Computing all-pairs shortest paths by leveraging low treewidth

Abstract

We present two new and ecient algorithms for computing all-pairs shortest paths. The algorithms operate on directed graphs with real (possibly negative) weights. They make use of directed path consistency along a vertex ordering d. Both algorithms run in O (n2) n 2wd time, where wd is the graph width induced by this vertex ordering. For graphs of constant treewidth, this yields O (n2) time, which is optimal. On chordal graphs, the algorithms run in O(nm) time. In addition, we present a variant that exploits graph separators to arrive at a run time of O (n2) n 2wd d + n2sd on general graphs, where sd wd is the size of the largest minimal separator induced by the vertex ordering d. We show empirically that on both constructed and realistic benchmarks, in many cases the algorithms outperform Floyd{Warshall's as well as Johnson's algorithm, which represent the current state of the art with a run time of O (n3) and O (nm + n2 log n), respectively. Our algorithms can be used for spatial and temporal reasoning, such as for the Simple Temporal Problem, which underlines their relevance to the planning and scheduling community. © 2012 AI Access Foundation.