Light Euclidean Spanners with Steiner Points
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The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ϵ)-spanner in ℝ^d is Õ(ϵ^-d) for any d = O(1) and any ϵ = Ω(n^-1/d-1) (where Õ hides polylogarithmic factors of 1/ϵ), and also shows the existence of point sets in ℝ^d for which any (1+ϵ)-spanner must have lightness Ω(ϵ^-d). 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 ℝ^2 with lightness O(ϵ^-1logΔ), where Δ is the spread of the point set. In the regime of Δ≪ 2^1/ϵ, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; 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 ϵ often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ϵ, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in O_ϵ(n) time for polynomially bounded spread, where O_ϵ hides a factor of poly(1/ϵ). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ϵ^-(d+1)/2 + ϵ^-2logΔ) for any 3≤ d = O(1) and any ϵ = Ω(n^-1/d-1).
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