Better Tradeoffs for Exact Distance Oracles in Planar Graphs

TL;DR

This paper introduces a new planar graph distance oracle that uses less space and maintains fast query times, with a flexible tradeoff between space and speed, improving upon previous methods.

Contribution

It presents an $O(n^{1.5})$-space distance oracle with $O( ext{log} n)$ query time and a novel space-query tradeoff for planar graphs, simplifying and enhancing prior work.

Findings

Achieves $O(n^{1.5})$ space with $O( ext{log} n)$ query time.

Provides a flexible space-query tradeoff for different space sizes.

Outperforms previous tradeoffs for space range $[n, n^{5/3}]$.

Abstract

We present an $O(n^{1.5})$-space distance oracle for directed planar graphs that answers distance queries in $O(\log n)$ time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses $O(n^{5/3})$-space and answers queries in $O(\log n)$ time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any $S\in [n,n^2]$, we show an oracle of size $S$ that answers queries in $\tilde O(\max\{1,n^{1.5}/S\})$ time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of $S$ and improves by polynomial factors over all the previously known tradeoffs for the range $S \in [n,n^{5/3}]$.