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

Abstract

We give a (1+ϵ)-approximate distance oracle with O(1) query time for an undirected planar graph G with n vertices and non-negative edge lengths. For ϵ>0 and any two vertices u and v in G, our oracle gives a distance d˜(u,v) with stretch (1+ϵ) in O(1) time. The oracle has size O(nlog⁡n((log⁡n)/ϵ+f(ϵ))) and pre-processing time O(nlog⁡n((log3⁡n)/ϵ2+f(ϵ))), where f(ϵ)=2O(1/ϵ). This is the first (1+ϵ)-approximate distance oracle with O(1) query time independent of ϵ and the size and pre-processing time nearly linear in n, and improves the query time O(1/ϵ) of previous (1+ϵ)-approximate distance oracle with size nearly linear in n.