n-level Graph Partitioning

Transcription

1 Vitaly Osipov, Peter Sanders - Algorithmik II 1 Vitaly Osipov: KIT Universität des Landes Baden-Württemberg und nationales Grossforschungszentrum in der Helmholtz-Gemeinschaft Institut für Theoretische Informatik

2 Overview Introduction ngp KasPar Contraction Local Search Experimental Evaluation Future Work 2 Vitaly Osipov:

3 Introduction Graph Partitioning G = (V, E, c, ω) ω : E R >0 c : V R 0, n = V, m = E. V 1 V k = V, s.t. V i V j = for i = j c(v i ) L max = (1 + ɛ)c(v )/k + max v V c(v) minimize cut i<j w(e ij ) E ij = {{u, v} E u V i, v V j } 3 Vitaly Osipov:

4 Introduction Multilevel Approaches Edge Contraction {u, v} x ω({x, z}) = ω({u, z}) + ω({v, z}). c(x) = c(u) + c(v) u v z z x contraction phase match contract input graph local improvement initial partitioning uncontract output partition refinement phase 4 Vitaly Osipov:

5 ngp n-gp(g,k,ɛ) begin if G is small then return initialpartition(g,k,ɛ) end pick the edge e = {u, v} with the highest rating contract(e) P :=n-gp(g,k,ɛ) activate(u) activate(v) localsearch() return P end 5 Vitaly Osipov:

6 Karlsrue Sequential Partitioner Contraction contract a single edge between two levels possibly n levels finegrained contraction consequtive levels are very similar no matching algorithm required use of different edge ratings uniform distribution of node weights priority queue defines the order of edges to be contracted Local Search efficent stopping creteria avoid quadratic runtime General Trial Trees improve quality by independent trials 6 Vitaly Osipov:

7 Contraction edge rating - expansion ({u, v}) = ω({u,v}) c(u)c(v) addressable priority queue based on pairing heaps dynamic graph data structure memory overhead Vitaly Osipov: m

8 Local Search Nodes unmarked, active, marked active nodes compute gains of moving from one block to another choose the maximum gain when moved active marked can t become active anymore unmarked neighbours of marked active gains in each step identically distributed, independent random variables expectation µ variance σ 2 compute m and σ 2 from previous steps stop after p steps if pµ 2 > ασ 2 + β 8 Vitaly Osipov: u v v u

9 Local Search Does it help? local search steps / n k=64 k=32 k=16 k=8 k=4 k= n without stoping criteria less than 1% improvement orders of magnitute slower 9 Vitaly Osipov:

10 Experimental evaluation Platform and Settings System one core of Intel Xeon Quad-core Processor featuring 2x4 MB of L2 cache and clocked at GHz of a 2 processor Intel Xeon X5355 node 16 GB of RAM gcc % imbalance 10 repetitions for the small networks 5 repetitions for the other averaging over multiple instances geometric mean 10 Vitaly Osipov:

11 Experimental evaluation two sets - small and large real world graphs Table: Geometric means (times, cut values) over all instances. code small graphs large graphs social networks best avg. t[s] best avg. t[s] best avg. t[s] KaSPar strong KaSPar fast KaSPar fast, α = KaPPa strong KaPPa fast kmetis Scotch On average 5.9% vs KaPPa strong (small instances) 8.1% vs KaPPa strong (large instances) 32% vs Metis (large instances) Repeating scotch as long as KasPar strong run and choosing the best result 12.1% larger cuts 11 Vitaly Osipov:

12 Experimental evaluation Walshaw s Benchmark 34 graphs for k {2, 4, 8, 16, 32, 64} and balance parameter ɛ {0, 0.01, 0.03, 0.05} 816 entries KasPar in one hour per entrie improved 155 values 42 for 1% 49 for 3% 64 for 5% if counting only large graphs (>44K nodes) and ɛ > 0 63% of the entries reproduces equaly good cuts in 83 other entries 12 Vitaly Osipov:

13 Future work better initial partitioner for large k exploit shared memory parallelism 13 Vitaly Osipov:

14 Thank you! 14 Vitaly Osipov: