Weighted Additive Spanners.

An $\alpha$-additive spanner of an undirected graph $G=(V, E)$ is a subgraph $H$ such that the distance between any two vertices in $G$ is stretched by no more than an additive factor of $\alpha$. It is previously known that unweighted graphs have 2-, 4-, and 6-additive spanners containing $\widetilde{O}(n^{3/2})$, $\widetilde{O}(n^{7/5})$, and $O(n^{4/3})$ edges, respectively. In this paper, we generalize these results to weighted graphs. We consider $\alpha=2W$, $4W$, $6W$, where $W$ is the maximum edge weight in $G$. We first show that every $n$-node graph has a subsetwise $2W$-spanner on $O(n |S|^{1/2})$ edges where $S \subseteq V$ and $W$ is a constant. We then show that for every set $P$ with $|P| = p$ vertex demand pairs, there are pairwise $2W$- and $4W$-spanners on $O(np^{1/3})$ and $O(np^{2/7})$ edges respectively. We also show that for every set $P$, there is a $6W$-spanner on $O(np^{1/4})$ edges where $W$ is a constant. We then show that every graph has additive $2W$- and $4W$-spanners on $O(n^{3/2})$ and $O(n^{7/5})$ edges respectively. Finally, we show that every graph has an additive $6W$-spanner on $O(n^{4/3})$ edges where $W$ is a constant.