New Results on Linear Size Distance Preservers

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 $O(n+n^{2\mathrm{/}3}p)$ edges.
• Any $p=\mathrm{\Omega }\left(\frac{n^2}{rs(n)}\right)$ 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 $\sigma \le n^{2\mathrm{/}3-o(1)}$.