More Compact Oracles for Approximate Distances in Planar Graphs

TL;DR

This paper presents new, more space-efficient approximate distance oracles for planar graphs that improve the tradeoff between space and query time, especially for moderate edge weights, advancing prior work by Thorup.

Contribution

The authors introduce the first distance oracle for planar graphs with subquadratic space-query time product dependency on 1/eps, and improve performance for graphs with moderate edge weights.

Findings

Achieved space ~O(n log n) with query time ~O(1/eps) for approximate distances.

Developed an oracle with space ~O(n log log n) and query time ~O(log log log n + 1/eps) for certain graph classes.

Improved the space-query time tradeoff over previous Thorup's oracle for planar graphs.

Abstract

Distance oracles are data structures that provide fast (possibly approximate) answers to shortest-path and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS'01, Thorup introduced approximate distance oracles for planar graphs. He proved that, for any eps>0 and for any planar graph on n nodes, there exists a (1+eps)-approximate distance oracle using space O(n eps^{-1} log n) such that approximate distance queries can be answered in time O(1/eps). Ten years later, we give the first improvements on the space-querytime tradeoff for planar graphs. * We give the first oracle having a space-time product with subquadratic dependency on 1/eps. For space ~O(n log n) we obtain query time ~O(1/eps) (assuming polynomial edge weights). The space shows a doubly logarithmic dependency on 1/eps only. We believe that the dependency on eps may be almost optimal. * For the case of moderate edge weights (average bounded by polylog(n), which appears to be the case for many real-world road networks), we hit a "sweet spot," improving upon Thorup's oracle both in terms of eps and n. Our oracle uses space ~O(n log log n) and it has query time ~O(log log log n + 1/eps). (Asymptotic notation in this abstract hides low-degree polynomials in log(1/eps) and log*(n).)