Constant time distance queries in planar unweighted graphs with subquadratic preprocessing time

Let G be an n-vertex planar, undirected, and unweighted graph. It was stated as open problems whether the Wiener index, defined as the sum of all-pairs shortest path distances, and the diameter of G can be computed in o(n(2)) time. We show that both problems can be solved in O(n(2) log log n/log n) time with O(n) space. The techniques that we apply allow us to build, within the same time bound, an oracle for exact distance queries in G. More generally, for any parameter S is an element of [(log n/log log n)(2), n(2/5)], distance queries can be answered in O (root S log S/log n) time per query with O(n(2)/root S) preprocessing time and space requirement. With respect to running time, this is better than previous algorithms when log S = o(log n). All algorithms have linear space requirement. Our results generalize to a larger class of graphs including those with a fixed excluded minor. (C) 2012 Elsevier B.V. All rights reserved.