Spanners in randomly weighted graphs: Euclidean case

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 study the size of spanners in the following scenario: we consider a random embedding of $G_{n,p}$ into the unit square with Euclidean edge lengths. For $\epsilon>0$ constant, we prove the existence w.h.p. of $(1+\epsilon)$-spanners for ${\mathcal X}_p$ that have $O_\epsilon(n)$ edges. These spanners can be constructed in $O_\epsilon(n^2\log n)$ time. (We will use $O_\epsilon$ to indicate that the hidden constant depends on $\epsilon$.) There are constraints on $p$ preventing it going to zero too quickly.