Efficient algorithms for shortest path queries in planar digraphs

This paper describes algorithms for answering shortest path queries in digraphs with small separators and, in particular, in planar digraphs. In this version of the problem, one has to preprocess the input graph so that, given an arbitrary pair of query vertices v and w, the shortest-path distance between v and w can be computed in a short time. The goal is to achieve balance between the preprocessing time and space and the time for answering a distance query. Previously, efficient algorithms for that problem were known only for the class of outerplanar digraphs and for the class of digraphs of constant treewidth. We describe efficient algorithms for this problem for any class of digraphs for which an O(root n) separator theorem holds. For such graphs our algorithm uses O(S) space and answers queries in O(n(2)/S) time, for any previously chosen S is an element of [n, n(2)]. For the class of planar digraphs improved-algorithms are described.