Spanners in randomly weighted graphs: independent edge lengths
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Given a connected graph G=(V,E) and a length function ℓ:E→ℝ we let d_v,w denote the shortest distance between vertex v and vertex w. A t-spanner is a subset E'⊆ E such that if d'_v,w denotes shortest distances in the subgraph G'=(V,E') then d'_v,w≤ t d_v,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 ≈1/2nlog n edges and that this is best possible. In particular, our result applies to the random graphs G_n,p for np≫log n.
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