Better Distance Preservers and Additive Spanners

TL;DR

This paper advances the understanding of distance preservers by establishing new bounds and introduces improved additive spanners with fewer edges and better error guarantees, enhancing graph distance approximation techniques.

Contribution

It provides new bounds for the size of distance preservers based on demand pairs and develops improved additive spanners with fewer edges and tighter error bounds.

Findings

New upper bound for distance preservers: O(n^{2/3}p^{2/3} + np^{1/3}) edges.

Established a construction of additive spanners with O(n) edges and +O(n^{3/7 + ε}) error.

Demonstrated extensions of the path-buying framework to larger clusters.

Abstract

We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work on distance preservers has exploited only a simple structural property of shortest paths, called consistency, stating that one can break shortest path ties such that no two paths intersect, split apart, and then intersect again later. We prove that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it. Specifically, our new upper bound is that any $p$ demand pairs in an $n$-node undirected unweighted graph have a distance preserver on $O(n^{2/3}p^{2/3} + np^{1/3})$ edges. We leave a conjecture that the right bound is $O(n^{2/3}p^{2/3} + n)$ or better. The second part of this paper leverages these distance preservers in a new construction of additive spanners, which are subgraphs that preserve all pairwise distances up to an additive error function. We give improved error bounds for spanners with relatively few edges; for example, we prove that all graphs have spanners on $O(n)$ edges with $+O(n^{3/7 + \varepsilon})$ error. Our construction can be viewed as an extension of the popular path-buying framework to clusters of larger radii.