# Extension of the parallel nested dissection algorithm to path algebra problems

## Abstract

This paper extends the author's parallel nested dissection algorithm of [PR] originally devised for solving sparse linear systems. We present a class of new applications of the nested dissection method, this time to path algebra computations, (in both cases of single source and all pair paths), where the path algebra problem is defined by a symmetric matrix A whose associated graph G with n vertices is planar. We substantially improve the known algorithms for path algebra problems of that general class: {fx470-1}

(In case of parallel algorithms we assume that G is given with its \({\text{O(}}\sqrt {\text{n}} )\)-separator family.) Further applications lead, in particular, to computing a maxflow and a mincut in an undirected planar network using O(log^{4}n) parallel steps, n^{1.5}/log n processors or alternatively O(log^{3}n) steps, n^{2}/log n processors, versus the known bounds, O(log^{2}n) and n^{4}, of [JV].

## Keywords

Short Path Planar Graph Parallel Algorithm Short Path Problem Path Algebra## Preview

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## References

- [B]
- [BGH]A. Borodin, J. von zur Gathen, and J. Hopcroft, “Fast Parallel Matrix and GCD Computations”,
*Proc. 23-rd Ann. ACM FOCS*, 65–71 (1982) and*Information and Control*53,3, 241–256 (1982).Google Scholar - [F]
- [Fr]G.N. Fredericson 1984, Fast Algorithms for Shortest Paths in Planar Graphs, with Applications, CSD TR 486,
*Dept. of Computer Sci., Purdue Univ.*, W. Lafayette, IN.Google Scholar - [GM]M. Gondran and M. Minoux 1984,
*Graphs and Algorithms*, Wiley-Interscience, New York.Google Scholar - [H]
- [HJ]R. Hassin and D.B. Johnson, An O(n log
^{2}n) Algorithm for Maximum Flow in Undirected Planar Networks,*SIAM J. on Computing*, to appear.Google Scholar - [JV]D.B. Johnson and S.W. Venkatesan 1982, Parallel Algorithms for Minimum Cuts and Maximum Flows in Planar Networks,
*Proc. 23-rd Ann. IEEE Symp. FOCS*, 244–254.Google Scholar - [KR]P. Klein and J. Reif 1985, An Efficiet Parallel Algorithm for Planarity, Tech. Report,
*Center for Research in Computer Technology, Aiken Computation Laboratory, Harvard Univ.*Cambridge, Mass.Google Scholar - [L]E.L. Lawler 1976,
*Combinatorial Optimization: Networks and Matroids*, Holt, Rinehard and Winston, N.Y.Google Scholar - [LT]R.J. Lipton and R.E. Tarjan 1979, A Separator Theorem for Planar Graphs,
*SIAM J. Applied Math.***36**, 2, 177–189.Google Scholar - [MNS]K. Matsumoto, T. Nishizeki, N. Saito 1985, An Efficient Algorithm for Finding Multicommodity Flows in Planar Networks,
*SIAM J. on Computing***14**, 2, 289–302.Google Scholar - [P]V. Pan 1984,
*How to Multiply Matrices Faster*, Lecture Notes in Computer Science, v. 179, Springer Verlag, Berlin.Google Scholar - [PR]V. Pan and J. Reif 1984, Fast and Efficient Solution of Linear Systems, Tech. Report TR-02-85,
*Center for Research in Computer Technology, Aiken Computation Laboratory, Harvard Univ.*, Cambridge, Mass., (extended abstract in*Proc. 17-th Ann. ACM STOC*, 143–152, Providence, R.I.).Google Scholar - [PR1]V. Pan and J. Reif 1985, Fast and Efficient Algorithms for Linear Programming and for the Linear Least Squares Problem, Techn. Report TR-11-85,
*Center for Research in Computer Technology, Aiken Computation Laboratory, Harvard Univ.*, Cambridge, Mass.Google Scholar - [T]
- [Ta]