arXiv 2007.116362020-07cs.CG

带Steiner点的轻欧氏生成子

Light Euclidean Spanners with Steiner Points

Hung Le, Shay Solomon

备注：: Add comments on reducing log(spread) to log(n)

摘要：

Le和Solomon的FOCS'19论文，对欧几里德空间的长期研究达到了高潮，证明了$\mathbb{R}^d$中贪婪的$(1+ε)$ -矩阵的亮度（归一化权重）对于任何$d = O(1)$和任何$ε= Ω(n^{-\frac{1}{d-1}})$都是$\tilde{O}(ε^{-d})$（其中$\tilde{O}$隐藏了$\frac{1}ε$的多对数因子），并且还表明了在$\mathbb{R}^d$中存在点集，对于这些点集，任何$(1+ε)$ -矩阵都必须具有亮度$Ω(ε^{-d})$。鉴于亮度上的这种紧密界限，自然产生的问题是是否可以使用Steiner点实现更好的亮度界限。我们的第一个结果是一个建设的施泰纳spectum在$\mathbb{R}^2$与亮度$O(ε^{-1} \log Δ)$，其中$Δ$是点集的蔓延。在$Δ\ll 2^{1/ε}$的体系中，这提供了对Le和Solomon [FOCS 2019]的亮度界限的改进;这种参数体系具有实际意义，因为点集出现在现实生活中的应用中（例如，对于各种随机分布）具有多项式有界的扩展，而在并行计算应用中，$ε$通常控制精度，并且它有时需要比$O(1/\log n)$小得多。此外，对于$1/ε$中的多项式有界扩展，该上限提供了对Le和Solomon的非Steiner界的二次改进[FOCS 2019]，然后我们证明了可以在$O_ε(n)$时间内构建这样的轻链，用于多项式有界扩展，其中$O_ε$隐藏了$\mathrm{poly}(\frac{1}ε)$的因子。最后，我们将构造扩展到更高的维度，证明了任何$3\leq d = O(1)$和任何$ε= Ω(n^{-\frac{1}{d-1}})$的亮度上限为$\tilde{O}(ε^{-(d+1)/2} + ε^{-2}\log Δ)$。

英文摘要：

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 $ε= Ω(n^{-\frac{1}{d-1}})$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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