论文标题
Euclidean Steiner Spanners：轻而稀疏
Euclidean Steiner Spanners: Light and Sparse
论文作者
论文摘要
轻度和稀疏性是欧几里得$（1+ \ varepsilon）$ spanners的两个自然参数。经典结果表明，当尺寸$ d \ in \ mathbb {n} $和$ \ varepsilon> 0 $是常数时，每个集合$ n $ s $ s $ s $的$ n $点都在$ d $ space中承认$（1+\ varepsilon）$ - 带有$ o（n）$ edges $ edges and Iffectional of thit ocuclidean of ecuclidean mst的$（n）$ edges和权重。在最近的突破中，Le and Solomon（2019）确立了对$ \ varepsilon> 0 $的确切依赖性，对于\ mathbb {n} $的常数$ d \，对$（1+ \ varepsilon）$的最小轻度和稀疏性可以改善一点点，并可以改善A a的最低光和稀疏性，并且可以改善A的点。 $（1+ \ varepsilon）$ - 扳手。他们给出了$ \ tilde {o}（\ varepsilon^{ - （d+1）/2}）$，用于尺寸的最小轻度$ d \ geq 3 $，和$ \ tilde {o} {o}（\ varepsilon^{ - （d-1）/2} $ d $ d $ d- $ d $ d- 在这项工作中，我们提高了欧几里得史坦纳$（1+ \ varepsilon）$ - 跨度的少量和稀疏性的几个范围。我们为轻度建立了$ω（\ varepsilon^{ - d/2}）$的下限，$ω（\ varepsilon^{ - （d-1）/2}）$，用于euclidean $ d $ d $ -d $ -d $ -d $ d $ d $ d $ d \ egeq geq geq 2 $的稀疏性。我们的下边界结构通过LE和所罗门概括了先前的构造，但是分析使用新的几何见解简化了先前的工作，重点关注边缘的方向。接下来，我们表明，对于（0,1] $中的每一个有限的点，每$ \ varepsilon \ in（0,1] $，都有一个欧几里得施泰纳$（1+ \ varepsilon）$ - 轻度$ o（\ varepsilon^{-1}）$ spanner of Light of Plower of plower of plowe of plowe deplore undies note undies unipect $ d = 2 $ demiles and demeriziz and。使用定向跨度和修改后的窗户分配方案来实现严格的体重分析。
Lightness and sparsity are two natural parameters for Euclidean $(1+\varepsilon)$-spanners. Classical results show that, when the dimension $d\in \mathbb{N}$ and $\varepsilon>0$ are constant, every set $S$ of $n$ points in $d$-space admits an $(1+\varepsilon)$-spanners with $O(n)$ edges and weight proportional to that of the Euclidean MST of $S$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\varepsilon>0$, for constant $d\in \mathbb{N}$, of the minimum lightness and sparsity of $(1+\varepsilon)$-spanners, and observed that Steiner points can substantially improve the lightness and sparsity of a $(1+\varepsilon)$-spanner. They gave upper bounds of $\tilde{O}(\varepsilon^{-(d+1)/2})$ for the minimum lightness in dimensions $d\geq 3$, and $\tilde{O}(\varepsilon^{-(d-1)/2})$ for the minimum sparsity in $d$-space for all $d\geq 1$. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+\varepsilon)$-spanners. We establish lower bounds of $Ω(\varepsilon^{-d/2})$ for the lightness and $Ω(\varepsilon^{-(d-1)/2})$ for the sparsity of such spanners in Euclidean $d$-space for all constant $d\geq 2$. Our lower bound constructions generalize previous constructions by Le and Solomon, but the analysis substantially simplifies previous work, using new geometric insight, focusing on the directions of edges. Next, we show that for every finite set of points in the plane and every $\varepsilon\in (0,1]$, there exists a Euclidean Steiner $(1+\varepsilon)$-spanner of lightness $O(\varepsilon^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
