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

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 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



a+b

A+B

a b

A B

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

match

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

steps

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



V1V2B

Gs t

∂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



s t

B

GV1

V2

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

match

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

match

... ...local improvement

partitioning

initialuncontractcontract

...

uncontract

local improvement...

match

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



match

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



match

contract

⇒

Exchanging good parts is easyCoarsest Level



>> <

>> <

G

v1

v2

v3

v4 >> <

>> <

G

v1

v2

v3

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

7900

8100

8300

k=64

normalized time tn

mea

n m

in c

ut Repetitions

KaFFPaE

Scalability



1 10 100 1000 10000

2450

2500

2550

2600

normalized time tn

mea

n m

in c

ut

(mm

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



input

[ESA10]

graphOutputPartition

contract

... ...

match

distr.evol. Alg.

partitioning

initial

local improvement

uncontract

[IPDPS10] [ESA11]

[ESA11] Cycles a la multigridW−F−V−

[IPDPS10]

[ALENEX12]

[SEA12]

n−level

todo

+

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: [email protected]@kit.edu

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