  A $t$-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 $\epsilon>0$,
any weighted graph $G$ on $n$ vertices admits a
$(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot
O_\epsilon(kn^{1/k}/\log k)$, 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 $O(\log
k)$.
