Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

TL;DR

This paper introduces near-optimal data structures for efficiently answering shortest distance queries from a vertex to the nearest vertex with a given label in planar graphs, improving previous bounds significantly.

Contribution

It presents the first non-trivial exact vertex-labeled distance oracles for planar graphs, matching the best known bounds for unlabeled cases and extending their applicability.

Findings

Achieves near-linear space and constant query time in the best case.

Provides a full tradeoff between space and query time for vertex-labeled distance oracles.

First such oracle for planar graphs beyond trees.

Abstract

Given an undirected $n$-vertex planar graph $G=(V,E,\omega)$ with non-negative edge weight function $\omega:E\rightarrow \mathbb R$ 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 $\lambda$ reports the shortest path distance from $u$ to the nearest vertex with label $\lambda$. 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 $\tilde{O}(s(n))$ space and $\tilde{O}(q(n))$ query time. Using the state-of-the-art distance oracle of Long and Pettie, our construction produces a vertex-labeled distance oracle using $n^{1+o(1)}$ space and query time $\tilde O(1)$ at one extreme, $\tilde O(n)$ 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.