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(log⁴n) parallel steps, n^1.5/log n processors or alternatively O(log³n) steps, n²/log n processors, versus the known bounds, O(log²n) and n⁴, of [JV].

Supported by NSF Grants MCS 8203232 and DCR-8507573.

This work was supported by Office of Naval Research Contract N00014-80-C-0647 and by NSF Grant DCR-850351.