Using cellular graph embeddings in solving all pairs shortest paths problems

An algorithm for generating a succinct encoding of all-pairs shortest path information in an n-vertex directed planar G with O(n) edges is presented. The edges have real-valued costs, but the graph contains no negative cycles. The time complexity is given in terms of a topological embedding measure defined in the paper. The algorithm uses a decomposition of the graph into outerplanar subgraphs satisfying certain separator properties, and a linear-time algorithm is presented to find this decomposition