Advanced Flow-Based Multilevel Hypergraph Partitioning SEA 2020 June 5, 2020 Lars Gottesb¨ uren , Michael Hamann, Sebastian Schlag, Dorothea Wagner I NSTITUTE OF T HEORETICAL I NFORMATICS · A LGORITHMICS G ROUP s t KIT – The Research University in the Helmholtz Association www.kit.edu
Hypergraph Partitioning Hypergraph H = ( V , E , c , ω ) vertex set V = { 1, ..., n } edge set E ⊆ P ( V ) \ ∅ incident edges Γ ( u ) = { e ∈ E | u ∈ e } vertex weights ϕ : V → R ≥ 1 edge weights ω : E → R ≥ 1 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Hypergraph Partitioning Hypergraph H = ( V , E , c , ω ) vertex set V = { 1, ..., n } edge set E ⊆ P ( V ) \ ∅ incident edges Γ ( u ) = { e ∈ E | u ∈ e } vertex weights ϕ : V → R ≥ 1 edge weights ω : E → R ≥ 1 hyperedge / net pin 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Hypergraph Partitioning Partition into k disjoint blocks Π = { V 1 , . . . , V k } blocks V i have roughly equal weight : � � ϕ ( V ) ϕ ( V i ) ≤ (1 + ε ) k 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Hypergraph Partitioning Partition into k disjoint blocks Π = { V 1 , . . . , V k } imbalance blocks V i have roughly equal weight : � � ϕ ( V ) ϕ ( V i ) ≤ (1 + ε ) k 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Hypergraph Partitioning Partition into k disjoint blocks Π = { V 1 , . . . , V k } imbalance blocks V i have roughly equal weight : � � ϕ ( V ) ϕ ( V i ) ≤ (1 + ε ) k minimize connectivity objective: con = � e ∈ E ( λ ( e ) − 1) ω ( e ) 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Hypergraph Partitioning Partition into k disjoint blocks Π = { V 1 , . . . , V k } imbalance blocks V i have roughly equal weight : � � ϕ ( V ) ϕ ( V i ) ≤ (1 + ε ) k minimize connectivity objective: con = � e ∈ E ( λ ( e ) − 1) ω ( e ) λ ( e ) = |{ V i | V i ∩ e � = ∅}| # blocks overlapping with e 1 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Applications q 2 q 1 Distributed Databases Route Planning VLSI Design HPC 2 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Multilevel Algorithms Input Hypergraph local search match or cluster contract uncontract initial partition 3 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Multilevel Algorithms Input Hypergraph local search match or cluster contract uncontract initial partition 3 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Classic Fiduccia-Mattheyses Algorithm 1: FM Local Search connectivity while improvement found do while ¬ done do find best move rollback pass perform best move rollback to best solution vertex moves slide kindly provided by Sebastian Schlag 4 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Classic Fiduccia-Mattheyses Algorithm 1: FM Local Search pass 1 pass 2 connectivity while improvement found do while ¬ done do find best move pass rollback perform best move rollback to best solution vertex moves slide kindly provided by Sebastian Schlag 4 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Classic Fiduccia-Mattheyses Algorithm 1: FM Local Search pass 1 pass 2 connectivity while improvement found do while ¬ done do find best move pass rollback perform best move rollback to best solution vertex moves ✗ get stuck in local optima ✗ large edges � zero gain moves ? OPT ? slide kindly provided by Sebastian Schlag 4 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Classic Fiduccia-Mattheyses Algorithm 1: FM Local Search pass 1 pass 2 connectivity while improvement found do max-flow-min-cut to the rescue while ¬ done do find best move pass rollback perform best move rollback to best solution vertex moves ✗ get stuck in local optima ✗ large edges � zero gain moves ? OPT ? slide kindly provided by Sebastian Schlag 4 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Classic Fiduccia-Mattheyses Algorithm 1: FM Local Search pass 1 pass 2 connectivity while improvement found do max-flow-min-cut to the rescue while ¬ done do find best move pass rollback perform best move rollback to best solution Issues? vertex moves only 2-way what are flows ✗ get stuck in local optima ✗ large edges � zero gain moves on hypergraphs? not balanced ? OPT ? slide kindly provided by Sebastian Schlag 4 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Flow-Based Refinement in KaHyPar B 1 B 2 V 1 V 2 V 1 V 2 s t V 3 V 4 select two adjacent blocks for refinement build graph-based flow model 5 5 2 2 2 2 2 2 1 1 s t s t 3 1 3 1 2 2 V 2 4 4 V 1 solve flow problem find more balanced minimum cut slide kindly provided by Sebastian Schlag 5 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Flow-Based Refinement in KaHyPar B 1 B 2 V 1 V 2 V 1 V 2 s t V 3 V 4 select two adjacent blocks for refinement build graph-based flow model 5 5 either: restrict flow model size so that balance is guaranteed 2 2 2 2 2 2 or: make it a little larger, hope for balance. if not � scale down again 1 1 s t s t 3 1 3 1 2 2 V 2 4 4 V 1 solve flow problem find more balanced minimum cut slide kindly provided by Sebastian Schlag 5 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Flow-Based Refinement in KaHyPar B 1 B 2 V 1 V 2 V 1 V 2 s t V 3 V 4 select two adjacent blocks for refinement build graph-based flow model 5 5 2 2 2 2 2 2 1 1 s t s t 3 1 3 1 2 2 V 2 4 4 V 1 solve flow problem find more balanced minimum cut slide kindly provided by Sebastian Schlag 5 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Flow-Based Refinement in KaHyPar B 1 B 2 V 1 V 2 V 1 V 2 s t V 3 V 4 select two adjacent blocks for refinement build graph-based flow model 5 5 2 2 2 2 2 2 1 1 s t s t 3 1 3 1 2 2 V 2 4 4 V 1 solve flow problem find more balanced minimum cut slide kindly provided by Sebastian Schlag 5 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
The new KaHyPar-HFC B 1 B 2 V 1 V 2 t s V 1 V 2 V 3 V 4 select two adjacent blocks for refinement flows directly on hypergraph 5 2 2 2 s t 1 s t naturally built-in 3 1 2 V 2 4 V 1 use FlowCutter find more balanced minimum cut [Hamann, Strasser JEA18] [Gottesb¨ uren, Hamann, Wagner ESA19] 6 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
The new KaHyPar-HFC B 1 B 2 V 1 V 2 t s V 1 V 2 V 3 V 4 select two adjacent blocks for refinement flows directly on hypergraph 5 2 2 2 s t 1 s t naturally built-in 3 1 2 V 2 4 V 1 use FlowCutter find more balanced minimum cut [Hamann, Strasser JEA18] [Gottesb¨ uren, Hamann, Wagner ESA19] 6 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
The new KaHyPar-HFC B 1 B 2 V 1 V 2 t s V 1 V 2 V 3 V 4 select two adjacent blocks for refinement flows directly on hypergraph 5 what’s new for FlowCutter? 2 2 2 weighted instances s t 1 s t naturally built-in new guidance 3 1 2 V 2 4 V 1 use FlowCutter find more balanced minimum cut [Hamann, Strasser JEA18] [Gottesb¨ uren, Hamann, Wagner ESA19] 6 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
Flows on Hypergraphs u x e v w 7 Lars Gottesb¨ uren – Advanced Flow-Based Multilevel Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group
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