Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

TL;DR

This paper introduces the skeleton dimension, a new graph parameter based on shortest-path trees, which explains small distance labels in networks more efficiently than previous highway dimension-based methods.

Contribution

It proposes the skeleton dimension as a more tractable and intuitive alternative to highway dimension for analyzing and computing small distance labels in networks.

Findings

Skeleton dimension provides comparable or stronger bounds on hub set size.

It allows for more efficient computation of labels.

The parameter applies to both directed and undirected graphs.

Abstract

The goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) $\subseteq$ V to each node u $\in$ V, in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) $\cap$ S(v) contains a node on the shortest uv-path. The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by Abraham et al. (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size.