Parallel Graph Partitioning for Complex...

1 Christian Schulz:Parallel Partitioning of Complex Networks

Department of InformaticsInstitute for Theoretical Computer Science

Institute for Theoretical Computer Science

Parallel Graph Partitioningfor Complex NetworksHenning Meyerhenke, Peter Sanders, Christian SchulzHigh-performance Graph Algorithms and Applications in Computational Science – Dagstuhl

www.kit.edu

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:FEM, linear equation systems, VLSI design, route planning, . . .

Multilevel Graph Partitioning



input graph

... ...

initial

contra

ctio

n p

hase

local improvement

uncontract

partitioning

contract

outputpartition

uncoars

enin

g p

hase

Successful in many systems (using matchings or AMG for coarsening)

KaHIPKarlsruhe High Quality Partitioning



input graph

OutputPartition

contract

... ...

match

partitioning

initial

local improvement

uncontract

[IPDPS10] [ESA11]

[ESA11] W−F−V−

[IPDPS10]

cycles a la multigrid

[SEA13]

[DIMACS12] [ALENEX12] evol. alg.distr.

highly balanced:

[ALENEX12]

[SEA12/14]

[SEA14]

A

C

B

+

edge ratings

flows etc.parallel

MultilevelGraph Partitioning

A B

C0 −1

−1

0

−1

A B

C0 −1

−1

0

−10 0

social separatorsbuffoon

Matching-based Coarsening



A+B

a+b

B

a b

A

Matching-based Coarsening



A+B

a+ba b

A B

Matching-based CoarseningProblem



bad for networks that are highly irregularsubstantial reduction is hard using matchingsmay contract wrong edges!

Basic Idea



aggressive contraction / simple and fast local searchmain idea: contract clusteringsclustering paradigm: internally dense and externally sparse

→

Basic IdeaContraction of Clusterings



A+B+C

a+b+ca bc

A

B

C

contraction: respect balance and cutavoid large blocks: size constraint Uconstruct coarse graph using hashingrecurse until graph is small

Label PropagationCut-based, Linear Time Clustering Algorithm [Raghavan et. al]



cut-based clustering using size-constraint label propagationstart with singletonstraverse nodes in random order or smallest degree firstmove node to cluster having strongest eligible connectionmodification eligible: w.r.t size constraint U

Scan

... ...

→

Label Propagation



Iteration Cut [%]0 1001 8.962 6.153 5.664 5.445 5.286 5.257 5.218 5.18... 5.09

Label PropagationSimple Local Search



Greedy Local Search:start with partition from coarser leveltraverse nodes in random ordermove node to cluster having strongest eligible connectioneligible: w.r.t size constraint U := (1 + ε) |V |k

Scan

... ...

→

Augmentations of the partitioning algorithm→ paper



Parallelization

Graph Distribution over PEs



Graph Distribution: a PE receives n/p vertices and their edges

Processor I Processor II

Communication

ghost nodes: adjacent nodes on other processor (communication!)interface nodes: nodes adjacent to ghost nodes

Label PropagationDistributed Memory



each PE has a static part of the graph, only block IDs can change

Overlap Computation and Communication (PE centric view):

Scan

V

Phase i Phase i+1Phase i−1

At the end of phase i :send block ID updates of phase i to neighboring PEsreceive block ID updates from neighboring PEs from phase i − 1

* while scanning in phase i , messages are routed through the network* algorithm converged→ nothing will be communicated

+ algorithms to maintain balance of clusters/blocks

Parallel Augmentations



input graph

... ...local improvement

partitioning

initialuncontractcontract

...

uncontract

local improvement...

contract

V-cycles:don’t contract cut edgesmodify label propagation to respect partition

Node Ordering for LP:perform smallest degree first only locally



Experiments

Instances



graph n m graph n mLarge Graphs

p2p-Gnutella04 6 405 29 215 citationCiteseer 268 495 ≈1.2Mwordassociation-2011 10 617 63 788 coAuthorsDBLP 299 067 977 676PGPgiantcompo 10 680 24 316 cnr-2000 325 557 ≈2.7Memail-EuAll 16 805 60 260 web-Google 356 648 ≈2.1Mas-22july06 22 963 48 436 coPapersCiteseer 434 102 ≈16.0Msoc-Slashdot0902 28 550 379 445 coPapersDBLP 540 486 ≈15.2Mloc-brightkite 56 739 212 945 as-skitter 554 930 ≈5.8Menron 69 244 254 449 amazon-2008 735 323 ≈3.5Mloc-gowalla 196 591 950 327 eu-2005 862 664 ≈16.1McoAuthorsCiteseer 227 320 814 134 in-2004 ≈1.3M ≈13.6Mwiki-Talk 232 314 ≈1.5M

Huge Graphsuk-2002 ≈18.5M ≈262M sk-2005 ≈50.6M ≈1.8Garabic-2005 ≈22.7M ≈553M uk-2007 ≈106M ≈3.3G

ε = 3%, k ∈ {2,4,8,16,32,64}, geom. mean of averages, 10 seeds

Experimental ResultsSelected



Algorithm impr. cut t [s]UFast 51.7% 1.5UFastV 54.7% 3.0UEcoV/B 60.9% 11.5UStrong 75.1% 296.4KaFFPaEco 22.2% 36.2KaFFPaStrong 66.2% 640.8kMetis 45.8% 0.4hMetis 60.5% 107.4Scotch baseline 10.6

UStrong best quality, running time ≤ KaFFPaStrongbest cut of hMetis 6% worse than avg. cut of UStrongUEcoV/B outperforms hMetis, order of magnitute less timeUFast better quality but slower

Parallel Solution QualityPerformance k = 2 blocks, 32PEs



instances: meshes and social networks/web graphs

ParMetis has ineffective coarsening (matching-based)due to memory consumption of coarsest graph (dist. among PEs)→ could not solve arabic-2005, sk-2005 and uk-2007

solved instances (ParMetis):fast and eco yield 19.2% and 27.4% improvementfast and eco slower on average

social networks/web graphs:fast: 38% less cut edges and > 2×fastereco: 45% less cut edges and slowerbest instance: 18×faster and 61.6% less cut edgesimpr. quality over FB or matching-based algorithms (other inst.)

Strong ScalingSocial Networks



10

100

1000

1 2 4 8 16 32 64 256 1K 2K

tota

l ti

me

[s]

number of PEs p

Fast sk-2007Fast arabic-2005

Fast uk-2002

Fast uk-2007Minimal uk-2007

uk-2007 can be partitioned in 15.2 seconds (seq. 10.5min)more scaling results in the paper

Conclusion



cluster contraction improves running time and solution qualityefficient parallelization dominates ParMetis on social networks∃(semi-)external variant of the algorithmmuch more results in the papers

→

References



Partitioning Complex Networks via Size-constrained ClusteringH. Meyerhenke, P. Sanders and C. SchulzArXiv:1402.3281

Parallel Graph Partitioning for Complex NetworksH. Meyerhenke, P. Sanders and C. SchulzArXiv:1404.4797

(Semi-)External Algorithms for Graph Partitioning and ClusteringY. Akhremtsev, P. Sanders and C. SchulzArXiv:1404.4887

Recent Advances in Graph PartitioningA. Buluc, H. Meyerhenke, I. Safro, P. Sanders and C. SchulzArXiv:1311.3144

Open ProblemsMostly from Recent Advances in GP



large values of k : multilevel looses attractivenessmulti-k -recursive parallel partitioning w.r.t. structure of clusterother metrics:

max. comm volume, max. quotient graph degree, pareto objectivesreal performance benchmarksfixed k questionable if application tolerates node failures→ repartitioning with changed kmapping blocks onto communication networksusing GP for graph tools such as Pregelexact ML algorithms for node separatorsusing GP has a hammer for optimization problems, e.g.

independent setgraph coloringmax clique



Thank you!

KaHIP: algo2.iti.kit.edu/documents/kahipon Twitter: twitter.com/ProjectKaHIP

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