论文标题
欧几里得轻的尖峰
Light Euclidean Spanners with Steiner Points
论文作者
论文摘要
Le和Solomon的FOCS'19论文最终导致了一系列关于欧几里得人士的研究，证明了贪婪的$（1+ε）$ - 在$ \ Mathbb {r} d $ is $ is $ \ tilde $} $} $} $（$ o} $ o（$ dim）中，贪婪的$（1+ε）$ - ω（n^{ - \ frac {1} {d-1}}）$（其中$ \ tilde {o} $隐藏$ \ frac {1}ε$）的polyrogarithmic因子，并且还显示$ \ mathbb {r}^d $的$（r}^d $ spacter yy $（1） $ω（ε^{ - d}）$。鉴于这种紧密的束缚在轻度上，一个自然的问题是，使用施泰纳点是否可以实现更好的光线。 我们的第一个结果是用$ \ mathbb {r}^2 $用轻度$ o（ε^{ - 1} \logΔ）$构造Steiner Spanners，其中$δ$是点集的差异。在$δ\ ll 2^{1/ε} $的制度中，这提供了对LE和所罗门的轻度界的改进[FOCS 2019]；这种参数制度具有实际感兴趣，因为在现实生活应用中（例如，对于各种随机分布）中引起的点集具有多项式界限的差异，而在Spanner应用程序中，$ε$通常会控制精度，并且有时需要小于$ O（1/\ log n）$。此外，对于以$ 1/ε$为界的传播，该上限为LE和所罗门的非steiner界提供了二次改进[FOCS 2019]，然后我们证明可以在$ o_is（n）$ o_is（n）$ o__ o_is $ o_im $ o_ip of aide a $ o_im $ hides的因素中构建这样的光扳手。 $ \ mathrm {poly}（\ frac {1}ε）$。最后，我们将构造扩展到更高的维度，证明了$ \ tilde {o}（ε^{ - （d + 1）/2} +ε^{ - 2} \logΔ）$的亮度上限，用于任何$ 3 \ leq d = o（1）$（1）$，任何$ 3 \ l lex = frac}}
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 $\mathbb{R}^d$ is $\tilde{O}(ε^{-d})$ for any $d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}ε$), and also shows the existence of point sets in $\mathbb{R}^d$ 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 $\mathbb{R}^2$ with lightness $O(ε^{-1} \log Δ)$, where $Δ$ is the spread of the point set. In the regime of $Δ\ll 2^{1/ε}$, 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 $\mathrm{poly}(\frac{1}ε)$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(ε^{-(d+1)/2} + ε^{-2}\log Δ)$ for any $3\leq d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$.
