Spanners in randomly weighted graphs: Euclidean case

Given a connected graph G = ( V , E ) $G=(V,E)$ and length function ℓ : → R $\ell :E\to {\mathbb{R}}$ we let d v w ${d}_{v,w}$ denote the shortest distance between vertex $v$ $w$ . A t $t$ -spanner is subset ′ ⊆ $E^{\prime} \subseteq E$ such that if ${d}_{v,w}^{^{\prime} }$ denotes distances in subgraph $G^{\prime} =(V,E^{\prime} )$ then ≤ }\le t{d}_{v,w}$ for all ∈ $v,w\in V$ We study size of spanners following scenario: consider random embedding X p ${{\mathscr{X}}}_{p}$ n ${G}_{n,p}$ into unit square with Euclidean edge lengths. For ϵ > 0 $\epsilon \gt 0$ constant, prove existence w.h.p. 1 + $(1+\epsilon -spanners have O ${O}_{\epsilon }(n)$ edges. These can be constructed 2 log }({n}^{2}\mathrm{log}n)$ time. (We will use to indicate hidden constant depends on ε $\varepsilon $ ). There are constraints $p$ preventing it going zero too quickly.