distributed evolutionary graph...

1 Peter Sanders, Christian Schulz:Distributed Evolutionary Graph Partitioning

Department of InformaticsInstitute for Theoretical Computer Science

Institute for Theoretical Computer Science

DistributedEvolutionary Graph PartitioningPeter Sanders, Christian SchulzWorkshop on EA & EA – Agra, India

KIT – Universitat des Landes Baden-Wurttemberg undnationales Großforschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu

Overview

IntroductionMultilevel AlgorithmsEvolutionary TechniquesExperimentsSummary

Simulation - FEM

Simulation space is discretizedinto a meshSolution of partial differentialequations are approximated bylinear equationsNumber of vertices canbecome quite large→ time and 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

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 is minimal.

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

Multilevel Graph Partitioning1. Contraction2. Local Search3. Global Search

Multi-Level Graph Partitioning

Successful in existing systems:Metis, Scotch, Jostle,. . . , KaPPa, KaSPar, KaFFPa, KaFFPaE

Contract Matchings

But how are the edges selected?

Graph PartitioningMatching Selection

Goals:1. large edge weights sparsify2. large #edges few levels3. uniform node weights “represent” input4. small node degrees “represent” input unclear objective gap to approx. weighted matchingwhich only considers 1.,2.

Our Solution:Apply approx. weighted matching to general edge rating function

input graph

contract

Graph PartitioningEdge Ratings

ω({u, v})

expansion({u, v}) :=ω({u, v})

c(u) + c(v)

expansion∗({u, v}) :=ω({u, v})c(u)c(v)

expansion∗2({u, v}) :=ω({u, v})2

c(u)c(v)

innerOuter({u, v}) :=ω({u, v})

Out(v) + Out(u)− 2ω(u, v)

where c = node weight, ω =edge weight,Out(u) := ∑{u,v}∈E ω({u, v})

FM Local Search

compute gain ∀ v ∈ V

g(v) = dext (v)− dint (v)store gain of boundarynodes (e.g. in a heap)

FM Local Search

compute gain ∀ v ∈ Vg(v) = dext (v)− dint (v)

store gain of boundarynodes (e.g. in a heap)

FM Local Search

compute gain ∀ v ∈ Vg(v) = dext (v)− dint (v)store gain of boundarynodes (e.g. in a heap)

FM Local Search

move highest gain verticesto opposite blockeach node at most onceupdate gain of neighbors

Step: 0Edge Cut: 5

FM Local Search

Step: 0 1Edge Cut: 5 4

FM Local Search

Step: 0 1 2Edge Cut: 5 4 6

FM Local Search

Step: 0 1 2 3Edge Cut: 5 4 6 8

FM Local Search

stop after limittake best edge cutwithin balance constraint

edgecut

Step: 0 1 2 3Edge Cut: 5 4 6 8

FM Local Search

stop after limittake best edge cutwithin balance constraint

edgecut

Step: 0 1 2 3Edge Cut: 5 4 6 8

FM Local Search – Discussion

+ Generalizable for multiple blocks+ Linear time− Unlikely to find improvements requiring ≥ 2 negative gain moves

More Localized Local Search

Idea: KaPPa, KaSPar ⇒ more local searches are betterTypical: k -way local search initialized with complete boundaryLocalization:

1. complete boundary⇒ maintained todo list T2. initialize search with single node v ∈rnd T3. iterate until T = ∅

each node moved at most once

Multilevel Graph Partitioning1. Contraction2. Local Search

Localization of Local Search helpsFlows as Local Search

3. Global Search

Flows as Local ImprovementTwo Blocks

∂1B ∂2B

area 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) c(V )2 − c(V2)

⇒ c(V2new) ≤ c(V2) + (1 + ε) c(V )2 − c(V2)

Flows as Local ImprovementTwo Blocks

obtain optimal cut in Bsince each cut in B yields a feasible partition→ improved two-partitionadvanced techniques possible and necessary

Example100x100 Grid

ExampleConstructed Flow Problem (using BFS)

ExampleApply Max-Flow Min-Cut

ExampleOutput Improved Partition

Local Improvement for k -partitionsUsing Flows?

on each pair of blocks

input graph

initial

outputpartition

local improvement

partitioning

contract uncontract

Multilevel Graph Partitioning1. Contraction2. Local Search

Localization of Local Search helpsFlows as Local Search

3. Global Search

Iterated Multilevel [Walshaw 2004]

contract

input graph

... ...local improvement

partitioning

initialuncontractcontract

uncontract

local improvement...

don’t contract cut edgesadapt previous solution as initial partitioningcuts can only improveV-cycles / F-cycles

Evolutionary Graph Partitioning

DistributedEvolutionary Graph Partitioning

Evolutionary Algorithms:highly inspired by biologypopulation of individualsselection, mutation, recombination, ...

Goal: Integrate KaFFPa in an Evolutionary StrategyEvolutionary Graph Partitioning:

individuals ↔ partitionsfitness ↔ edge cut

Parallelization→ quality records in a few minutes for small graphs

Combine

contract

two individuals P1, P2:don’t contract cut edges of P1 or P2

until no matchable edge is leftcoarsest graph↔ Q-graph of overlay→ exchanging good parts is easyinital solution: use better of both parents

ExampleTwo Individuals P1, P2

ExampleOverlay of P1, P2

ExampleMultilevel Combine of P1, P2

contract

Exchanging good parts is easyCoarsest Level

v4 >> <

>> large weight, < small weightstart with the better partition (red, P2)move v4 to the opposite blockintegrated into multilevel scheme (+local search on each level)

ExampleResult of P1, P2

Parallelization

each PE has its own island (a local population)locally: perform combine and mutation operationscommunicate analog to randomized rumor spreading

1. rumor↔ currently best local partition2. local best partition changed → send it to O(log P) random PEs3. asynchronous communication (MPI Isend)

Experiments

Experimental ResultsComparison with Other Systems

Geometric mean, imbalance ε = 0.03:11 graphs (78K–18M nodes) ×k ∈ {2,4,8,16,64}

Algorithm large graphsBest Avg. t[s]

KaFFPa strong 12 053 12 182 121.22KaSPar strong 12 450 +3% 87.12KaFFPa eco 12 763 +6% 3.82Scotch 14 218 +20% 3.55KaFFa fast 15 124 +24% 0.98kMetis 15 167 +33% 0.83

Repeating Scotch as long as KaSPar strong run and choosing thebest result 12.1% larger cutsWalshaw instances, road networks, Florida Sparse MatrixCollection, random Delaunay triangulations, random geometricgraphs

QualityEvolutionary Graph Partitioning

blocks k KaFFPaEimprovement overreps. of KaFFPa

2 0.2%4 1.0%8 1.5%

16 2.7%32 3.4%64 3.3%

128 3.9%256 3.7%

overall 2.5%

2h time, 32 cores per graph and k , geom. mean

Quality

1 5 20 100 500

normalized time tn

ut Repetitions

KaFFPaE

Scalability

1 10 100 1000 10000

normalized time tn

c)p = 1p = 2p = 4

p = 8p = 16p = 32

p = 64p = 128p = 256

Walshaw Benchmark

runtime is not an issue614 instances (ε ∈ {1%,3%,5%})focus on partition quality

overall quality records (at submission time):

ε ≤1% 78%3% 92%5% 94%

Summary

[ESA10]

graphOutputPartition

contract

... ...

distr.evol. Alg.

partitioning

initial

local improvement

uncontract

[IPDPS10] [ESA11]

[ESA11] Cycles a la multigridW−F−V−

[IPDPS10]

[ALENEX12]

[SEA12]

n−level

edge ratings

flows etc.parallel

GraphpartitioningMultilevel

Outlook

Further Material in the Paper(s)F-cycles, High Quality Matchings, ....Different combine and mutation operatorsSpecialization to road networks (Buffoon)Many more details and experiments ...

Future Workother objective functions

currently via selection criterionconnectivity? f (P) := f (P) + χ{P not connected} · |E |

integrate other partitionersgraph clusteringopen source release

Thank you!

Contact: christian.schulz@kit.edusanders@kit.edu

distributed evolutionary graph...

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