相关论文: Optimal Euclidean spanners: really short, thin and…

In this paper we devise a novel \emph{unified} construction of Euclidean spanners that trades between the maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+eps)-spanner with maximum…

In STOC'95 Arya et al. (1995) conjectured that for any constant dimensional $n$-point Euclidean space, a $(1+\eps)$-spanner with constant degree, hop-diameter $O(\log n)$ and weight $O(\log n) \cdot \omega(MST)$ can be built in $O(n \log…

We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $\omega(T) = O(\log n) \cdot \omega(MST(M))$. Moreover, there is a designated point $rt$ such that for every…

Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic "compactness'' measures of a Euclidean spanner $E$ are the size (number of edges) $|E|$ and the weight (sum of edge…

In STOC'95 \cite{ADMSS95} Arya et al.\ showed that for any set of $n$ points in $\mathbb R^d$, a $(1+\epsilon)$-spanner with diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges) can be built in $O(n…

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and…

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…

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in $\mathbb{R}^d$. In a recent breakthrough, Le and Solomon (2019)…

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 $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any $d =…

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…

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners…

Given a set $S$ of $n$ points in the plane and a parameter $\varepsilon>0$, a Euclidean $(1+\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that contains, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)\|pq\|$. We…

Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any $d$-dimensional $n$-point Euclidean space, there…

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…

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as…

Recently Elkin and Solomon gave a construction of spanners for doubling metrics that has constant maximum degree, hop-diameter O(log n) and lightness O(log n) (i.e., weight O(log n)w(MST). This resolves a long standing conjecture proposed…

A Euclidean noncrossing Steiner $(1+\epsilon)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+\epsilon$ times the Euclidean…

We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch <\sqrt{2} and subquadratic size are not possible, even if we add Steiner points. On the other hand, if we…

Lightness, sparsity, and hop-diameter are the fundamental parameters of geometric spanners. Arya et al. [STOC'95] showed in their seminal work that there exists a construction of Euclidean $(1+\varepsilon)$-spanners with hop-diameter…

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…