Lightweight Near-Additive Spanners

An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of $H$ to the weight of a minimum spanning tree). In this paper we focus on near-additive spanners, where $\alpha=1+\varepsilon$ for arbitrarily small $\varepsilon>0$. We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter $k\ge 1$, we obtain an $(1+\varepsilon,O(k/\varepsilon)^k\cdot W(\cdot,\cdot))$-spanner with lightness $\tilde{O}(n^{1/k})$ (where $W(\cdot,\cdot)$ indicates for every pair $u, v \in V$ the heaviest edge in some shortest path between $u,v$). In addition, we can also bound the number of edges in our spanner by $O(kn^{1+3/k})$.