Minimum Weight Euclidean (1+ε)-Spanners
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Given a set S of n points in the plane and a parameter ε>0, a Euclidean (1+ε)-spanner is a geometric graph G=(S,E) that contains, for all p,q∈ S, a pq-path of weight at most (1+ε)pq. We show that the minimum weight of a Euclidean (1+ε)-spanner for n points in the unit square [0,1]^2 is O(ε^-3/2 √(n)), and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline O(ε^-2√(n)), obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on n points in [0,1]^2, and a tight bound for the lightness of Euclidean (1+ε)-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to Euclidean d-space for every dimension d∈ℕ: The minimum weight of a Euclidean (1+ε)-spanner for n points in the unit cube [0,1]^d is O_d(ε^(1-d^2)/dn^(d-1)/d), and this bound is the best possible. For the n× n section of the integer lattice, we show that the minimum weight of a Euclidean (1+ε)-spanner is between Ω(ε^-3/4· n^2) and O(ε^-1log(ε^-1)· n^2). These bounds become Ω(ε^-3/4·√(n)) and O(ε^-1log(ε^-1)·√(n)) when scaled to a grid of n points in the unit square. In particular, this shows that the integer grid is not an extremal configuration for minimum weight Euclidean (1+ε)-spanners.
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