Contiguous Graph Partitioning For Optimal Total Or Bottleneck Communication [article]

Peter Ahrens
2021 arXiv   pre-print
Graph partitioning schedules parallel calculations like sparse matrix-vector multiply (SpMV). We consider contiguous partitions, where the m rows (or columns) of a sparse matrix with N nonzeros are split into K parts without reordering. We propose the first near-linear time algorithms for several graph partitioning problems in the contiguous regime. Traditional objectives such as the simple edge cut, hyperedge cut, or hypergraph connectivity minimize the total cost of all parts under a balance
more » ... onstraint. Our total partitioners use O(Km + N) space. They run in O((Kmlog(m) + N)log(N)) time, a significant improvement over prior O(K(m^2 + N)) time algorithms due to Kernighan and Grandjean et. al. Bottleneck partitioning minimizes the maximum cost of any part. We propose a new bottleneck cost which reflects the sum of communication and computation on each part. Our bottleneck partitioners use linear space. The exact algorithm runs in linear time when K^2 is O(N^C) for C < 1. Our (1 + ϵ)-approximate algorithm runs in linear time when Klog(c_high/(c_lowϵ)) is O(N^C) for C < 1, where c_high and c_low are upper and lower bounds on the optimal cost. We also propose a simpler (1 + ϵ)-approximate algorithm which runs in a factor of log(c_high/(c_lowϵ)) from linear time. We empirically demonstrate that our algorithms efficiently produce high-quality contiguous partitions on a test suite of 42 test matrices. When K = 8, our hypergraph connectivity partitioner achieved a speedup of 53× (mean 15.1×) over prior algorithms. The mean runtime of our bottleneck partitioner was 5.15 SpMVs.
arXiv:2007.16192v4 fatcat:lrjtllq7e5emfclfu3wfcejhne