Numerische Mathematik

, Volume 50, Issue 4, pp 377–404

The analysis of a nested dissection algorithm

  • John R. Gilbert
  • Robert Endre Tarjan
Convergence of the SSOR Method for Nonlinear Systems of Simultaneous Equations

Summary

Nested dissection is an algorithm invented by Alan George for preserving sparsity in Gaussian elimination on symmetric positive definite matrices. Nested dissection can be viewed as a recursive divide-and-conquer algorithm on an undirected graph; it usesseparators in the graph, which are small sets of vertices whose removal divides the graph approximately in half. George and Liu gave an implementation of nested dissection that used a heuristic to find separators. Lipton and Tarjan gave an algorithm to findn1/2-separators in planar graphs and two-dimensional finite element graphs, and Lipton, Rose, and Tarjan used these separators in a modified version of nested dissection, guaranteeing bounds ofO (n logn) on fill andO(n3/2) on operation count. We analyze the combination of the original George-Liu nested dissection algorithm and the Lipton-Tarjan planar separator algorithm. This combination is interesting because it is easier to implement than the Lipton-Rose-Tarjan version, especially in the framework of existïng sparse matrix software. Using some topological graph theory, we proveO(n logn) fill andO(n3/2) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree withn1/2-separators. For planar and finite element graphs, the leading constant factor is smaller than that in the Lipton-Rose-Tarjan analysis. We also construct a class of graphs withn1/2-separators for which our algorithm does not achieve anO(n logn) bound on fill.

Subject Classifications

AMS(MOS) 05C10 65F05 65F50 CR G.1.3 G.2.2 

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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • John R. Gilbert
    • 1
  • Robert Endre Tarjan
    • 2
  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA
  2. 2.Computer Science DepartmentPrinceton UniversityPrincetonUSA

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