On Euclidean Steiner $(1+\epsilon)$-Spanners
Sujoy Bhore2020, arXiv: Computational Geometry
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Abstract
Lightness and sparsity are two natural parameters for Euclidean $(1+\epsilon)$-spanners. Classical results show that, when the dimension $d\in \mathbb{N}$ and $\epsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits an $(1+\epsilon)$-spanners with $O(n)$ edges and weight proportional to that of the Euclidean MST of $S$. Tight bounds on the dependence on $\epsilon>0$ for constant $d\in \mathbb{N}$ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a $(1+\epsilon)$-spanner. They gave upper bounds of $\tilde{O}(\epsilon^{-(d+1)/2})$ for the minimum lightness in dimensions $d\geq 3$, and $\tilde{O}(\epsilon^{-(d-1))/2})$ for the minimum sparsity in $d$-space for all $d\geq 1$. They obtained lower bounds only in the plane ($d=2$). Le and Solomon (ESA 2020) also constructed Steiner $(1+\epsilon)$-spanners of lightness $O(\epsilon^{-1}\log\Delta)$ in the plane, where $\De...
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