Sparse source-wise and pair-wise distance preservers

Abstract

We introduce and study the notions of pair-wise and source-wise preservers. Given an undirected N-vertex graph G = (V, E) and a subset P of pairs of vertices, let G' = (V, H), H ⊆ E, be called a pair-wise preserver of G with respect to P if for every pair {u, w} ∈ P, distcG'(u, w) = distG(u, w). For a set S ⊆ V of sources, a pair-wise preserver of G with respect to the set of all pairs P = (2s) of sources is called a source-wise preserver of G with respect to S. We prove that for every undirected possibly weighted N-vertex graph G and every subset P of P = O(N1/2) pairs of vertices of G, there exists a linear-size pair-wise preserver of G with respect to P. Consequently, for every subset S ⊆ V of S = O(N1/4) sources, there exists a linear-size source-wise preserver of G with respect to S. On the negative side we show that neither of the two exponents (1/2 and 1/4) can be improved even when the attention is restricted to unweighted graphs. Our lower bounds involve constructions of dense convexly independent sets of vectors with small Euclidean norms. We believe that the link between the areas of Discrete Geometry and spanners that we establish is of independent interest, and might be useful in the study of other problems in the area of low-distortion embeddings.