Efficient Vertex-Label Distance Oracles for Planar Graphs

TL;DR

This paper introduces an efficient data structure for approximate distance queries from vertices to labeled vertices in directed planar graphs, with fast query times and manageable preprocessing costs.

Contribution

It presents a novel vertex-label distance oracle for directed planar graphs that provides near-optimal approximation with efficient preprocessing and query times.

Findings

Preprocessing time is $O(rac{1}{ ext{ extepsilon}^2} n ext{lg}^3 n ext{lg}(nN))$.

Query time is $O( ext{lg lg} n ext{lg lg}(nN) + ext{ extepsilon}^{-1})$.

Corrects earlier claims about undirected planar graph label distance oracles.

Abstract

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.