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On Pairwise Spanners

Given an undirected $n$-node unweighted graph $G = (V, E)$, a spanner with stretch function $f(\cdot)$ is a subgraph $H\subseteq G$ such that, if two nodes are at distance $d$ in $G$, then they are at distance at most $f(d)$ in $H$. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the $u$-$v$ distance only for pairs $(u,v)$ in a given set $\cP \subseteq V\times V$. Such $\cP$-spanners were studied before [Coppersmith,Elkin'05] only in the special case that $f(\cdot)$ is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same $\cP$) and of the best known spanners (with the same $f(\cdot)$). In more detail, for arbitrary $\cP$, we show that there exists a $\mathcal{P}$-spanner of size $O(n(|\cP|\log n)^{1/4})$ with $f(d)=d+4\log n$. Alternatively, for any $\eps>0$, there exists a $\cP$-spanner of size $O(n|\cP|^{1/4}\sqrt{\frac{\log n}{\eps}})$