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′⊆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∈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 ϵ>0 $\epsilon \gt 0$ constant, we prove the existence w.h.p. of (1+ϵ) $(1+\epsilon )$‐spanners for Xp ${{\mathscr{X}}}_{p}$ that have Oϵ(n) ${O}_{\epsilon }(n)$ edges. These spanners can be constructed in Oϵ(n2logn) ${O}_{\epsilon }({n}^{2}\mathrm{log}n)$ time. (We will use Oϵ ${O}_{\epsilon }$ to indicate that the hidden constant depends on ε $\varepsilon $). There are constraints on p $p$ preventing it going to zero too quickly.