Experimental Studies of Symbolic Shortest-Path Algorithms

Abstract

Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function. To solve problems in such implicitly given graphs, specialized symbolic algorithms are needed which are restricted to the use of functional operations offered by the OBDD data structure. In this paper, two symbolic algorithms for the single-source shortest-path problem with nonnegative positive integral edge weights are presented which represent symbolic versions of Dijkstra’s algorithm and the Bellman-Ford algorithm. They execute \(\mathcal{O}(N\cdot{\rm log}(NB))\) resp. \(\mathcal{O}(NM\cdot{\rm log}(NB))\) OBDD-operations to obtain the shortest paths in a graph with N nodes, M edges, and maximum edge weight B. Despite the larger worst-case bound, the symbolic Bellman-Ford-approach is expected to behave much better on structured graphs because it is able to handle updates of node distances effectively in parallel. Hence, both algorithms have been studied in experiments on random, grid, and threshold graphs with different weight functions. These studies support the assumption that the Dijkstra-approach behaves efficient w. r. t. space usage, while the Bellman-Ford-approach is dominant w. r. t. runtime.