Small Stretch Pairwise Spanners and D - spanners Student

An (α, β)-spanner of an undirected unweighted connected graph G = (V,E) is a subgraph H such that: dH(u, v) ≤ α · dG(u, v) + β, for all pairs (u, v) ∈ V × V , where dH(u, v) and dG(u, v) are the distances between u and v in H and G respectively. The quantities α and β are non negative real numbers and are called the multiplicative stretch and additive stretch of the spanner respectively. If α = 1, the spanner is called additive. In this report, we focus our attention to additive spanners. Additive spanners are well studied. We study a natural generalization of the additive spanner problem where we look to approximate the distances of only a specified set of pairs of nodes. Given a graph G = (V,E) and a set P ⊆ V ×V , an (α, β) P-spanner, or a pairwise spanner, of G is a subgraph H such that dH(u, v) ≤ α · dG(u, v) + β for all (u, v) ∈ P. We obtain polynomial time constructions for the following pairwise spanners: a (1, 2) P-spanner with Õ(n|P|1/3) edges when P ⊆ V × V is arbitrary, a (1, 2) P-spanner with Õ(n|P|1/4) edges when P = S × V for some S ⊆ V. In the special case when P contains exactly those pairs of nodes which are at a distance at least D in G, an (α, β) P-spanner of G is also called as an (α, β) D-spanner. For any integer k ≥ 1, we present polynomial time algorithms to construct: a (1, 4k) D-spanner with Õ(n3/2/Dk/(2k+2)) edges. A part of this work has appeared in the proceedings of ICALP 2013 as a paper titled Small Stretch Pairwise Spanners [KV13].