# Unstructured path problems and the making of semirings

## Abstract

The solution of the algebraic path problem, for instance with the algorithm by Floyd and Warshall, is part of the classical repertoire on algorithms. This solution presupposes that path costs are computed in a closed semiring or a similar algebraic structure. The associativity and distributivity laws in such algebraic structures exclude many possible path costs. In the seventies, several authors have developed algebraic methods in order to overcome such restrictions.

Recently, applications have created the need for handling such unstructured path costs. Motivated by such applications, we investigate the efficiency of algorithms solving path problems with unstructured costs. The resulting procedure allows for developing efficient algorithms for solving the algebraic path problem with respect to a wide variety of cost measures, including finding shortest paths with discounting, and counting paths or computing the expected length of paths with desired properties.

## Keywords

Short Path Cost Structure Path Cost Neutral Element Edge Label## Preview

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