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 $9n$ to $5.5n$.
(II) Using a new construction, we show that every $n$-vertex unit disk graph has a $3$-hop spanner with at most $11n$ edges.
(III) Every $n$-vertex unit disk graph has a $2$-hop spanner with $O(n\log n)$ edges. This is the first nontrivial construction of $2$-hop spanners.
(IV) For every sufficiently large positive integer $n$, there exists a set $P$ of $n$ points on a circle, such that every plane hop spanner on $P$ has hop stretch factor at least $4$. Previously, no lower bound greater than $2$ was known.
(V) For every finite point set on a circle, there exists a plane (i.e., crossing-free) $4$-hop spanner. As such, this provides a tight bound for points on a circle.
(VI) The maximum degree of $k$-hop spanners cannot be bounded from above by a function of $k$ for any positive integer $k$.
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…
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.…
We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.
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 means of compressing the $\binom{n}{2}$ pairwise distances of a $d$-dimensional Euclidean space into $O(n) = O_{ε,d}(n)$ spanner edges, so that the spanner distances preserve the original distances to within a factor of $1+ε$, for any $ε> 0$. Moreover, one can compute such spanners in optimal $O(n \log n)$ time. Once the spanner has been computed, it serves…
It is a standing open question to decide whether the Yao-Yao structure for unit disk graphs (UDGs) is a length spanner of not. This question is highly relevant to the topology control problem for wireless ad hoc networks. In this paper we make progress towards resolving this question by showing that the Yao-Yao structure is a length spanner for UDGs of bounded aspect ratio. We also propose a new local algorithm, called Yao-Sparse-Sink, based on the Yao-Sink method introduced by Li, Wan, Wang and Frieder, that computes a (1+e)-spanner of bounded degree for a given UDG and for given e > 0. The Y…
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…
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…
$\newcommand{\Emph}[1]{\it{#1}} \newcommand{\FF}{\mathcal{F}}\newcommand{\region}{\mathsf{r}}\newcommand{\restrictY}[2]{#1 \cap {#2}}$For a set of points $P \subseteq \mathbb{R}^2$, and a family of regions $\FF$, a $\Emph{local~t-spanner}$ of $P$, is a sparse graph $G$ over $P$, such that, for any region $\region \in \FF$, the subgraph restricted to $\region$, denoted by $\restrictY{G}{\region} = G_{P \cap \region}$, is a $t$-spanner for all the points of $\region \cap P$. We present algorithms for the construction of local spanners with respect to several families of regions, such as homothet…
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…
Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robus…
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 …
It is an open problem whether Yao-Yao graphs $\mathsf{YY}_k$ (also known as sparse-Yao graphs) are all spanners when the integer parameter $k$ is large enough. In this paper we show that, for any integer $k\geq 42$, the Yao-Yao graph $\mathsf{YY}_{2k}$ is a $t_k$-spanner, with stretch factor $t_k=6.03+O(k^{-1})$ when $k$ tends to infinity. Our result generalizes the best known result which asserts that all $\mathsf{YY}_{6k}$ are spanners for $k$ large enough [Bauer and Damian, SODA'13]. Our proof is also somewhat simpler.
Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph $G$ with $n$ vertices and a positive constant $ε< 1$ finds a $(1+ε)$-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju, which r…
