engineering graph partitioning...

1 Vitaly Osipov, Peter Sanders, Christian Schulz:Engineering Graph Partitioning Algorithms

Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II

EngineeringGraph Partitioning AlgorithmsVitaly Osipov, Peter Sanders, Christian Schulz

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

Overview



IntroductionMultilevel AlgorithmsAdvanced TechniquesEvolutionary 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, . . .

Multi-Level Graph Partitioning



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

Advanced TechniquesTalk Today



Edge RatingsHigh Quality MatchingsFlow Based RefinementsMore Localized Local SearchF-cycles for Graph Partitioningn-Level Approach

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)

expansion∗2−adist(u, v) :=expansion∗2(u, v)

φu,v

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

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

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

Global SearchIterated Multilevel [Walshaw 2004]



contract

input graph

match

... ...local improvement

partitioning

initialuncontractcontract

...

uncontract

local improvement...

match

don’t contract cut edgesno new initial partitioning neededrandom tiebreaking neededlocal improvement has to assure ≤ cuts

Global SearchV-Cycles



Co

arse

nin

gU

nco

arsenin

g

Graph not partitioned

Graph partitioned

Global SearchF-Cycles



Un

coarsen

ingC

oar

sen

ing

Graph not partitioned

Graph partitioned

Sequential Graph PartitioningKarlsruhe Sequential Partitioner



1. Contractioncontract a single edge between two levels

possibly n levelsfinegrained contraction consequtive levels are very similarno matching algorithm required

use of different edge ratings uniform distribution of node weightspriority queue defines the order of edges to be contracted

2. Local Searchefficent stopping criterion avoid quadratic runtime

3. GeneralTrial Trees improve quality by independent trials

KaSParLocal Search



Nodes unmarked, active, markedactive nodes

compute gains of moving from one block toanotherchoose the maximum gain

when movedactive markedcan’t become active anymoreunmarked neighbours of marked active

v

u

⇒

v

u

KaSParLocal Search - Stopping Criteria



touching each node on each level leadsto Ω(n2) runtime⇒ need flexible stopping criteriagains in each step identicallydistributed, independent randomvariables

expectation µvariance σ2

compute µ and σ2 from previous stepsstop after p steps if pµ2 > ασ2 + β

v

u

⇒

v

u

KaSParMemory Overhead



addressable PQ based on pairing heaps used for contractiondynamic graph data structure

1

2

3

4

5

104

105

106

107

108

me

mo

ry o

verh

ea

d

m

Sequential Graph PartitioningKarlsruhe Fast Flow Partitioner



Focus: Local and Global SearchLocal Search:

1. Flow Based Improvement Methods2. More Localized FM Algorithm (inspired by KaSPar)

Global Search:1. V-cycles and F-cycles similar to methods used in multigrid

V1V2B

Gs t

∂1B ∂2B

contract

input graph

match

... ...local improvement

partitioning

initialuncontractcontract

...

uncontract

local improvement...

match

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

Evolutionary AlgorithmsGeneral Structure



procedure steady-state-EAcreate initial population Pwhile stopping criterion not fulfilled

select parents P1, P2 from Pcombine P1 with P2 to create offspring omutate offspring oevict individual in population using o

return the fittest individual that occurred

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)

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

Walshaw Benchmark



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

Algorithm < ≤KaPPa 131 189KaSPar 155 238KaFFPa 317 435KaFFPaE 300 470

overall quality records ≤:ε ≤1% 78%3% 92%5% 94%

Summary



input graph

OutputPartition

contract

... ...

match

distr.evol. Alg.

partitioning

initial

local improvement

uncontract

Cycles a la multigridW−F−V−

[ALENEX12]

+

flows etc.

MultilevelGraph Partitioning

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) + χP not connected · |E |

integrate other partitionersgraph clusteringopen source release



Thank you!

Contact: [email protected]@kit.edu

engineering graph partitioning...

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