# The analysis of a nested dissection algorithm

## Authors

- Received:

DOI: 10.1007/BF01396660

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

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## 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 uses*separators* 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 find*n*^{1/2}-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 of*O* (*n* log*n*) on fill and*O*(*n*^{3/2}) 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 prove*O*(*n* log*n*) fill and*O*(*n*^{3/2}) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree with*n*^{1/2}-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 with*n*^{1/2}-separators for which our algorithm does not achieve an*O*(*n* log*n*) bound on fill.