Minimum weight Euclidean (1+ɛ)-spanners - Tóth, Csaba D. (European journal of combinatorics, 2024)
Given a set S of n points in the plane and a parameter ɛ>0, a Euclidean (1+ɛ)-spanner is a geometric graph G=(S,E) that contains, for all p,q∈S, a pq-path of weight at most (1+ɛ)‖pq‖. We show that the minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit square [0,1]2 is O(ɛ−3/2n), and this bound is the best possible. The upper bound is based on a new spanner algorithm in the plane. It improves upon the baseline O(ɛ−2n), 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+ɛ)-spanners, which is the ratio of the spanner weight to the weight of the MST. Our result generalizes to Euclidean d-space for every constant dimension d∈N: The minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit cube [0,1]d is Od(ɛ(1−d2)/dn(d−1)/d), and this bound is the best possible. For the n×n section of the integer lattice in the plane, we show that the minimum weight of a Euclidean (1+ɛ)-spanner is between Ω(ɛ−3/4⋅n2) and O(ɛ−1log(ɛ−1)⋅n2). These bounds become Ω(ɛ−3/4⋅n) and O(ɛ−1log(ɛ−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+ɛ)-spanners.