Fast Algorithms for Solving Path Problems.

Let G V,E be a directed graph with a distinguished source vertex s. The single-source path expression problem is to find, for each vertex v, a regular expression Ps,v which represents the set of all paths in G from s to v. A solution to this problem can be used to solve shortest path problems, solve sparse systems of linear equations, and carry out global flow analysis. We describe a method to compute path expressions by dividing G into components, computing path expressions on the components by Gaussian elimination, and combining the solutions. This method requires 0 m alpha om,n time on a reducible flow graph, where n is the vertices in G, m is the number of edges in G, and alpha is a functional inverse of Ackermanns function. The method makes use of an algorithm for evaluating functions defined on paths in trees. A simplified version of the algorithm, which runs in 0m log n time on reducible flow graphs, is quite easy to implement and efficient in practice. Author