Efficient Vertex-Label Distance Oracles for Planar Graphs

We consider distance queries in vertex-labeled planar graphs. For any fixed $0 < \epsilon \leq 1/2$ we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex $u$ and a label $\lambda$ return a $(1+\epsilon)$-approximation of the distance from $u$ to its closest vertex with label $\lambda$. For a directed planar graph with $n$ vertices, such that the ratio of the largest to smallest arc length is bounded by $N$, the preprocessing time is $O(\epsilon^{-2}n\lg^{3}{n}\lg(nN))$, the data structure size is $O(\epsilon^{-1}n\lg{n}\lg(nN))$, and the query time is $O(\lg\lg{n}\lg\lg(nN) + \epsilon^{-1})$. We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect.