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 $O(n + n^{2/3} p)$ 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 $\sigma \le n^{2/3 - o(1)}$. 
Greg Bodwin
cs.DS