The solution of the Mgebralc path problem, for instance with the algorithm by Floyd and WarshM1, is part of the classical repertoire on Mgorithms. This solution presupposes that path costs are computed in a dosed 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. Motiwted 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.
I n t r o d u c t i o nThe algebraic path problem is defined as follows. We are given a directed graph G = ( { 1 , . . . , n}, E) with an edge labeling A : E --~ R into a set R. Edge values can be composed with a binary operation Q, in order to yield costs of paths of edges.
~(;) = ("'(~(v0,,,)O~(v,,v2))OX(,~,v3))-")e~(,k-l,vk)The operation (~ has a neutral element (~ 0 = 0. The operation ® has a neutral element 1 which is also the cost of the empty patl~LS Po,i from each vertex i, 1 < i < n, to itself. We call C = (R, ~, ®, 0,1) an algebraic cost structure. There are basically two approaches to solving algebraic path problems in the literature. The first approach--pursued by several authors in different application areas, see e.g. [5, 6, 9, 10, 13, 15, 16, 19J--requires the algebraic cost structure C = (R, (D, ®, 0,1) to fulfill the following additional requirements.A I : Q is associative.A 2 : 0 is absorptive over (D, Le., for all r E R, we have r ® 0 = 0 Q r = 0.
