Efficient algorithms for solving systems of linear equations and path problems

Abstract

Efficient algorithms are presented for solving systems of linear equations defined on and for solving path problems[11] for treewidth k graphs [20] and for α-near-planar graphs [22]. These algorithms include the following: 1. O(nk2) and O(n3/2) time algorithms for solving a system of linear equations and for solving the single source shortest path problem, 2. O(n2k) and O(n2log n) time algorithms for computing A−1 where A is an n×n matrix over a field or for computing A* where A is an n×n matrix over a closed semiring, and 3. O(n2k) and O(n2log n) time algorithms for the all pairs shortest path problems. One corollary of these results is that the single source and all pairs shortest path problems are solvable in O(n) and O(n2) steps, respectively, for any of the decomposable graph classes in[5].