Using Cellular Graph Embeddings in Solving All Pairs Shortest Paths Problems

An algorithm is presented for generating a succinct encoding of all pairs shortest path information in an n-vertex directed graph G with O(n) edges. The edges have real-valued costs, but the graph contains no negative cycles. The algorithm runs in O('Y(G)n + ('Y(G) )'log 'Y(G)) time, where 'Y(G) is a topological emhedding measure that we define. The algorithm uses a decomposition of the graph into O(-y(G)) outerplanar subgraphs satisfying certain separator properties, and a linear-time algorithm is presented to find this decomposition.