Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

TL;DR

This paper introduces a new distance oracle for planar graphs that uses subquadratic space and provides exact shortest-path distances in constant query time, a significant advancement over previous methods.

Contribution

It presents the first truly subquadratic size distance oracle for planar graphs with constant query time for exact shortest-path distances.

Findings

Space complexity is $O(n^{5/3+\varepsilon})$ for any $oldsymbol{\varepsilon > 0}$.

Queries are answered in worst-case time $O(\log (1/oldsymbol{\varepsilon}))$.

No prior truly subquadratic exact distance oracles with constant query time existed.

Abstract

Given an undirected, unweighted planar graph $G$ with $n$ vertices, we present a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of $G$ in constant time. For any $\varepsilon > 0$, our distance oracle takes up $O(n^{5/3+\varepsilon})$ space and is capable of answering shortest-path distance queries exactly for any pair of vertices of $G$ in worst-case time $O(\log (1/\varepsilon))$. Previously no truly sub-quadratic size distance oracles with constant query time for answering exact all-pairs shortest paths distance queries existed.