Spanners in randomly weighted graphs: Independent edge lengths

Given a connected graph G=(V,E) and length function ℓ:E→R we let dv,w denote the shortest distance between vertex v w. A t-spanner is subset E′⊆E such that if dv,w′ denotes distances in subgraph G′=(V,E′) then dv,w′≤tdv,w for all v,w∈V. We show large class of graphs with suitable degree expansion properties independent exponential mean one edge lengths, there w.h.p. 1-spanner uses ≈12nlogn edges this best possible. In particular, our result applies to random Gn,p np≫logn.