Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure

SEA’17 · June 23, 2017 Tobias Heuer and Sebastian Schlag

INSTITUTEOF 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, ω) e1 e4 v2 v Vertex set V = {1, ..., n} v 12 11 v3

Edge set E ⊆ P (V ) \ ∅ v10 v1 v13 Node weights c : V → R≥1 v8 v7 e v 5 v e 9 4 Edge weights ω : 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, ω) e1 e4 v2 v Vertex set V = {1, ..., n} v 12 11 v3

Edge set E ⊆ P (V ) \ ∅ v10 v1 v13 Node weights c : V → R≥1 v8 v7 e v 5 v e 9 4 Edge weights ω : E → R≥1 3

v P P v6 5 e2 |P| = e∈E |e| = v∈V 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, ω) into k disjoint blocks

Π = {V1, ... , Vk } such that:

blocks Vi are roughly equal-sized: c(V ) c(Vi ) ≤ (1 + ε) 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, ω) into k disjoint blocks

Π = {V1, ... , Vk } such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ε) 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, ω) into k disjoint blocks

Π = {V1, ... , Vk } such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ε) 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, ω) into k disjoint blocks

Π = {V1, ... , Vk } such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ε) k connectivity objective is minimized: P V1 V2 e∈cut(λ−1) ω(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, ω) into k disjoint blocks

Π = {V1, ... , Vk } such that: imbalance blocks Vi are roughly equal-sized: parameter c(V ) c(Vi ) ≤ (1 + ε) k connectivity objective is minimized: P V1 V2 e∈cut(λ−1) ω(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 ω(e) r(u, v) := |e|−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 ω(e) of heavy nets ... r(u, v) := |e|−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 X 2: reduce number of nets easier initial partitioning X foo ⇒hypergraph-tailored rating functions: P ω(e) of heavy nets ... r(u, v) := |e|−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 ω(e) of heavy nets ... r(u, v) := |e|−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: