smooth boundaries

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A parallelisable multi-level banded diffusion scheme for

computing balanced partitions with smooth boundaries

François Pellegrini

To cite this version:

François Pellegrini. A parallelisable multi-level banded diffusion scheme for computing balanced

par-titions with smooth boundaries. EuroPar, Aug 2007, Rennes, France. pp.195-204,

�10.1007/978-3-540-74466-5_22�. �hal-00301427�

A parallelisable multi-level banded diusion

s heme for omputing balan ed partitions with

smooth boundaries

FrançoisPellegrini

ENSEIRB,LaBRIandINRIAFuturs UniversitéBordeauxI

351, oursdelaLibération,33405TALENCE,FRANCE pelegrinlabri.fr

Abstra t. Graphpartitioningalgorithmshaveyettobeimproved, be- ausegraph-basedlo aloptimizationalgorithmsdonot omputesmooth andglobally-optimalfrontiers,whileglobaloptimizationalgorithmsare

tooexpensivetobeofpra ti aluseonlargegraphs.Thispaperpresents awaytointegrateaglobaloptimization,diusionalgorithminabanded

multi-level framework, whi h dramati ally redu es problem size while yieldingbalan edpartitions withsmoothboundaries. Sin eallof these

algorithms do parallelize well, high-quality parallel graph partitioners builtusingthese algorithmswillhavethe samequalityas state-of-the-artsequentialpartitioners.

1 Introdu tion

Graphpartitioning is an ubiquitouste hniquewhi h hasappli ations in many

eldsof omputers ien eandengineering,su hasworkloadbalan inginparallel

omputing, databasestorage,VLSI designorbio-informati s.Itis mostlyused

to help solving domain-dependent optimization problems modeled in terms of

weightedor unweightedgraphs,wherending good solutionsamountsto

om-puting, eventuallyre ursivelyin adivide-and- onquerframework,small vertex

oredge utsthatbalan eevenlytheweightsofthegraphparts.

Manyalgorithmshavebeenproposed to omputee ientpartitions of any

graphs, su h as graph orevolutionary algorithms, spe tral methods, orlinear

optimization methods. Basi ally, all of these methods belong to two distin t

lasses: global methods, whi h onsider all of the graph data, and lo al

opti-mizationheuristi s,whi htrytoimprovelo allyapreexistingpartition.Global

methods oftenyield betterresults, but their ostsdramati allyin reasesalong

with problem size, whi h makes them pra ti ally impossibleto use for graphs

omprisingseveraltensmillionverti es,whi harethegraphsnowbeing

onsid-eredinmanys ienti engineeringproblems.

Themulti-levelapproa h [5,6℄hasbeenaquitesu essful attemptto

om-bine bothapproa hes.It onsistsin repeatedly omputingaset ofin reasingly

hal-00301427, version 1 - 21 Jul 2008

Author manuscript, published in "EuroPar, Rennes : France (2007)"

DOI : 10.1007/978-3-540-74466-5_22

Coarsening

phase

Uncoarsening

phase

Initial partitioning

Projected partition

Refined partition

Fig.1.Multi-levelframeworkfor omputingabipartitionofagraph.

mat hings whi h ollapseverti esand edges,until the oarsest graphobtained

isnolargerthanafewhundredsofverti es,then omputingaseparatoronthis

oarsestgraph,andproje tingba kthisseparator,from oarsertonergraphs,

up to the original graph. Most often, a lo al optimization algorithm, su h as

Kernighan-Lin[7℄orFidu ia-Mattheyses[4℄(FM),isusedin theun oarsening

phasetorenethepartitionthat isproje tedba kat everylevel,su hthatthe

granularityofthesolutionistheoneoftheoriginalgraphandnottheoneofthe

oarsestgraph,asillustratedin Figure 1.This approa h improvesqualityover

plaingraphalgorithms,andspeedoverplainglobaloptimizationalgorithms,by

taking thebest of bothworlds.Globaloptimization algorithms anbeusedon

smallgraphstogivethegeneraldire tionofthepartitiontoset,andinexpensive

lo aloptimization algorithms anbeusedatlow ostonnergraphswithtens

ofmillionverti es.

However,thequalityofpartitionsprodu edbythisapproa hisnotasgood

astheonethatwouldbeyieldedbyplainglobaloptimizationalgorithms.

Coars-eningartifa ts,aswellasthemeshingtopologyoftheoriginalgraphs,traplo al

optimization algorithmsin lo al optimaoftheir ostfun tions,su h that

fron-tiersareoftenmadeofnon-optimalsetsofsegments,asillustratedinFigure5.a.

This paper des ribesan e ient way to integrate diusion s hemes into a

multi-levelframework,soasto omputepartitionswithsmallandsmooth

fron-tiers in atime equivalent in magnitude to the one of state-of-the-artlo al

op-timization algorithms.Itisorganizedasfollows.Afterpresentingrelatedworks

inSe tion 2,weintrodu ein Se tion3ourmulti-levelbandeddiusions heme,

andshowsomepartitioningandmappingresults,obtainedwithS ot h5.0,in

Se tion 4.Then omesthe on lusion.

2 Related works

Manyauthorshadalreadynoti ed thatpartitionsyieldedbylo aloptimization

thatsu hpartitionswerenotttedfortheirpurpose,assubdomainswithlonger

frontiersorirregularshapesresultedin alargernumberofiterationstoa hieve

onvergen e.Tomeasurethequalityofea hoftheparts,severalauthorsdened

a metri alled aspe t ratio, whi h an be thought in 2D as ameasure of the

perimeterofapartwithrespe ttothesquarerootofitsarea.Themore ompa t

apartis,thesmalleritsaspe tratiovalueis,asidealpartsareof ir ularshape

in theEu lideanspa e.

In [3℄, Diekmann et al. eviden ed su h a behavior, and proposed both a

measureoftheaspe tratiooftheparts,aswellasasetofheuristi sto reateand

renethepartitions,withtheobje tiveofde reasingtheiraspe tratio.Among

thesealgorithmsisabubble-growingalgorithm.Thisalgorithmisbasedonthe

observationthatsetsofsoapbubblesself-organizesoastominimizethesurfa eof

theirinterfa es,whi hisindeedwhatisexpe tedfromapartitioningalgorithm.

Consequently, the authors' idea was to grow, from as many seed verti es as

the desirednumber of parts,a olle tionof expanding bubbles, byperforming

breadth-rst traversals rooted at these seedverti es.On e everygraph vertex

has been assigned to some part, ea h part omputes its enter based on the

graph distan e metri . These enter verti es are taken as new seeds and the

expansion pro ess is started again, until it onverges, that is, until enters of

subdomains nolonger move.An important drawba kof this method is that it

doesnotguaranteethat all partswill hold thesamenumberofverti es,whi h

requires to all other heuristi sin turn to perform loadbalan ing. Also,all of

thegraphverti esmustbevisitedmanytimes,whi hmakesthisalgorithmquite

expensive,allthemoreitis ombinedwith ostlyalgorithmssu h assimulated

annealing,andthe omputationoftheaspe tratiorequires someknowledgeon

thegeometryofthegraphs,whi hisnotalwaysavailable.

In[8℄,MeyerhenkeandS hambergerfurtherexplore thebubblemodel, and

devise a way to grow the bubbles by solving, possibly in parallel, systems of

linearequations, insteadof iteratively omputingbubble enters. This method

yields partitions of high quality too,but is veryslow, even in parallel [9℄, and

theloadbalan ing problemis alsonotaddressed,whi hrequires to resorttoa

greedyloadbalan ing algorithmafterwards.

In[13℄,Wanetal.exploreadiusivemodel, alledtheinuen emodel,where

verti esimpa t theirneighbors by diusing them information ontheir urrent

state.Thismodelalsodoesnothandleloadbalan ingproperly.

3 Multi-level banded diusion s heme

Inspiteoftheirbetterquality,alloftheabovediusions hemeshavetwo

draw-ba ks:rst,theydonotnaturallybalan eloadsbetweenpartsandse ond,they

are expensive as they involve all of the graph verti es. The method that we

3.1 The jug of the Danaides

The diusion s heme that we propose an apply to an arbitrary number of

parts, but for the sake of larity we will des ribe it in the ontext of graph

bipartitioning, thatis, withtwoparts only.Wemodelthegraphto bipartition

in the following way, depi tedin Figure 2.Nodes are representedasbarrels of

innite apa ity,whi hleak su hthat oneunitof liquidatmostdripsperunit

of time. When graph verti es are weighted, always with integer weights, the

maximumquantityofliquidtobelostperunitoftimeisequaltotheweightof

thevertex.Graphedges aremodeledbypipes ofse tionequalto theirweight.

In bothparts, a sour e vertex is hosen, to whi h a sour e pipe is onne ted,

whi howsin

|V |

2

unitsofliquidperunitoftime.Twosortsofliquidsareinfa t inje tedinthesystem:s ot hintherstpipe,andanti-s ot hinthese ondpipe,

su h thatwhen somequantity ofs ot h mixes withthe samequantityof

anti-s ot h, bothvanish. Toease thewritingof the algorithmin thebipartitioning

ase,s ot his representedbypositivequantities andanti-s ot hisrepresented

bynegativeones,sothatmutualdestru tionnaturallytakespla ewhenadding

anytwoquantities ofoppositesigns.

ThediusionalgorithmperformsasoutlinedinFigure3.Forea htimestep,

andforea hvertex,theamountofliquid(whethers ot horanti-s ot h)whi h

remains after some has leaked is spread a ross the onne ting pipes towards

theneighboring barrels,a ordingtotheir relativese tions.This pro ess ould

beiterated until onvergen e,but in fa t itis only performed fora numberof

stepssu ienttoa hievesignstability.Indeed,wearenotinterestedin omplete

onvergen e,but in thestabilityof thesignsofall ontentquantities borne by

graph verti es, whi h indi ate whether s ot h or anti-s ot h dominatesin the

barrels,that is,ifsomevertexbelongstopart

0

or

1

.

Sin e

|V |

unitsofbothliquidsareinje tedonthewholeperunitoftime,and sin eallof thebarrels anleak thesameoverallamountin thesametime, the

systemisboundto onverge,allthemorethatliquid andisappearby ollision

ofs ot handanti-s ot h.Asin thebubbles hemes,what isexpe ted isthata

smoothfrontwillbe reatedbetweenthetwoparts.Thepurposeofthealgorithm

reset ontentsofnewarrayto

0

;

old[s

0

] ← old[s

0

] − |V |/2

; /* Refill sour e barrels */

old[s

1

] ← old[s

1

] + |V |/2

; for(allverti es

v

ingraph){

c ← old[v]

; /* Get ontents of barrel */

if(

|c| > weight[v]

) { /* If not all ontents have leaked */

c ← c − weight[v] ∗ sign(c

); /* Compute what will remain */

σ ←

P

_e=(v,v

′

)

weight[e]

; /* Sum weights of all adja ent edges */ for(alledges

e = (v, v

′

₎

){ /* For all edges adja ent to v */

f ← c ∗ weight[e]/σ

; /* Fra tion to be spread to v' */

new[v

′

_{] ← new[v}

′

_{] + f}

; /* A umulate spreaded ontributions */ }

} }

swapoldandnewarrays; }

Fig.3. Sket hof thejug-of-the-Danaides diusion algorithm. S ot h,representedas positive quantities,ows fromthe sour eof part

1

, while anti-s ot h,representedas negativequantities,owsfromthesour eofpart

0

.Forea hstep,the urrentandnew ontentsofeveryvertexarestoredinarraysoldandnew,respe tively.

the ut.Infa t,unlikeallofthealgorithmspresentedinthepreviousse tion,our

methodprivilegesloadbalan ingover utminimization.Forthislatter riterion,

werelyonanadditionalfeatureofours heme,asexplainedbelow.

3.2 Band graphsin a multi-levels heme

Ourdiusionalgorithm,assu h,presentstwoweaknesses:nothingissaidabout

the sele tion of the seed verti es, and performing su h iterations over all of

thegraphsverti esisveryexpensive omparedtolo aloptimizationalgorithms

whi honly onsiderverti esintheimmediatevi inityofthefrontiers.

Toaddress these twoproblems on urrently, we usea method we have

de-velopedin [1℄, illustratedin Figure 4. It onsistsin usingamulti-level s heme

in whi h renementalgorithms are notapplied to thefull graphsbut to band

graphsthat ontainverti esthatare at mostat somesmall distan e, typi ally

3

,from theproje tedseparator.Inthese band graphs,twoadditionalan hor

verti esrepresentalloftheremovedverti esofea hpart,andare onne tedto

the last band layersof verti es of ea h of the parts.The vertexweight of the

an hor verti es is equalto the sum of the vertex weightsof all of the verti es

theyrepla e,topreservethebalan eofthetwobandparts.

Theunderlyingreasoningofthispre- onstrainedbandings hemeisthatsin e

everyrenement is lassi ally performed bymeans of alo al algorithm,whi h

perturbs only in a limited way the position of the proje ted separator, lo al

renementalgorithmsneedonlytobepassedasubgraphthat ontainsthe

atedaroundtheproje tednerseparator,withan horverti esrepresentingallofthe

removedverti es inea h part. After some optimizationalgorithm (whether lo al or global) isapplied,therenedbandseparatorisproje tedba kto thefullgraph, and

theun oarseningpro essgoeson.

when performing Fidu ia-Mattheysesrenementon bandgraphsthat ontain

onlyverti esthat areat distan e at most

3

from theproje tedseparators,the qualityofthenest separatornotonlyremains onstant,but evensigni antly

improvesinmost ases.Ourinterpretation isthat thispre- onstrainedbanding

preventslo aloptimizationalgorithmsfromexploringandbeingtrappedinlo al

optimathatwouldbetoofarfrom theglobaloptimumsket hedat the oarsest

levelofthemulti-levelpro ess.

Su h a banded s heme is ideal for using our diusion s heme, as an hor

verti esrepresentanatural hoi etobetakenasseedverti es.Indeed,themost

important problem for bubble-growing algorithms is the determination of the

seedverti esfromwhi hbubblesaregrown,whi hrequiresexpensivepro esses

involvingallofthe graphverti es[3,8℄. Sin ean horverti esare onne ted to

alloftheverti esofthelastlayers,thediusedliquidsowasafrontasifthey

originated from the farthest verti es from the frontier, whi h is indeed what

wouldhappeniftheyowedfrom the enter ofabubblehavingthefrontieras

itsperimeter.

3.3 Parallelization

Ourdiusionalgorithmhastheadditionalinterestofbeinghighlys alable.Ifwe

assumethatfullgraphs,aswellasbandgraphs,aredistributeda rosspro essors

su hthateverypro essorholdsafra tionofthegraphverti esalongwiththeir

adja en y lists, like what is done for instan e in PT-S ot h [2℄, the parallel

versionofS ot h,theparallelversionofthealgorithmis straightforward.

Ev-erypro essorperforms its lo al update and omputes the ontributions it has

to spreadtodistantneighbors,after whi hthese ontributionsaresenttotheir

destination pro essorsin orderto beaggregated.Inorderto over

ommuni a-tionby omputations,verti esthathavedistantneighbors anbepro essedrst,

then ommuni ationsarestarted,andverti eswithpurely lo aladja en ylists

ex eptthread.

|V |

and

|E|

arethevertexandedge ardinalities,inthousands. Graph Size(

×10

3

)Average

|V |

|E|

degree altr4

26

163

12.50

audikw1

944

38354

81.28

auto

449

3315

14.77

bmw32

227

5531

48.65

body

45

164

7.26

bra ket

63

367

11.71

Graph Size(

×10

3

)Average

|V |

|E|

degree onesphere1m

1055

8023

15.21

o ean

143

410

5.71

oilpan

74

1762

47.77

pwt

37

145

7.93

thread

30

2220

149.32

4 Experimental results

Thediusion algorithmdis ussedabovehasbeenimplemented,asasequential

graph bipartitioningmethod, in version5.0 theS ot h [10℄ graph

partition-ingandstati mappingsoftware.Its k-wayimplementationisnotyetavailable,

be auseit requires more oding, in luding a k-wayband extra tion algorithm

whi hdoesnotexist to date.All of thene essaryoating-pointarithmeti has

beenimplementedinsinglepre ision.

Thetests were runon aLenovoThinkPad T60laptop, with anIntel

dual- ore T2400 pro essor running at 1.8 MHz and 1 Gb of memory. As we ran

sequentialtestsonly,thedual- orefeatureofthepro essorisnotrelevant.The

testgraphswehaveusedinourexperimentsarelistedinTable1.Thesegraphs

were partitioned into

2

to

128

parts, and the three quality metri s that we onsider arethenumberof utedges, alled Cut,aloadimbalan e ratioequal

tothesizeofthelargestpartdividedbytheaveragesize, alledMaCut,andthe

maximumdiameteroftheparts,referredtoasMDi,whi hisanindire tmetri

oftheshapeofthepartition,andisusableeveninthe aseofgraphsofunknown

or nonexistentgeometry.This lattermetri is insu ient, asitdoesnotreally

apturethesmoothness oftheinterfa es,sin eirregularlyshapedparts anstill

havesmalldiameters;thebest proofwouldhavebeentorunaniterativesolver

and measure onvergen erates basingonthenumbersof iterations.Thiswork

isin progress.

Three diusion heuristi s were ompared againstthe lassi alstrategy

im-plementedinS ot h4.0,referredtoasRMFinthefollowing,whi hperforms

re ursivebipartitioning with bipartitions omputed in a multi-level way, using

FM renement.

Therstmethod, RMBD,usesthesamere ursivebipartitioningand

multi-levelstrategy, but banded diusion is performedduring the multi-level

rene-ment steps. Theresults a hieved with this method validate our approa h: the

obtainedpartitions haveverysmoothboundaries(see Figure5.b), and are

ad-equately balan ed if the number of diusion iterations is su iently high, as

Table 2. Evolution ofthe utsize (

∆

Cut),of theload imbalan e ratio(

∆

MaCut) andofthemaximumdiameteroftheparts(

∆

MDi)produ edbyvariouspartitioning heuristi swithrespe ttotheRMFstrategy,averagedoveralltestgraphsandnumbers of parts.Figures belowpartitioning strategy namesindi atethe numberof diusion stepsperformed. Method RMBD RMBDF RMBaDF 500 200 100 40 500 40 40

∆

Cut(%) +19.51+20.01+18.15+21.49+2.26+3.10 -3.17

∆

MaCut(%) +0.58 +1.12 +1.80 +9.76 -0.95 -0.29 -0.21

∆

MDi(%) +3.86 +1.92 +4.69 +5.43+2.26+3.10 -3.24

∆

Time(

×

) 21.31 9.33 5.33 2.93 21.47 2.99 3.07

Whenperforming

100

diusionsteps,theaverageMaCutvalueforRMBDis

1.046

,only

1.80

%higherthantheoneofRMF.However,themaximumdiameter Mdiisnotsigni antlyredu ed,andisevenin reasedonaverageby

4.69%

with respe ttoRMF.Thismethodisalso

5.33

timesslowerthanRMFandin reases the utbyabout

20%

,whi hmakesitoflittlepra ti aluse.

Wehavethereforeexperimentedase ondmethod,RMBDF,wherethe

las-si al FM algorithm is applied to the band graphafter the diusion algorithm.

The idea of this strategy is to benet from the global optimization

apabil-ities brought by the diusion algorithm, while lo ally optimizing the frontier

afterward.Evenwhenperforming

40

diusion stepsonly,thesmoothnessofthe boundariesis preservedand parts aremorebalan ed, whilethe utis only

in- reasedby

3.10%

with respe t to RMF. This strategyis also only three times slowerthanRMF,whi hisextremelyfastforadiusion-basedalgorithm.

Inordertofavortheminimization ofdiameters,wehavemodiedour

diu-sion method soasto double at ea h stepthe amountof liquidborne byevery

vertex,inanavalan he-likepro ess.ThismethodisreferredtoasaD.Itisno

longerboundto onverge,andindeed ausesoverowsforlargenumbersof

dif-fusionsteps,butgivesgoodresultsforsmallnumbersofiterations.Asamatter

of fa ts,we an see in Table2that theRMBaDF methodis themost e ient

oneonaverage,andyields better resultsthanthe lassi alRMF method while

stillprovidingsmoothboundaries,aseviden edin Figure5. .

Forthesakeof omparison,we ompareinTable3someofourresultsagainst

the ones obtainedwith K-MeTiS . K-MeTiS uses dire t k-waypartitioning

in-stead of re ursive bipartitioning, whi h usually makes it more e ient when

the numberof parts in reases,and alsomu h faster (from

10

to

20

times). As analyzed in [11℄,the performan e of re ursivebipartitioning methods tends to

de rease when thenumberof parts in reases,whi h should limitthee ien y

ofRMBDFmethodsforlargenumbersofparts.Afullk-waydiusionalgorithm

oftheparts(MDi),betweenthreeheuristi s:multi-levelwithFMrenement(RMF,

asimplementedinS ot h4.0),multi-levelwithbandeddiusionandFMrenements (RMBaDF),andK-MeTiS .

Test Numberofparts

ase 2 4 8 16 32 64 128 altr4 RMF Cut 1688 3197 4978 7788 11905 17656 24478 MDi 50 52 40 33 25 21 14 RMBaD(40)F Cut 1621 3203 5017 7776 11980 17669 24831 MDi 48 46 41 30 25 18 14 KMeTiS Cut 1670 3233 4981 8115 12147 17355 24058 MDi 48 45 41 34 26 22 14 bmw32 RMF Cut 17271 54424 84222 120828181844267427394418 MDi 93 116 130 106 74 120 68 RMBaD(40)F Cut 16032 54446 83422 124945 183454 275594 411154 MDi 91 130 96 84 68 63 56 KMeTiS Cut 15529 55506 92658 125686 193169 286111 420965 MDi 87 108 99 87 70 61 68

5 Con lusion and future work

In thispaper,wehavepresentedadiusion algorithm whi h, used in a

multi-level bandedframework,resultsin smootherpartition frontiers andmore

om-pa tparts.Usedinourbanded ontext,thisalgorithmisfastenoughtobeused

on very large graphs,as it is only about three times slower than lassi al

lo- al optimization s hemes. The 2-waysequential versionhas been integratedin

version5.0ofS ot h.

Thisalgorithmisalsoeasilyparallelizableandhighlys alable,whi hmakes

itaverygood andidatefortherealizationofafastande ientparallelgraph

partitioner,takingadvantageoftheparallel multi-levelandbandgraph

extra -tionroutinesalreadydevelopedinPT-S ot hin the ontextofsparsematrix

reordering.

Even morethan lassi alFM-likealgorithms, this algorithm is onstrained

bythegreedynatureofthere ursivebipartitionings heme,whi hpreventsthe

globalimprovementoffrontiers omputedatpreviousstages.Afullk-wayversion

ofthealgorithmisthereforeunderdevelopment,whi hextendsthe2-waymodel

by onsidering

k

dierentliquidshavingthesamemutualannihilationproperties, su hthat when

p

dierentliquidsare mixed in thesamebarrel,onlythemost abundantoneremains. Thisbehaviorisequivalent totheoneof ouralgorithm

in the2-way ase.Usinganativek-ways hemeshouldalsosigni antlyredu e

running times ompared to re ursive bipartitioning. A parallel versionis also

Fig.5.Partitionofgraphaltr4into

8

partsusingthreedierent strategies.The seg-mented frontiers produ ed by FM-like algorithms are learly eviden ed inFigure a. RMBDprodu esthesmoothestboundaries,asshowninFigureb.RMBaDFtakesthe

bestofbothworlds,inFigure .

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