Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

TL;DR

This paper introduces linear-space approximate distance oracles for planar, bounded-genus, and minor-free graphs, achieving efficient storage and preprocessing with acceptable query times, suitable for memory-constrained environments.

Contribution

It presents novel linear-space distance oracles for specific graph classes, with faster preprocessing and adaptable space requirements, improving over previous methods.

Findings

Space complexity is reduced to O(n), independent of epsilon and graph class.

Preprocessing time is faster than previous constructions, e.g., O(n lg^2 n) for planar graphs.

Query time is acceptable, e.g., O(eps^{-2} lg^2 n) for planar graphs.

Abstract

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself.