Path algebras: a multiset-theoretic approach

Abstract

This thesis develops an algebraic theory for path problems such as that of finding the shortest or more generally, the k shortest paths in a network, enumerating elementary 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 the development of algebraic methods for solving them. r A number of solution methods are derived, which are analogous tothe classical techniques of solving linear algebraic equations, and the applicability of these methods to different classes of path0problems is examined in detail.The thesis also presents in particular an algebra which is appropriate for the formulation and solution of k-shortest-paths problems. This algebra is a generalization of Giffler's Schedule Algebra for computing all the numerical labels of paths in a network. It is shown formally that these labels can be calculated by using direct methods of linear algebra and an algorithm similar to the long-division procedure of ordinary arithmetic. Such a method is then modified to yield an algorithm for finding k shortest elementary paths in a network.

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