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1 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
High Quality Graph PartitioningPeter Sanders, Christian Schulz
KIT – Universitat des Landes Baden-Wurttemberg undnationales Großforschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu
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Overview
2 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
IntroductionMultilevel AlgorithmsAdvanced TechniquesEvolutionary TechniquesExperimentsSummary
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ε-Balanced Graph Partitioning
3 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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, . . .
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Multi-Level Graph Partitioning
4 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
Successful in existing systems:Metis, Scotch, Jostle,. . . , KaPPa, KaSPar, KaFFPa, KaFFPaE
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Advanced TechniquesTalk Today
5 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
Edge RatingsHigh Quality MatchingsFlow Based RefinementsMore Localized Local SearchF-cycles for Graph Partitioning
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Graph PartitioningMatching Selection
6 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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Graph PartitioningEdge Ratings
7 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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})
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Flows as Local ImprovementTwo Blocks
8 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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)
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Flows as Local ImprovementTwo Blocks
9 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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Example100x100 Grid
10 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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ExampleConstructed Flow Problem (using BFS)
11 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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ExampleApply Max-Flow Min-Cut
12 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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ExampleOutput Improved Partition
13 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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Local Improvement for k -partitionsUsing Flows?
14 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
on each pair of blocks
input graph
...
initial
...
outputpartition
local improvement
partitioning
match
contract uncontract
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
![Page 21: High Quality Graph Partitioning - Peter Sanders, Christian Schulz](https://reader035.vdocument.in/reader035/viewer/2022062317/5867803f1a28ab6d408bcb0a/html5/thumbnails/21.jpg)
More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
![Page 23: High Quality Graph Partitioning - Peter Sanders, Christian Schulz](https://reader035.vdocument.in/reader035/viewer/2022062317/5867803f1a28ab6d408bcb0a/html5/thumbnails/23.jpg)
More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
![Page 24: High Quality Graph Partitioning - Peter Sanders, Christian Schulz](https://reader035.vdocument.in/reader035/viewer/2022062317/5867803f1a28ab6d408bcb0a/html5/thumbnails/24.jpg)
More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
![Page 25: High Quality Graph Partitioning - Peter Sanders, Christian Schulz](https://reader035.vdocument.in/reader035/viewer/2022062317/5867803f1a28ab6d408bcb0a/html5/thumbnails/25.jpg)
More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
![Page 26: High Quality Graph Partitioning - Peter Sanders, Christian Schulz](https://reader035.vdocument.in/reader035/viewer/2022062317/5867803f1a28ab6d408bcb0a/html5/thumbnails/26.jpg)
More Localized Local Search
15 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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DistributedEvolutionary Graph Partitioning
16 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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Combine
17 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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ExampleTwo Individuals P1, P2
18 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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ExampleOverlay of P1, P2
19 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
⇒
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ExampleMultilevel Combine of P1, P2
20 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
match
contract
⇒
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Exchanging good parts is easyCoarsest Level
21 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
>> <
>> <
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)
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ExampleResult of P1, P2
22 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
⇒
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Parallelization
23 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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)
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Experimental ResultsComparison with Other Systems
24 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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QualityEvolutionary Graph Partitioning
25 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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Quality
26 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
1 5 20 100 500
7900
8100
8300
k=64
normalized time tn
mea
n m
in c
ut Repetitions
KaFFPaE
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Walshaw Benchmark
27 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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%
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Summary
28 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
input graph
OutputPartition
contract
... ...
match
distr.evol. Alg.
partitioning
initial
local improvement
uncontract
Cycles a la multigridW−F−V−
[ALENEX12]
+
flows etc.
MultilevelGraph Partitioning
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Outlook
29 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
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
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30 Peter Sanders, Christian Schulz:High Quality Graph Partitioning
Department of InformaticsInstitute for Theoretical Computer Science, Algorithmics II
Thank you!
Contact: [email protected]@kit.edu