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 P(s, 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. A method is described for computing path expressions by dwidmg G mto components, computing path expressions on the components by Gaussian elimination, and combining the solutions This method requires O(ma(m, n)) time on a reducible flow graph, where n Is the number of vertices m G, m is the number of edges in G, and a is a functional inverse of Ackermann's function The method makes use of an algonthm for evaluating functions defined on paths in trees. A smapllfied version of the algorithm, which runs in O(m log n) time on reducible flow graphs, is quite easy to implement and efficient m practice.