engineering multilevel graph partitioning...

1 December 1, 2010Peter Sanders, Christian Schulz – Institute for Theoretical Computer Science, Algorithmics IIEngineering Multilevel Graph Partitioning Algorithms

KIT – die Kooperation von Forschungszentrum Karlsruhe GmbH und Universitat Karlsruhe (TH)

Institute for Theoretical Computer Science, Algorithmics II

Engineering Multilevel Graph PartitioningAlgorithmsPeter Sanders, Christian Schulz


Overview



IntroductionMultilevel AlgorithmsLocal Improvement using FlowsMore Localized SearchGlobal SearchExperimental EvaluationFuture Work

Simulation - FEM



Simulation space isdiscretized into a meshSolution of partitial differentialequations are approximatedby linear equationsNumber of vertices canbecome quite large→ timeand memoryParallel processing required

The Common Parallel Approach



Mesh partitioned via dual graph1. Each volume (data, calculation) is

represented by a vertex2. Interdependencies are represented by

edges

All PE’s get same amount of workCommunication is expensive

Graph Partitioning Problem:Partition a graph into (almost) equally sized blocks, such that thenumber of edges connecting vertices from different blocks isminimal.

ε-Balanced Graph Partitioning



Partition graph G = (V , E , c : V → R>0, ω : E → R>0)into k disjoint blocks s.t.

total node weight of each block ≤ 1 + ε

ktotal node weight

total weight of cut edges as small as possible

Applications:finite element simulations, VLSI design, route planning, . . .

Multi-Level Graph Partitioning



Successful in existing systems:DiBaP, Chaco, Jostle, Metis, Scotch,. . . , KaPPa, KaSPar

Flows as Local ImprovementConstruction



tsG

Bb1

b2

corridor B, such that each (s, t)-min cut is ε-balanced cut in Ge.g. 2 times BFS (left, right)stop the BFS, if size would exceed (1 + ε) n

2 −ω(b2)⇒ ω(b2new) ≤ ω(b2) + (1 + ε) n

2 −ω(b2)

Local Improvement for k -partitionsUsing Flows?



on each pair of blocks

input graph

...

initial

...

outputpartition

local improvement

partitioning

match

contract uncontract

Local Improvement for k -partitionsUsing Flows?



on each pair of blocks

Scheduling Pairs of BlocksSketch



Idea: refine only blocks in which change still happensblock can be active or inactiverefine pairs of blocks with at least one active blockblocks become active next round if something improved

More Localized Local Search



Idea: KaPPa and KaSPar⇒ more local searches are betterTypical: k -way local search initialized with complete boundaryeach node moved at most onceLocalization:1. complete boundary⇒ maintained todo list T2. initialize search with single node v ∈rnd T3. iterate until T = ∅

Global SearchIterated Multilevel



match

Global SearchIterated Multilevel



contract

input graph

match

... ...local improvement

partitioning

initialuncontractcontract

...

uncontract

local improvement...

match

no new initial partitioning neededrandom tiebreaking neededlocal improvement has to assure ≤ cuts

Global SearchV-Cycles



Global SearchW-Cycles



Global SearchF-Cycles



ExperimentsToday’s Testset



Medium sized instancesgraph n mrgg17 217 1 457 506rgg18 218 3 094 566Delaunay17 217 786 352Delaunay18 218 1 572 792bcsstk29 13 992 605 4964elt 15 606 91 756fesphere 16 386 98 304cti 16 840 96 464memplus 17 758 108 384cs4 33 499 87 716pwt 36 519 289 588bcsstk32 44 609 1 970 092body 45 087 327 468t60k 60 005 178 880wing 62 032 243 088brack2 62 631 733 118finan512 74 752 522 240bel 463 514 1 183 764nld 893 041 2 279 080af shell9 504 855 17 084 020

ExperimentsRemove Components



Repetition Avg. Best. tKaFFPa Strong 2683 2617 8.93-KWay 2682 2614 9.23-MoreLocalizedSearch 2729 2656 5.55-FCycle 2748 2668 3.27-Flow 2934 2823 1.66

Effectiveness Avg. Best.KaFFPa Strong 2636 2616-KWay 2636 2618-MoreLocalizedSearch 2668 2650-FCycle 2669 2653-Flow 2799 2775

Outlook



Walshaw Benchmark: 317 improvements (of about 600 possible)118 same cut as previous bestTR: http://arxiv.org/abs/1012.0006Many more experiments

Future Work



1. better initial partitioning for large k2. integration into meta-heuristics (even parallel)3. exploit shared memory parallelism4. output vertex separators5. graph clustering

10th DIMACS ImplementationChallengeAnnouncement



Graph Partitioning and ClusteringOrganizers:1. David Bader Georgia Institute of Technology2. Peter Sanders and Dorothea Wagner, KIT

1st phase: call for instances and applications deadline ”soon”2nd phase: paper submission3rd phase: challenge workshop4th phase: proceedings



Thank you!

Contact: [email protected]

engineering multilevel graph partitioning...

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