Spanners in randomly weighted graphs: Euclidean case

Given a connected graph G=(V,E) $G=(V,E)$ and a length function l:E -> R $\ell :E\to {\mathbb{R}}$ we let dv,w ${d}_{v,w}$ denote the shortest distance between vertex v $v$ and vertex w $w$. A t $t$-spanner is a subset E 'subset of E $E<^>{\prime} \subseteq E$ such that if dv,w ' ${d}_{v,w}<^>{<^>{\prime} }$ denotes shortest distances in the subgraph G '=(V,E ') $G<^>{\prime} =(V,E<^>{\prime} )$ then dv,w '<= tdv,w ${d}_{v,w}<^>{<^>{\prime} }\le t{d}_{v,w}$ for all v,w is an element of V $v,w\in V$. We study the size of spanners in the following scenario: we consider a random embedding Xp ${{\mathscr{X}}}_{p}$ of Gn,p ${G}_{n,p}$ into the unit square with Euclidean edge lengths. For epsilon>0 $\epsilon \gt 0$ constant, we prove the existence w.h.p. of (1+epsilon) $(1+\epsilon )$-spanners for Xp ${{\mathscr{X}}}_{p}$ that have O epsilon(n) ${O}_{\epsilon }(n)$ edges. These spanners can be constructed in O epsilon(n2logn) ${O}_{\epsilon }({n}<^>{2}\mathrm{log}n)$ time. (We will use O epsilon ${O}_{\epsilon }$ to indicate that the hidden constant depends on epsilon $\varepsilon $). There are constraints on p $p$ preventing it going to zero too quickly.