Semirings and path spaces

AbstractThis paper develops a unified algebraic theory for a class of path problems such as that of finding the shortest or, more generally, the k shortest paths in a network; the enumeration of elemementary or simple paths in a graph. It differs from most earlier work in that the algebraic structure appended to a graph or a network of a path problem is not axiomatically given as a starting point of the theory, but is derived from a novel concept called a “path space”. This concept is shown to provide a coherent framework for the analysis of path problems, and hence the development of algebraic methods for solving them