Symbolic Solutions of Shortest-Path Problems and Their Applications

Abstract

The paper proposes a symbolic technique for shortest-path problems. This technique is based on a presentation of a shortest-path algorithm as a symbolic expression. Literals of this expression are arc tags of a graph, and they are substituted for corresponding arc weights which appear in the algorithm. The search for the most efficient algorithm is reduced to the construction of the shortest expression. The advantage of this method, compared with classical numeric algorithms, is its stability and faster reaction to data renewal. These problems are solved with reference to two kinds of n-node digraphs: Fibonacci graphs and complete source-target directed acyclic graphs. $O(n^{2})$ and $O\left( 2^{\left\lceil \log _{2}n\right\rceil ^{2}-\left\lceil \log _{2}n\right\rceil }\right)$ complexity algorithms, respectively, are provided in these cases.