Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure SEA’17 · June 23, 2017 Tobias Heuer and Sebastian Schlag INSTITUTE OF THEORETICAL INFORMATICS · ALGORITHMICS GROUP KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Hypergraphs Generalization of graphs ) hyperedges connect ≥ 2 nodes Graphs ) dyadic (2-ary) relationships Hypergraphs ) (d-ary) relationships Hypergraph H = (V , E, c, !) e1 e4 v2 v Vertex set V = f1, ..., ng v 12 11 v3 Edge set E ⊆ P (V ) n ; v10 v1 v13 Node weights c : V ! R≥1 v8 v7 e v 5 v e 9 4 Edge weights ! : E ! R≥1 3 v6 v5 e2 1 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraphs Generalization of graphs ) hyperedges connect ≥ 2 nodes Graphs ) dyadic (2-ary) relationships Hypergraphs ) (d-ary) relationships pin net Hypergraph H = (V , E, c, !) e1 e4 v2 v Vertex set V = f1, ..., ng v 12 11 v3 Edge set E ⊆ P (V ) n ; v10 v1 v13 Node weights c : V ! R≥1 v8 v7 e v 5 v e 9 4 Edge weights ! : E ! R≥1 3 v P P v6 5 e2 jPj = e2E jej = v2V d(v) 1 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraph Partitioning Problem Partition hypergraph H = (V , E, c, !) into k disjoint blocks Π = fV1, ::: , Vk g such that: blocks Vi are roughly equal-sized: c(V ) c(Vi ) ≤ (1 + ") k connectivity objective is minimized: V1 V2 V 4 V3 2 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraph Partitioning Problem Partition hypergraph H = (V , E, c, !) into k disjoint blocks Π = fV1, ::: , Vk g such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ") k connectivity objective is minimized: V1 V2 V 4 V3 2 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraph Partitioning Problem Partition hypergraph H = (V , E, c, !) into k disjoint blocks Π = fV1, ::: , Vk g such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ") k connectivity objective is minimized: V1 V2 V 4 V3 2 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraph Partitioning Problem Partition hypergraph H = (V , E, c, !) into k disjoint blocks Π = fV1, ::: , Vk g such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ") k connectivity objective is minimized: P V1 V2 e2cut(λ−1) !(e)=6 connectivity: # blocks connected by net e V 4 V3 2 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Hypergraph Partitioning Problem Partition hypergraph H = (V , E, c, !) into k disjoint blocks Π = fV1, ::: , Vk g such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ") k connectivity objective is minimized: P V1 V2 e2cut(λ−1) !(e)=6 connectivity: # blocks connected by net e V 4 V3 2 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Applications VLSI Design Scientific Computing 0 1 2 3 4 5 6 7 0 ××× 1 × × 2 ×××× Application 3 ××× 4 ××× Domain 5 ×× 6 ×××××××× 7 ×× Hypergraph Model facilitate Goal minimize floorplanning & placement communication 3 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group The Multilevel Framework input hypergraph output partition ··· ··· match / cluster local search contract uncontract ··· ··· initial partitioning 4 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group This Talk: Coarsening Phase input hypergraph output partition ··· ··· match / cluster local search contract uncontract ··· ··· initial partitioning 5 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Common Strategy: avoid global decisions local, greedy algorithms Objective: identify highly connected vertices using... 1 foreach vertex v do 2 cluster[v] := argmax rating(v,u) neighbor u 6 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Common Strategy: avoid global decisions local, greedy algorithms Objective: identify highly connected vertices using... 1 foreach vertex v do 2 cluster[v] := argmax rating(v,u) neighbor u Main Design Goals:[Karypis,Kumar 99] 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning 3: maintain structural similarity good coarse solutions 6 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo )hypergraph-tailored rating functions: P !(e) r(u, v) := je|−1 net e containing u,v 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo )hypergraph-tailored rating functions: P !(e) r(u, v) := je|−1 net e large number ... containing u,v 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo )hypergraph-tailored rating functions: P !(e) of heavy nets ... r(u, v) := je|−1 net e large number ... containing u,v 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo )hypergraph-tailored rating functions: P !(e) of heavy nets ... r(u, v) := je|−1 net e ... with small size large number ... containing u,v 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search X 2: reduce number of nets easier initial partitioning X foo )hypergraph-tailored rating functions: P !(e) of heavy nets ... r(u, v) := je|−1 net e ... with small size large number ... containing u,v 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search X 2: reduce number of nets easier initial partitioning X foo )hypergraph-tailored rating functions: P !(e) of heavy nets ... r(u, v) := je|−1 net e ... with small size foolarge number ... containing u,v 3:foomaintain structural similarity good coarse solutions )prefer clustering over matching )ensure ∼balanced vertex weights 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search X 2: reduce number of nets easier initial partitioning X foo )hypergraph-tailored rating functions: P !(e) of heavy nets ... r(u, v) := je|−1 net e ... with small size foolarge number ... containing u,v 3:foomaintain structural similarity good coarse solutions )prefer clustering over matching ensure ∼balanced vertex weights ) enough? 7 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group What could possibly go wrong? ... a lot: n1 u v fu, vg input obscured clusters 8 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group What could possibly go wrong? ... a lot: n1 u v fu, vg input obscured clusters maximal matching random tie-breaking ISSUES prefer unclustered heavy neighbors 8 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group What could possibly go wrong? ... a lot: n1 u v fu, vg input obscured clusters maximal matching random tie-breaking ISSUES prefer unclusteredfoo heavy neighbors )Problem: relying only on local information! 8 Sebastian Schlag – Community-aware Coarsening Schemes for Hypergraph Partitioning Institute of Theoretical Informatics Algorithmics Group Our Approach: Community-aware Coarsening n1 u v fu, vg input obscured clusters 9 Sebastian