Geometric Spanners Research Articles

Bounded-Degree Plane Geometric Spanners in Practice

The construction of bounded-degree plane geometric spanners has been a focus of interest since 2002 when Bose, Gudmundsson, and Smid proposed the first algorithm to construct such spanners. To date, 11 algorithms have been designed with various tradeoffs in degree and stretch-factor. We have implemented these sophisticated spanner algorithms in C ++ using the CGAL library and experimented with them using large synthetic and real-world pointsets. Our experiments have revealed their practical behavior and real-world efficacy. We share the implementations via GitHub for broader uses and future research. We design and engineer EstimateStretchFactor , a simple practical algorithm, which can estimate stretch-factors (obtains lower bounds on the exact stretch-factors) of geometric spanners—a challenging problem for which no practical algorithm is known yet. In our experiments with bounded-degree plane geometric spanners, we found that EstimateStretchFactor estimated stretch-factors almost precisely. Further, it gave linear runtime performance in practice for the pointset distributions considered in this work, making it much faster than the naive Dijkstra-based algorithm for calculating stretch-factors.

Emanation Graph: A Plane Geometric Spanner with Steiner Points

Emanation Graph: A Plane Geometric Spanner with Steiner Points

On path-greedy geometric spanners

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose edges are line segments between the corresponding points. Geometric t-spanners are t-spanners of the complete geometric graph on a given point set. Besides approximating the distance between points, we may ask a geometric t-spanner to be planar, have low degree, or low total edge length.One famous algorithm used to generate spanners is path-greedy, which scans pairs of vertices in non-decreasing order of edge length and adds the edge between them unless the current set of added edges already connects them with a path that t-approximates the edge length. Graphs from this algorithm are called path-greedy spanners. This work analyzes properties of path-greedy geometric spanners under different conditions. Specifically, we answer an open problem regarding the planarity and degree of path-greedy t-spanners for points in convex position in 2D. Further, we show a simple and efficient way to reduce the degree of a geometric spanner by adding extra points.

Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing \(\Gamma \) of G in the plane such that, for any two vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is the spanning ratio of \(\Gamma \). First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar straight-line drawing with spanning ratio 1 are NP-complete, \(\exists \mathbb R\)-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 1 to spanning ratio \(1+\epsilon \) allows us to draw every graph. Namely, we prove that, for every \(\epsilon >0\), every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than \(1+\epsilon \). Third, our drawings with spanning ratio smaller than \(1+\epsilon \) have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

On the spanning and routing ratios of the directed Θ6-graph

The family of Θk-graphs is an important class of sparse geometric spanners with a small spanning ratio. Although they are a well-studied class of geometric graphs, no bound is known on the spanning and routing ratios of the directed Θ6-graph. We show that the directed Θ6-graph of a point set P, denoted Θ→6(P), is a 7-spanner and there exist point sets where the spanning ratio is at least 4−ε, for any ε>0. It is known that the standard greedy Θ-routing algorithm may have an unbounded routing ratio on Θ→6(P). We design a simple, online, local, memoryless routing algorithm on Θ→6(P) whose routing ratio is at most 8 and show that no algorithm can have a routing ratio better than 6−ε.

Local Geometric Spanners

Local Geometric Spanners

δ-Greedy t-spanner

We introduce a new geometric spanner, δ-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The δ-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong (1+ε)-spanner, for every ε>0. The δ-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of n points in the plane in O(n2log⁡n) time.The δ-Greedy spanner has an additional parameter δ, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For δ=t, the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear.Finally, we show that, for a set of n points placed independently at random in a unit square, the expected construction time of the δ-Greedy algorithm is O(nlog⁡n). Our analysis indicates that the δ-Greedy spanner gives the best results among the known spanners of expected O(nlog⁡n) time for random point sets. Moreover, analysis implies that by setting δ=t, the δ-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected O(nlog⁡n) time.

Geometric spanner games

Consider a scenario in which several agents are located in the Euclidean space, and the agents want to create a network in which everyone has fast access to all or some other agents. Geometric t-spanners are examples of such a network providing fast connections between the nodes of the network for some fixed value t, i.e. the length of the shortest path between any two nodes in the network is at most t times their Euclidean distance. Geometric t-spanners have been extensively studied in the area of computational geometry where they are created by a central authority. In this paper, we investigate a situation in which selfish agents want to create such a network in the absence of a central authority. To this end, we introduce two different non-cooperative games: the t-spanner game and the sink t-spanner game, where it is vital for the agents to have a fast connection to all other agents or to a special node, respectively. We study the existence of the Nash equilibrium in both games and present some structural properties of the resulting networks in the Nash equilibrium.

Continuous Yao graphs

In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points S⊂R2 and an angle 0<θ⩽2π, we define the continuous Yao graphcY(θ) with vertex set S and angle θ as follows. For each p,q∈S, we add an edge from p to q in cY(θ) if there exists a cone with apex p and aperture θ such that q is a closest point to p inside this cone.We study the spanning ratio of cY(θ) for different values of θ. Using a new algebraic technique, we show that cY(θ) is a spanner when θ⩽2π/3. We believe that this technique may be of independent interest. We also show that cY(π) is not a spanner, and that cY(θ) may be disconnected for θ>π, but on the other hand is always connected for θ⩽π. Furthermore, we show that cY(θ) is a region-fault-tolerant geometric spanner for convex fault regions when θ<π/3. For half-plane faults, cY(θ) remains connected if θ⩽π. Finally, we show that cY(θ) is not always self-approaching for any value of θ.

Lattice spanners of low degree

Let [Formula: see text] denote the degree [Formula: see text] dilation of a point set [Formula: see text] in the domain of plane geometric spanners. If [Formula: see text] is the infinite square lattice, it is shown that [Formula: see text] and [Formula: see text]. If [Formula: see text] is the infinite hexagonal lattice, it is shown that [Formula: see text] and [Formula: see text]. All our constructions are planar lattice tilings constrained to degree [Formula: see text] or [Formula: see text].

Lower Bounds on the Dilation of Plane Spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least [Formula: see text], improving the previous best lower bound of [Formula: see text] for the worst-case dilation of plane spanners. (II) For every [Formula: see text], there exists an [Formula: see text]-element point set [Formula: see text] such that the degree [Formula: see text] dilation of [Formula: see text] equals [Formula: see text] in the domain of plane geometric spanners. In the same domain, we show that for every [Formula: see text], there exists a an [Formula: see text]-element point set [Formula: see text] such that the degree [Formula: see text] dilation of [Formula: see text] equals [Formula: see text] The previous best lower bound of [Formula: see text] holds for any degree. (III) For every [Formula: see text], there exists an [Formula: see text]-element point set [Formula: see text] such that the stretch factor of the greedy triangulation of [Formula: see text] is at least [Formula: see text].

Distribution-Sensitive Construction of the Greedy Spanner

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take varOmega (n^2) time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an O(n log ^2 n log ^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an O((n + |E|) log ^2 n log log n) expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.

A lower bound for computing geometric spanners

It is known that the problem of computing (Steiner) spanners on a set of n points has an Ω(nlog⁡n) lower bound. However, the proof is based on an example of points on the real line. Therefore, if we assume that the points belong to the plane or higher dimensions, and moreover, they are in general position, then the lower bound example does not work.In this paper, we show that the complexity of computing geometric spanners, possibly containing Steiner points, for a set of n points in d-dimensional Euclidean space (Rd) that are in general position is Ω(nlog⁡n), in the algebraic computation tree model. To this end, we reduce the spanner construction to a variant of the closest pair problem which has an Ω(nlog⁡n) lower bound.

Optimal Euclidean Spanners

The degree, the (hop-)diameter, and the weight are the most basic and well-studied parameters of geometric spanners. In a seminal STOC'95 paper, titled “Euclidean spanners: short, thin and lanky”, Arya et al. [1995] devised a construction of Euclidean (1+ε)-spanners that achieves constant degree, diameter O (log n ), weight O (log 2 n ) ċ ω( MST ), and has running time O ( n ċ log n ). This construction applies to n -point constant-dimensional Euclidean spaces. Moreover, Arya et al. conjectured that the weight bound can be improved by a logarithmic factor, without increasing the degree and the diameter of the spanner, and within the same running time. This conjecture of Arya et al. became one of the most central open problems in the area of Euclidean spanners. Nevertheless, the only progress since 1995 towards its resolution was achieved in the lower bounds front: Any spanner with diameter O (log n ) must incur weight Ω(log n ) ċ ω( MST ), and this lower bound holds regardless of the stretch or the degree of the spanner [Dinitz et al. 2008; Agarwal et al. 2005]. In this article we resolve the long-standing conjecture of Arya et al. in the affirmative. We present a spanner construction with the same stretch, degree, diameter, and running time, as in Arya et al.'s result, but with optimal weight O (log n ) ċ ω( MST ). So our spanners are as thin and lanky as those of Arya et al., but they are really short! Moreover, our result is more general in three ways. First, we demonstrate that the conjecture holds true not only in constant-dimensional Euclidean spaces, but also in doubling metrics . Second, we provide a general trade-off between the three involved parameters, which is tight in the entire range . Third, we devise a transformation that decreases the lightness of spanners in general metrics , while keeping all their other parameters in check. Our main result is obtained as a corollary of this transformation.

Geometric Spanning Cycles in Bichromatic Point Sets

Given a set of points in the plane each colored either red or blue, we find non-self-intersecting geometric spanning cycles of the red points and of the blue points such that each edge of the red spanning cycle is crossed at most three times by the blue spanning cycle and vice-versa.

Introduction to the Special Issue: Combinatorics and Geometry

This special issue is an outgrowth of the Mexican Conference on Combinatorics and Geometry held in Oaxaca, Mexico, on November 11–15, 2013 to celebrate Jorge Urrutia’s 60th birthday. Jorge is one of those rare individuals endowedwith awonderful mathematical sense of creativity. All he needs is a pencil and paper but is always willing to help with a proof, always available to interact anytime and anywhere, always tireless to elaborate on a new mathematical paradigm, and always delighted and excited when a new idea leads to a beautiful new theorem. For many of us Jorge has been not only a teacher, a friend, and a research collaborator but also an inspirational figure whose love for mathematics has been a model for making the subject approachable and lively and for contributing to a deeper understanding of its beautiful, undiscovered secrets. As Isaac Newton said “I keep the subject constantly before me and wait till the first dawnings open little by little into full light”. Most of the papers included in this issue were presented at the conference and were all rigorously refereed upon submission to this special issue. They include a variety of interesting topics reflecting the varied interests of the participants: Reversibility of convex polygons, Trapping disks by four lines, Extremal antipodal polygons,Monitoring the Plane with Rotating Radars, Geometric biplane graphs, Geometric Spanning Cycles in Bichromatic Point Sets, Cross-Intersecting Families of Vectors, Empty Triangles in Good Drawings of the Complete Graph, and Isoperimetric enclosures. We hope you will enjoy reading and learn from them as much beautiful mathematics as we did.

The [formula omitted]-graph is a spanner

Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most 50+225≈9.960. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most 50+225 times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of 12(115−17)≈3.798.

FLOC-SPANNER: An O(1) Time, Locally Self-Stabilizing Algorithm for Geometric Spanner Construction in a Wireless Sensor Network

We present a distributed algorithm for creation of geometric spanners in a wireless sensor network. Given any connected network, we show that the algorithm terminates in [Formula: see text] time, irrespective of network size. Our algorithm uses an underlying clustering algorithm as a foundation for creating spanners and only relies on the periodic heartbeat messages associated with cluster maintenance for the creation of the spanners. The algorithm is also shown to stabilize locally in the presence of node additions and deletions. The performance of our algorithm is verified using large scale simulations. The average path length ratio for routing along the spanner for large networks is shown to be less than 2.

On plane geometric spanners: A survey and open problems

Given a weighted graph G=(V,E) and a real number t⩾1, a t-spanner of G is a spanning subgraph G′ with the property that for every edge xy in G, there exists a path between x and y in G′ whose weight is no more than t times the weight of the edge xy. We review results and present open problems on different variants of the problem of constructing plane geometric t-spanners.

Robust Geometric Spanners

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 robust spanners with a near-linear number of edges.