New Results on Linear Size Distance Preservers

TL;DR

This paper establishes new upper and lower bounds on the number of edges required for distance preservers in both directed and undirected graphs, revealing how graph size and pairwise distances influence sparsity.

Contribution

It provides novel bounds on the size of distance preservers, including tight upper bounds for directed graphs and bounds involving the Ruzsa-Szemerédi function for undirected graphs.

Findings

Directed graphs have distance preservers with O(n + n^{2/3} p) edges.

Undirected graphs require O(p) edges for certain pair sets, related to the Ruzsa-Szemerédi function.

Lower bounds show that some subsets require superlinear edges to preserve all distances.

Abstract

Given $p$ node pairs in an $n$-node graph, a distance preserver is a sparse subgraph that agrees with the original graph on all of the given pairwise distances. We prove the following bounds on the number of edges needed for a distance preserver: - Any $p$ node pairs in a directed weighted graph have a distance preserver on $O(n + n^{2/3} p)$ edges. - Any $p = \Omega\left(\frac{n^2}{rs(n)}\right)$ node pairs in an undirected unweighted graph have a distance preserver on $O(p)$ edges, where $rs(n)$ is the Ruzsa-Szemer\'edi function from combinatorial graph theory. - As a lower bound, there are examples where one needs $\omega(\sigma^2)$ edges to preserve all pairwise distances within a subset of $\sigma = o(n^{2/3})$ nodes in an undirected weighted graph. If we additionally require that the graph is unweighted, then the range of this lower bound falls slightly to $\sigma \le n^{2/3 - o(1)}$.