Light euclidean spanners with steiner points

The FOCS’19 paper of Le and Solomon [59], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1 + )-spanner in R^d is Õ(^−d) for any d = O(1) and any = Ω(n^{− d−1 1} ) (where Õ hides polylogarithmic factors of ¹ ), and also shows the existence of point sets in R^d for which any (1 + )-spanner must have lightness Ω(^−d).¹ Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in R² with lightness O(⁻¹ log ∆), where ∆ is the spread of the point set.² In the regime of ∆ 2^1/, this provides an improvement over the lightness bound of [59]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications often controls the precision, and it sometimes needs to be much smaller than O(1/ log n). Moreover, for spread polynomially bounded in 1/, this upper bound provides a quadratic improvement over the non-Steiner bound of [59], We then demonstrate that such a light spanner can be constructed in O(n) time for polynomially bounded spread, where O hides a factor of poly(¹ ). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(⁻(d+1)/² + ⁻² log ∆) for any 3 ≤ d = O(1) and any = Ω(n^{− d}−1).