A Symbolic Approach to the All-Pairs Shortest-Paths Problem

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, a symbolic algorithm for the all-pairs shortest-paths (APSP) problem in loopless directed graphs with strictly positive integral edge weights is presented. It requires Θ ( log(NB) ) OBDD-operations to obtain the lengths and edges of all shortest paths in graphs with N nodes and maximum edge weight B. It is proved that runtime and space usage are polylogarithmic w. r. t. N and B on graph sequences with characteristic bounded-width functions. This convenient property is closed under certain graph composition operations. Moreover, an alternative symbolic approach for general integral edge weights is sketched which does not behave efficiently on general graph sequences with bounded-width functions. Finally, two variants of the APSP problem are briefly discussed.