Light euclidean steiner spanners in the plane

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ N of the minimum lightness of a (1 + ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1 + ε)-spanners of lightness O(ε⁻¹ log ∆) in the plane, where ∆ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε⁻(d+1)/²) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε⁻d/²) for the lightness of Steiner (1 + ε)-spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1 + ε)-spanner of lightness O(ε⁻¹); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.