Path Problems in Graphs and Matrix Multiplication

Abstract

In this chapter we concentrate on path problems in graphs. Typical examples are the problems of computing shortest or longest paths or computing the k shortest path between all pairs of points in a graph. The best known algorithms for these problems differ only slightly. In fact, they are all special cases of an algorithm for solving general path problems on graphs. General path problems over closed semi-rings and Kleene’s algorithm for solving them are dealt with in section 1, special cases are then treated in section 2. The algebraic point of view allows us to formulate the connection between general path problems and matrix multiplication in an elegant way: Matrix multiplication in a semi-ring and solution of a general path problem have the same order of complexity. In section 4 we consider fast algorithms for multiplication of matrices over a ring. This is then applied to boolean matrices. Section 7 contains a lower bound on the complexity of boolean matrix product.