Constant query time (1+ε)-approximate distance oracle for planar graphs

We give a (1 + epsilon)-approximate distance oracle with O(1) query time for an undirected planar graph G with n vertices and non-negative edge lengths. For epsilon > 0 and any two vertices u and v in G, our oracle gives a distance (d) over bar (u, v) with stretch (1 + epsilon) in O(1) time. The oracle has size O(n log n((log n)/epsilon + f (epsilon))) and pre-processing time O(n log n((log(3) n)/epsilon(2) + f(epsilon))), where f (epsilon) = 2(O(1/epsilon)). This is the first (1 + epsilon)-approximate distance oracle with O(1) query time independent of epsilon and the size and pre-processing time nearly linear in n, and improves the query time O(1/epsilon) of previous (1 + epsilon)-approximate distance oracle with size nearly linear in n. (C) 2018 Elsevier B.V. All rights reserved.