Spanners in randomly weighted graphs: independent edge lengths

TL;DR

This paper demonstrates that in certain large random graphs with independent exponential edge lengths, a near-optimal 1-spanner with about half the number of edges as the complete graph exists with high probability, and this is proven to be optimal.

Contribution

The paper establishes the existence and optimality of sparse 1-spanners in a broad class of random graphs with independent exponential edge lengths.

Findings

Existence of 1-spanners with approximately 0.5n log n edges in certain random graphs.

Such 1-spanners are shown to be optimal in the given graph classes.

Results apply to G_{n,p} graphs with np much larger than log n.

Abstract

Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex $w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes shortest distances in the subgraph $G'=(V,E')$ then $d'_{v,w}\leq t d_{v,w}$ for all $v,w\in V$. 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 $\approx \frac12n\log n$ edges and that this is best possible. In particular, our result applies to the random graphs $G_{n,p}$ for $np\gg \log n$.