Solomon and Elkin constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight $(1+ε)$-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after $O(\log n)$ hops and requires $O(Δ\log n)$ bits of storage per vertex where $Δ$ is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.
We introduce a family of directed geometric graphs, denoted $\paz$, that depend on two parameters $λ$ and $θ$. For $0\leq θ<\fracπ{2}$ and ${1/2} < λ< 1$, the $\paz$ graph is a strong $t$-spanner, with $t=\frac{1}{(1-λ)\cosθ}$. The out-degree of a node in the $\paz$ graph is at most $\lfloor2π/\min(θ, \arccos\frac{1}{2λ})\rfloor$. Moreover, we show that routing can be achieved locally on $\paz$. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $λ$ and $θ$ indicate that for random point sets, the spanning r…
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…
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…
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…
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…
In this paper we devise an optimal construction of fault-tolerant spanners for doubling metrics. Specifically, for any $n$-point doubling metric, any $\eps > 0$, and any integer $0 \le k \le n-2$, our construction provides a $k$-fault-tolerant $(1+\eps)$-spanner with optimal degree $O(k)$ within optimal time $O(n \log n + k n)$. We then strengthen this result to provide near-optimal (up to a factor of $\log k$) guarantees on the diameter and weight of our spanners, namely, diameter $O(\log n)$ and weight $O(k^2 + k \log n) \cdot ω(MST)$, while preserving the optimal guarantees on the degree $O…
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…
Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+ε)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating pro…
A spanner is reliable if it can withstand large, catastrophic failures in the network. More precisely, any failure of some nodes can only cause a small damage in the remaining graph in terms of the dilation, that is, the spanner property is maintained for almost all nodes in the residual graph. Constructions of reliable spanners of near linear size are known in the low-dimensional Euclidean settings. Here, we present new constructions of reliable spanners for planar graphs, trees and (general) metric spaces.
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, …
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}…
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…
A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$θ_6$-graph is general…
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $Ω(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tre…
