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

## References

1. 1.
Djidjev, H.N.: On the problem of partitioning planar graphs. SIAM J. Algebraic Discrete Methods3, 229–240 (1982)Google Scholar
2. 2.
George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal.10, 345–363 (1973)Google Scholar
3. 3.
George, A., Liu, J.W.-H.: An automatic neste dissection algorithm for irregular finite element problems. SIAM J. Numer. Anal.15, 1053–1069 (1978)Google Scholar
4. 4.
George, A., Liu, J.W.-H.: Computer Solution of Large Sparse Positive Definite Systems. Englewood Cliffs, NJ: Prentice-Hall 1981Google Scholar
5. 5.
Gilbert, J.R.: Graph Separator Theorems and Sparse Gaussian Elimination. Ph.D. thesis, Stanford University 1980Google Scholar
6. 6.
Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms5, 391–407 (1984)Google Scholar
7. 7.
Gilbert, J.R., Rose, D.J., Edenbrandt, A.: A separator theorem for chordal graphs. SIAM J. Algebraic Discrete Methods5, 306–313 (1984)Google Scholar
8. 8.
Gilbert, J.R., Schreiber, R.: Nested dissection with partial pivoting. Sparse Matrix Symposium. Fairfield Glade, Tennessee 1982Google Scholar
9. 9.
10. 10.
Hoey, D., Leiserson, C.E.: A layout for the shuffle-exchange network. Carnegie-Mellon University Computer Science Department technical report CMU-CS-80-139 (1980)Google Scholar
11. 11.
Jordan, C.: Sur les assemblages de lignes. J. Reine Angew. Math.70, 185–190 (1869)Google Scholar
12. 12.
Leighton, F.T.: A layout strategy for VLSI which is provably good. Proc. 14th Ann. ACM Symp. Theory Comput. pp. 85–98 (1982)Google Scholar
13. 13.
Leiserson, C.E.: Area-efficient graph layouts (for VLSI). Proc. 21st Ann. Symp. Found. Comput. Sci. pp. 270–281 (1980)Google Scholar
14. 14.
Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM J. Numer. Anal.16, 346–358 (1979)Google Scholar
15. 15.
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math.36, 177–189 (1979)Google Scholar
16. 16.
Miller, G.L.: Finding small simple cycle separators for 2-connected planar graphs. Proc. 16th Ann. ACM Symp. Theory Comput. pp. 376–382 (1984)Google Scholar
17. 17.
Nash-Williams, C.St.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc.39, 12 (1964)Google Scholar
18. 18.
Parter, S.: The use of linear graphs in Gauss elimination. SIAM Rev.3, 119–130 (1961)Google Scholar
19. 19.
Roman, J.: Calculs de complexité relatifs à une méthode de dissection emboîtée. Numer. Math.47, 175–190 (1985)Google Scholar
20. 20.
Rose, D.J.: A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Graph Theory and Computing (R.C. Read, ed.), pp. 183–217. New York: Academic Press 1972Google Scholar
21. 21.
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput.5, 266–283 (1976)Google Scholar
22. 22.
Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebraic Discrete Methods2, 77–79 (1981)Google Scholar