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 $Δ\geq Ω(\sqrt{n})$ is the \emph{spread} of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness $\tilde{O}(\varepsilon^{-(d+1)/2})$ in dimensions $d\geq 3$. Recently, Bhore and Tóth (2020) established a lower bound of $Ω(\varepsilon^{-d/2})$ for the lightness of Steiner $(1+\varepsilon)$-spanners in $\mathbb{R}^d$, for $d\ge 2$. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions $d\geq 2$.
In this work, we show that for every finite set of points in the plane and every $\varepsilon>0$, 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.
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…
Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Aalto University（阿尔托大学）；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）；Max Planck Institute for Informatics（马克斯·普朗克计算机科学研究所）；University of Copenhagen（哥本哈根大学）
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 describe an algorithm that takes as input n points in the plane and a parameter ε, and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + ε)-approximation to the geometric distances between the given points. For point sets in which the Delaunay triangulation has bounded sharpest angle, our algorithm's output has O(n) vertices, its weight is O(1) times the minimum spanning tree weight, and the algorithm's running time is bounded by O(n \sqrt{log log n}). We use this result in a similarly fast deterministic a…
Given a set P of points in the plane, an Euclidean t-spanner for P is a geometric graph that preserves the Euclidean distances between every pair of points in P up to a constant factor t. The weight of a geometric graph refers to the total length of its edges. In this paper we show that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, of weight at most w is NP-hard for every real constant t > 1, both whether planarity of the t-spanner is required or not.
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-…
Let $S$ be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane $\frac{3+4π}{3}$-spanner of $S$ whose vertex degree is at most 3. Let $Λ$ be the vertex set of a finite non-uniform rectangular lattice in the plane. We present an algorithm that constructs a plane $3\sqrt{2}$-spanner for $Λ$ whose vertex degree is at most 3. For points that are in the plane and in general position, we show how to compute plane degree-3 spanners with a linear number of Steiner points.
We show that the greedy spanner algorithm constructs a $(1+ε)$-spanner of weight $ε^{-O(d)}w(\mathrm{MST})$ for a point set in metrics of doubling dimension $d$, resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in $d$-dimension Euclidean space has a $(1+ε)$-spanner of weight at most $ε^{-O(d)}w(\mathrm{MST})$. Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.
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…
A geometric $t$-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most $t$ times the Euclidean distance between the points. Informally, a spanner is $\mathcal{O}(k)$-robust if deleting $k$ vertices only harms $\mathcal{O}(k)$ other vertices. We show that on any one-dimensional set of $n$ points, for any $\varepsilon>0$, there exists an $\mathcal{O}(k)$-robust $1$-spanner with $\mathcal{O}(n^{1+\varepsilon})$ edges. Previously it was only known that $\mathcal{O}(k)$-robust spanners with $\mathcal{O}(n^2)$ edges exists and that there…
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…
Given two sets of points in the plane, $P$ of $n$ terminals and $S$ of $m$ Steiner points, a Steiner tree of $P$ is a tree spanning all points of $P$ and some (or none or all) points of $S$. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, $P$ and $S$, and a positive integer $k \le m$, find a bottleneck Steiner tree of $P$ with at most $k$ Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP…
Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane s…
Let $P$ and $S$ be two disjoint sets of $n$ and $m$ points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to $S$, in which each point of $P$ is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in $O((n+m)\log m)$ time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.
For any constants $d\ge 1$, $ε>0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$, there exists $F^+\supseteq F$, $|F^+| \le (1+ε)|F|$ such that, for any pair $p,q\in P\setminus F^+$, the graph $G-F$ contains a path from $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclidean distance between $p$ and $q$. In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736, 2013) the graph $G$ is a $(1+ε)k$-robust $t$-spanner o…
Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $kρ$-approximation algorithm is known, where $ρ\le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$.
Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space $\mathbb{R}^d$ to the $1$-dimensional line. They used LSO's to solve a host of problems. Later, Buchin, Har-Peled, and Ol{á}h [2019,2020] used the LSO of Chan {\em et al. } to construct very sparse \emph{reliable spanners} for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90\% of the nodes fail. In a follow-up work, Har-Peled, Men…
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 Ge…
Given a metric space $\mathcal{M}=(X,δ)$, a weighted graph $G$ over $X$ is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$, $δ(u,v)\le d_G(u,v)\le t\cdot δ(u,v)$, where $d_G$ is the shortest path metric in $G$. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one at a time (i.e., after $i$ steps, we saw $S_i = \{s_1, \ldots , s_i\}$). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to re…
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 …
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 …
We give a randomized O(n polylog n)-time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed eps > 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1 + eps)-approximation to the minimum-length forest that connects every requested pair of terminals.
Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S, E) is called a t-spanner for the additive weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.d_w(p, q) for a real number t > 1. Here, d_w(p,q) is the additive weighted distance between p and q. For so…
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 degree O(k), diameter O(log_k n + alpha(k)), weight O(k \cdot log_k n \cdot log n) \cdot w(MST(S)), and O(n) edges. Note that for k= n^{1/alpha(n)} this gives rise to diameter O(alpha(n)), weight O(n^{1/alpha(n)} \cdot log n \cdot alpha(n)) \cdot w(MST(S)) and maximum degree O(n^{1/alpha(n)}), which improves upon a classical result of Arya et al. \cite{ADMSS95}…
It has long been known that $d$-dimensional Euclidean point sets admit $(1+ε)$-stretch spanners with lightness $W_E = ε^{-O(d)}$, that is total edge weight at most $W_E$ times the weight of the minimum spaning tree of the set [DHN93]. Whether or not a similar result holds for metric spaces with low doubling dimension has remained an important open problem, and has resisted numerous attempts at resolution. In this paper, we resolve the question in the affirmative, and show that doubling spaces admit $(1+ε)$-stretch spanners with lightness $W_D = (ddim/ε)^{O(ddim)}$. Important in its own right, …
Given the coordinates of four terminals in the Euclidean plane we present explicit formulas for Steiner point coordinates for Steiner minimal tree problem. We utilize the obtained formulas for evaluation of the influence of terminal coordinates on the loci of Steiner points.
$t$-spanners are used to approximate the pairwise distances between a set of points in a metric space. They have only a few edges compared to the total number of pairs and they provide a $t$-approximation on the distance of any two arbitrary points. There are many ways to construct such graphs and one of the most efficient ones, in terms of weight and the number of edges of the resulting graph, is the greedy spanner. In this paper, we study the edge crossings of the greedy spanner for points in the Euclidean plane. We prove a constant upper bound for the number of intersections with larger edg…
Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer k \ge 1, and a real number ε> 0, we present algorithms for computing a spanner network G(S, E) for the metric space (S, d_w) induced by the weighted points in S. The weighted distance function d_w on the set S of points is defined as follows: for any p, q \in S, d_w(p, q) is equal to w(p) + d_π(p, q) + w(q) if p \ne q, otherwise, d_w(p, q) is 0. Here, d_π(p, q) is the Euclidean distance between p and q if points in S are in \mathbb{R}^d, otherwise, it is the geodesic (Euclid…
Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance difference of at most $2n-4$, where $n$ is the number of terminals. We show that this difference is best possible in the Euclidean plane, but not in Minkowski planes with parallelogram unit balls. We also intro…
The unit disk graph (UDG) is a widely employed model for the study of wireless networks. In this model, wireless nodes are represented by points in the plane and there is an edge between two points if and only if their Euclidean distance is at most one. A hop spanner for the UDG is a spanning subgraph $H$ such that for every edge $(p,q)$ in the UDG the topological shortest path between $p$ and $q$ in $H$ has a constant number of edges. The hop stretch factor of $H$ is the maximum number of edges of these paths. A hop spanner is plane (i.e. embedded planar) if its edges do not cross each other.…
A spanner graph on a set of points in $R^d$ contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. In this paper we investigate new models and aim to interpret why good spanners 'emerge' in reality, when they are clearly built in pieces by agents with their own interests and the construction is not coordinated. Our main result is to show that if edges are built in an arbitrary order but an edge is built if and only if its endpoints are not 'close' to the endpoints of an existing edge, the graph is a $(1 + \eps)$-spanner with a lin…
Given a real number $t>1$, a geometric $t$-spanner is a geometric graph for a point set in $\mathbb{R}^d$ with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most $t$. An imprecise point set is modeled by a set $R$ of regions in $\mathbb{R}^d$. If one chooses a point in each region of $R$, then the resulting point set is called a precise instance of~$R$. An imprecise $t$-spanner for an imprecise point set $R$ is a graph $G=(R,E)$ such that for ea…
Given N points in the plane $P_1 P_2...P_N$ and a location $Ω$, the union of discs with diameters $[ΩP_i], i = 1, 2,...N$ covers the convex hull of the points. The location $Ω_s$ minimizing the area covered by the union of discs, is shown to be the Steiner center of the convex hull of the points. Similar results for $d$-dimensional Euclidean space are conjectured.
Let $P \subset \mathbb{R}^2$ be a planar $n$-point set such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) \leq r_p$. Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H \subseteq G$ with vertex set $P$ so that for any two vertices $p,q \in P$, we have $d_H(p, q) \leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths). We sho…
