(1 + ε)-Distance Oracles for Vertex-Labeled Planar Graphs

Abstract

We consider vertex-labeled graphs, where each vertex v is attached with a label from a set of labels. The vertex-to-label distance query desires the length of the shortest path from the given vertex to the set of vertices with the given label. We show how to construct an oracle for a vertex-labeled planar graph, such that $O(\frac{1}{\epsilon}n\log n)$ storing space is needed, and any vertex-to-label query can be answered in $O(\frac{1}{\epsilon}\log n\log \Delta)$ time with stretch 1 + ε. Here, Δ is the hop-diameter of the given graph. For the case that Δ = O(logn), we construct a distance oracle that achieves $O(\frac{1}{\epsilon}\log n)$ query time, without changing space usage.