A new approach to all pairs shortest paths in planar graphs

An algorithm is presented for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with real-valued edge costs but no negative cycles. The algorithm runs in &Ogr;(pn) time, where n is the number of vertices in G, and p is the minimum cardinality of a subset of the faces that cover all vertices, taken over all planar embeddings of G. Linear-time algorithms are presented for various subproblems including that of finding an appropriate embedding of G and a corresponding face-on-vertex covering of cardinality &Ogr;(p), and of generating all pairs shortest path information in a directed outerplanar graph.