Better distance preservers and additive spanners

Abstract

We make improvements to the upper bounds on several popular types of distance preserving graph sketches. The first part of our paper concerns pairwise distance preservers, which are sparse subgraphs that exactly preserve the pairwise distances for a set of given pairs of vertices. Our main result here is that all unweighted, undirected n-node graphs G and all pair sets P have distance preservers on |H| = O(n2/3|P|2/3 + n|P|1/3) edges. This improves the known bounds whenever |P| = ω(n3/4).We then develop a new graph clustering technique, based on distance preservers, and we apply this technique to show new upper bounds for additive (standard) spanners, in which all pairwise distances must be preserved up to an additive error function, and for subset spanners, in which only distances within a given node subset must be preserved up to an error function. For both of these objects, we obtain the new best tradeoff between spanner sparsity and error allowance in the regime where the error is polynomial in the graph size.We leave open a conjecture that O(n2/3|P|2/3 + n) pairwise distance preservers are possible for undirected unweighted graphs. Resolving this conjecture in the affirmative would improve and simplify our upper bounds for all the graph sketches mentioned above.