A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners ahve arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimention). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known.
In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α-snowflake of a metric (X,δ) is the metric (X,$δ^α$) for 0<α<1. By utilizing an approach completely different than those used for Euclidean spaces, we demonstrate that snowflake metrics admit light spanners. Further, we show that the spanner is of diameter O($\log$n), a result not possible for Euclidean spaces. As an immediate corollary to our spanner, we obtain dramatic improvments in algorithms for the traveling salesman problem in this setting, achieving a polynomial-time approximation scheme with near-linear runtime. Along the way, we show that all ${\ell}_p$ spaces admit light spanners, a result of interest in its own right.
From Hierarchical Partitions to Hierarchical Covers: Optimal Fault-Tolerant Spanners for Doubling Metrics
在本文中，我们设计了一个最佳结构的容错空间加倍度量。具体地说，对于任意$n$ -点加倍度量、任意$\eps > 0$和任意整数$0 \le k \le n-2$，我们的构造提供了一个在最优时间$O(n \log n + k n)$内具有最优度$O(k)$的$k$ -容错$(1+\eps)$ -模型.然后，我们加强这一结果，以提供接近最佳的（$\log k$的一个因素）保证我们的spectum的直径和重量，即直径$O(\log n)$和重量$O(k^2 + k \log n) \cdot ω(MST)$，同时保持最佳的保证程度$O(k)$和运行时间$O(n \log n + k n)$。我们的结果解决了这一领域的几个基本的开放问题，最终结束了从Arya等人的STOC'95论文开始的一系列研究。以及Levcopoulos等人的STOC'98论文。在得到这一结果的过程中，我们发展了一种新的技术，用于在加倍度量中构造空间。我们的构造是基于一个新的\emph{hierarchical cover}的度量，而大多数以前的构造spiritual加倍和欧几里德度量（如网树的spiritual）是基于\emph{hierarchical partitions}的度量。我们证明了权力的层次覆盖的背景下，几何speries通过提高国家的最先进的结果在这方面。
Vertex Fault-Tolerant Spanners for Weighted Points in Polygonal Domains
给定$n$点的集合$S$、将非负权重与$S$中的每个点相关联的权重函数$w$、正整数$k \ge 1$和实数$ε> 0$，我们设计以下算法来计算由$S$中的加权点引起的度量空间的$k$ -顶点容错拓扑网络$G(S, E)$：（1）当$S$中的点位于简单多边形中时，我们提出了一种利用乘法拉伸$\sqrt{10}+ε$计算$G$的算法，其中$G$中的边数（$G$的大小）为$O(k n (\lg{n})^2)$。(2)当$S$中的点位于具有$h$个障碍物的多边形区域$\cal P$的自由空间中时，我们提出了一种计算$G$的算法，该算法具有乘法拉伸$6+ε$和大小$O(\sqrt{h} k n(\lg{n})^2)$ . (3)当$S$中的点位于多面体地形上时，我们设计了一个计算$G$的算法，该算法具有乘法拉伸$6+ε$和大小$O(k n (\lg{n})^2)$。
