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

DOI: 10.1007/BF01396660

Gilbert, J.R. & Tarjan, R.E. Numer. Math. (1986) 50: 377. doi:10.1007/BF01396660

## 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)05C1065F0565F50CRG.1.3G.2.2