Small Stretch Pairwise Spanners

Abstract

Let G = (V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (α,β) spanner of G if for each (u,v) ∈ V ×V, the u-v distance in H is at most α·δ _G (u,v) + β. The following is a natural relaxation of the above problem: we care for only certain distances, these are captured by the set $\mathcal{P} \subseteq V \times V$ and the problem is to construct a sparse subgraph H, called an (α,β) $\mathcal{P}$-spanner, where for every $(u,v) \in \mathcal{P}$, the u-v distance in H is at most α·δ _G (u,v) + β.

We show how to construct a (1,2) $\mathcal{P}$-spanner of size $\tilde{O}(n\cdot|\mathcal{P}|^{1/3})$ and a (1,2) (S×V)-spanner of size $\tilde{O}(n\cdot(n|S|)^{1/4})$. A D-spanner is a $\mathcal{P}$-spanner when $\mathcal{P}$ is described implicitly via a distance threshold D as $\mathcal{P} = \{(u,v): \delta(u,v) \ge D\}$. For a given D ∈ ℤ⁺, we show how to construct a (1,4) D-spanner of size $\tilde{O}(n^{3/2}/{D^{1/4}})$ and for D ≥ 2, a (1,4logD) D-spanner of size $\tilde{O}(n^{3/2}/{\sqrt{D}})$.

This work was supported by IMPECS (Indo-German Max Planck Center for Computer Science).