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