Linear Size Distance Preservers

The famous shortest path tree lemma states that, for any node s in a graph G = (V,E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s}×V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node pairs, where P has some different structure than {s}×V or possibly no guaranteed structure at all. Trivial lower bounds of a path or a clique show that such a subgraph will need Ω(n + p) edges in the worst case. The question is then to determine when these trivial lower bounds are sharp; that is, when do graphs have linear size distance preservers on O(n+ p) edges? In this paper, we make the first new progress on this fundamental question in over ten years. We show: 1. All G,P has a distance preserver onO(n) edges whenever p = O(n), even if G is directed and/or weighted. These are the first nontrivial preservers of size O(n) known for directed graphs. 2. All G,P has a distance preserver on O(p) edges whenever p = Ω (