We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $ω(T) = O(\log n) \cdot ω(MST(M))$. Moreover, there is a designated point $rt$ such that for every point $v$, $dist_T(rt,v) \le (1+ε) \cdot dist_M(rt,v)$, for an arbitrarily small constant $ε> 0$. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is \emph{tight up to constant factors} in the entire range of parameters. These results enable us to settle a long-standing open question in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter $O(\log n)$ and weight $O(\log n) \cdot ω(MST(M))$. Ten years later in SODA'05 Agarwal et al. showed that this result is tight up to a factor of $O(\log \log n)$. We close this gap and show that the result of Arya et al. is tight up to constant factors.
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 exists a $(1+ε)$-spanner with $nO(ε^{-d+1})$ edges and lightness $O(ε^{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $ε$ and $d$ for small $d$ can be improved has remained elusive, even for $d = 2$. This question naturally arises in any application of Euclidean spanners where precision is a necessity. The state-of-the-…
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 $O(\log n) \cdot ω(MST)$ can be built in $O(n \log n)$ time. Recently Elkin and Solomon (technical report, April 2012) proved this conjecture of Arya et al. in the affirmative. In fact, the proof of Elkin and Solomon is more general in two ways. First, it applies to arbitrary doubling metrics. Second, it provides a complete tradeoff between the three involved parameters that is tight (up to constant factors) in the…
In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(n\log n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(n\log^2 n)$-size 3-hop spanner for $n$ axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs? We address both questions simultaneously, by presenting new constructions of con…
In this paper, we study the online Euclidean spanners problem for points in $\mathbb{R}^d$. Suppose we are given a sequence of $n$ points $(s_1,s_2,\ldots, s_n)$ in $\mathbb{R}^d$, where point $s_i$ is presented in step~$i$ for $i=1,\ldots, n$. The objective of an online algorithm is to maintain a geometric $t$-spanner on $S_i=\{s_1,\ldots, s_i\}$ for each step~$i$. First, we establish a lower bound of $Ω(\varepsilon^{-1}\log n / \log \varepsilon^{-1})$ for the competitive ratio of any online $(1+\varepsilon)$-spanner algorithm, for a sequence of $n$ points in 1-dimension. We show that this bo…
In STOC'95 \cite{ADMSS95} Arya et al.\ showed that for any set of $n$ points in $\mathbb R^d$, a $(1+ε)$-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 $O(n \log n)$ time. Moreover, it was shown in \cite{ADMSS95,NS07} that for any $k \ge 4$, one can build in $O(n (\log n) 2^k α_k(n))$ time a $(1+ε)$-spanner with diameter at most $2k$ and $O(n 2^k α_k(n))$ edges. The function $α_k$ is the inverse of a certain function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $α_0(n) = \lceil n/2 …
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) established the precise dependencies on $\varepsilon>0$ and $d\in \mathbb{N}$ of the minimum lightness of $(1+\varepsilon)$-spanners, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner $(1+\varepsilon)$-spanners of lightness $O(\varepsilon^{-1}\logΔ)$ in the plane, where $Δ\geq Ω(\sqrt{n}…
A unit disk graph $G$ on a given set $P$ of points in the plane is a geometric graph where an edge exists between two points $p,q \in P$ if and only if $|pq| \leq 1$. A spanning subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for every edge $pq\in G$, there is a path between $p,q$ in $G'$ with at most $k$ edges. We obtain the following results for unit disk graphs in the plane. (I) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most $5.5n$ edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from $9n$ …
Let ${\cal P}$ be a set of $n$ points embedded in the plane, and let ${\cal C}$ be the complete Euclidean graph whose point-set is ${\cal P}$. Each edge in ${\cal C}$ between two points $p, q$ is realized as the line segment $[pq]$, and is assigned a weight equal to the Euclidean distance $|pq|$. In this paper, we show how to construct in $O(n\lg{n})$ time a plane spanner of ${\cal C}$ of maximum degree at most 4 and stretch factor at most 20. This improves a long sequence of results on the construction of plane spanners of ${\cal C}$. Our result matches the smallest known upper bound of 4 by …
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 substant…
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 $S$. Tight bounds on the dependence on $\varepsilon>0$ for constant $d\in \mathbb{N}$ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a $(1+\varepsilon)$-spa…
In this thesis, we study two different graph problems. The first problem revolves around geometric spanners. Here, we have a set of points in the plane and we want to connect them with straight line segments, such that there is a path between each pair of points that does not make a large detour. If we achieve this, the resulting graph is called a spanner. We focus our attention on $Θ$-graphs, which are constructed by connecting each point with its nearest neighbour in a fixed number of cones. Although this construction is very straight-forward, it has proven challenging to fully determine the…
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. O…
An emanation graph of grade $k$ on a set of points is a plane spanner made by shooting $2^{k+1}$ equally spaced rays from each point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. Emanation graphs of grade one were studied by Mondal and Nachmanson in the context of network visualization. They proved that the spanning ratio of such a graph is bounded by $(2+\sqrt{2})\approx 3.414$. We improve this upper bound to $\sqrt{10} \approx 3.162$ and show this to be tight, i.e., there exist emanation graphs with spanning ratio $\s…
We present tight bounds on the spanning ratio of a large family of ordered $θ$-graphs. A $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$. An ordered $θ$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \geq 1$, ordered $θ$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(θ/2) / (\cos(θ/2) - \sin(θ/2))$. We also show that for any integer $k \geq 2$, ordered $θ$-graphs with $4k + 2$ cones have a…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Tel Aviv University（特拉维夫大学）；California State University Northridge（北岭加州州立大学）；Tufts University（塔夫茨大学）；State Key Laboratory for Novel Software Technology, Nanjing University（南京大学软件新技术国家重点实验室）
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $\V$ be a finite set system defined over an $n$-point set $X$; we view $\V$ as a set of indicator vectors over the $n$-dimensional unit cube. A $δ$-separated set of $\V$ is a subcollection $\W$, s.t. the Hamming distance between each pair $\uu, \vv \in \W$ is greater than $δ$, where $δ> 0$ is an integer parameter. The $δ$-packing number is then defined as the cardinality of the largest $δ$-separated subcollection of $\V$. Haussler showed an asymptotically tight bound of…
