Spanners in randomly weighted graphs: independent edge lengths

Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes shortest distances in the subgraph $G'=(V,E')$ then $d'_{v,w}\leq t d_{v,w}$ for all $v,w\in V$. We show that for a large class of graphs with suitable degree and expansion properties with independent exponential mean one edge lengths, there is w.h.p.~a 1-spanner that uses $\approx \frac12n\log n$ edges and that this is best possible. In particular, our result applies to the random graphs $G_{n,p}$ for $np\gg \log n$.