Distances in Planar Graphs are Almost for Free!

TL;DR

This paper demonstrates that exact distance oracles for planar graphs can be constructed efficiently, with near-linear preprocessing time and fast query times, nearly matching the best possible tradeoffs.

Contribution

It introduces a method to build exact distance oracles for planar graphs in near-linear time, improving over previous higher time complexities.

Findings

Constructed distance oracles in $n^{1+o(1)}$ time.

Achieved query times of $ ext{polylog}(n)$ for exact distances.

Matched the best known space and query time tradeoffs with improved construction time.

Abstract

We prove that, up to subpolynomial or polylogarithmic factors, there is no tradeoff between preprocessing time, query time, and size of exact distance oracles for planar graphs. Namely, we show how given an $n$-vertex weighted directed planar graph $G$, one can compute in $n^{1+o(1)}$ time and space a representation of $G$ from which one can extract the exact distance between any two vertices of $G$ in $\log^{2+o(1)}(n)$ time. Previously, it was only known how to construct oracles with these space and query time in $n^{3/2+o(1)}$ time [STOC 2019, SODA 2021, JACM 2023]. We show how to construct these oracles in $n^{1+o(1)}$ time.