Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Given an undirected n-vertex planar graph G = (V, E, ω) with non-negative edge weight function ω : E → R and given an assigned label to each vertex, a vertex-labeled distance oracle is a data structure which for any query consisting of a vertex u and a label λ reports the shortest path distance from u to the nearest vertex with label λ. We show that if there is a distance oracle for undirected n-vertex planar graphs with non-negative edge weights using s(n) space and with query time q(n), then there is a vertex-labeled distance oracle with Õ(s(n))¹ space and Õ(q(n)) query time. Using the state-of-the-art distance oracle of Long and Pettie [12], our construction produces a vertex-labeled distance oracle using n¹⁺o⁽¹⁾ space and query time Õ(1) at one extreme, Õ(n) space and n^o(1) query time at the other extreme, as well as such oracles for the full tradeoff between space and query time obtained in their paper. This is the first non-trivial exact vertex-labeled distance oracle for planar graphs and, to our knowledge, for any interesting graph class other than trees.