Title: Light Spanners
Authors: Michael Elkin, Ofer Neiman and Shay Solomon
Abstract:
A \emph{$t$-spanner} of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$.
The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important
measures of the spanner's quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters $k\ge 1$ and $\eps>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\eps)$-stretch
spanner of weight at most $w(MST(G))\cdot O_\eps(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$.
Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.