A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering

doi:10.1006/jpdc.1997.1403

Journal of Parallel and Distributed Computing

Volume 48, Issue 1, 10 January 1998, Pages 71-95

https://doi.org/10.1006/jpdc.1997.1403 Get rights and content

Abstract

In this paper we present a parallel formulation of the multilevel graph partitioning and sparse matrix ordering algorithm. A key feature of our parallel formulation (that distinguishes it from other proposed parallel formulations of multilevel algorithms) is that it partitions the vertices of the graph into $p$ parts while distributing the overall adjacency matrix of the graph among allpprocessors. This mapping results in substantially smaller communication than one-dimensional distribution for graphs with relatively high degree, especially if the graph is randomly distributed among the processors. We also present a parallel algorithm for computing a minimal cover of a bipartite graph which is a key operation for obtaining a small vertex separator that is useful for computing the fill reducing ordering of sparse matrices. Our parallel algorithm achieves a speedup of up to 56 on 128 processors for moderate size problems, further reducing the already moderate serial run time of multilevel schemes. Furthermore, the quality of the produced partitions and orderings are comparable to those produced by the serial multilevel algorithm that has been shown to outperform both spectral partitioning and multiple minimum degree.

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^☆: This work was supported by NSF CCR-9423082, by the Army Research Office Contract DA/DAAH04-95-1-0538, by the IBM Partnership Award, and by Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory Cooperative Agreement Number DAAH04-95-2-0003/Contract DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. Access to computing facilities was provided by AHPCRC. Minnesota Supercomputer Institute, Cray Research Inc., and by the Pittsburgh Supercomputing Center. Related papers are available via WWW at URL:http://www.cs.umn.edu/∼karypis.

²: E-mail: [email protected].

³: E-mail: [email protected].

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Regular ArticleA Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering☆

Abstract

J. Parallel Distrib. Comput.

J. Parallel Distrib. Compt.

A parallel implementation of multilevel recursive spectral bisection for application to adaptive unstructured meshes

Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing

A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems

Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing

A heuristic for reducing fill in sparse matrix factorization

6th SIAM Conf. Parallel Processing for Scientific Computing

Parallel algorithms for dynamically partitioning unstructured grids

Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing

A linear time heuristic for improving network partitions

Proc. 19th IEEE Design Automation Conference

Finding clusters in VLSI circuits

Proceedings of IEEE International Conference on Computer Aided Design

Computer Solution of Large Sparse Positive Definite Systems