An Optimal Algorithm for Computing Angle-Constrained Spanners

Carmi, Paz; Smid, Michiel

doi:10.1007/978-3-642-17517-6_29

Cited by 9 publications

(7 citation statements)

References 20 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is somewhat surprising, since many variations on sparse geometric spanners have been studied, including spanners of low degree [6,19,36], spanners of low weight [14,24,26], spanners of low diameter [8,9], planar spanners [5,21,23,29], spanners of low chromatic number [13], fault-tolerant spanners [2,22,31,32], low-power spanners [4,34,37], kinetic spanners [1,3], angle-constrained spanners [20], and combinations of these [7,10,15,16,17,18]. The closest related work is that on fault-tolerant spanners [2,22,31,32], but r-fault-tolerance is analogous to the traditional definition of r-connectivity in graph theory and suffers the same shortcoming: every r-fault-tolerant spanner has Ω(rn) edges.…”

Section: Introductionmentioning

confidence: 99%

Robust geometric spanners

Bose

Dujmović

Morin

et al. 2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Robust geometric spanners

Bose

Dujmović

Morin

et al. 2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…The first spanner (the roads), denoted by G 2 = (V , E 2 ), is a 2-spanner of V , and has O(n) edges [8,18,20]. The next spanner (the highways), denoted by G 1 = (H, E 1 ), is a (1 + 1/k 1/(d−1) )spanner of H, and has O(k|H|) = O(n) edges [10,17].…”

Section: The Graph Gmentioning

confidence: 99%

“…The first spanner (the roads), denoted by G 2 = (V , E 2 ), is a 2-spanner of V , and has O(n) edges [8,18,20]. The next spanner (the highways), denoted by [10,17].…”

Section: The Graph Gmentioning

confidence: 99%

Average Stretch Factor: How Low Does It Go?

Dujmović

Morin

Smid

2015

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

Abstract. In a geometric graph, G, the stretch factor between two vertices, u and w, is the ratio between the Euclidean length of the shortest path from u to w in G and the Euclidean distance between u and w. The average stretch factor of G is the average stretch factor taken over all pairs of vertices in G. We show that, for any constant dimension, d, and any set, V , of n points in R d , there exists a geometric graph with vertex set V , that has O(n) edges, and that has average stretch factor 1 + o n (1). More precisely, the average stretch factor of this graph is 1 + O((log n/n) 1/(2d+1) ). We complement this upper-bound with a lower bound: There exist n-point sets in R 2 for which any graph with O(n) edges has average stretch factor 1 + Ω(1/ √ n). Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exists point sets, V , such that any graph with worst-case stretch factor 1 + o n (1) has a superlinear number of edges.

show abstract

“…This is the first paper to consider combining low spanning ratio with high global connectivity. This is somewhat surprising, since many variations on sparse geometric spanners have been studied, including spanners of low degree [6,19,36], spanners of low weight [14,24,26], spanners of low diameter [8,9], planar spanners [5,21,23,29], spanners of low chromatic number [13], fault-tolerant spanners [2,22,31,32], lowpower spanners [4,34,37], kinetic spanners [1,3], angle-constrained spanners [20], and combinations of these [7,10,15,16,17,18]. The closest related work is that on faulttolerant spanners [2,22,31,32], but r-fault-tolerance is analogous to the traditional definition of r-connectivity in graph theory and suffers the same shortcoming: every r-fault-tolerant spanner has Ω(rn) edges.…”

Section: Introductionmentioning

confidence: 99%

Robust Geometric Spanners

Bose¹,

Dujmović²,

Morin³

et al. 2013

SIAM J. Comput.

Self Cite

View full text Add to dashboard Cite

Abstract. 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.Key words. spanners, stretch-factor, spanning-ratio, treewidth, connectivity, expansion AMS subject classifications. 68M10, 05C10, 65D18 DOI. 10.1137/1208744731. Introduction. The cost of building a network, such as a computer network or a network of roads, is closely related to the number of edges in the underlying graph that models this network. This gives rise to the requirement that this graph be sparse. However, sparseness typically has to be counterbalanced with other desirable graph (that is, network design) properties such as reliability and efficiency.The classical notion of graph connectivity provides some guarantee of reliability. In particular, an r-connected graph remains connected as long as fewer than r vertices are removed. However, these graphs are not sparse for large values of r; an r-connected graph with n vertices has at least rn/2 edges.For many applications, disconnecting a small number of nodes from the network is an inconvenience for the nodes that are disconnected but has little effect on the rest of the network. In contrast, disconnecting a large part (say, a constant fraction) of the network from the rest is catastrophic. For example, it may be tolerable that the failure of one network component cuts off internet access for the residents of a small village. However, the failure of a single component that eliminates all communications between North America and Europe would be disastrous.This global notion of connectivity is captured in graph theory by expanders and graphs of high treewidth, each of which can have a linear number of edges. These two properties of graphs have an enormous number of applications and have been the subject of intensive research for decades. See, for example, the book by Kloks [30] or the surveys by Bodlaender [11,12] on treewidth and the survey by Hoory, Linial, and Wigderson [27] on expanders.In this paper, we consider how to combine this global notion of connectivity with another desirable property of geometric graphs: low spanning ratio (a.k.a., low stretch factor or low dilation), the property of approximately preserving Euclidean distances

show abstract

An Optimal Algorithm for Computing Angle-Constrained Spanners

Cited by 9 publications

References 20 publications

Robust geometric spanners

Robust geometric spanners

Average Stretch Factor: How Low Does It Go?

Robust Geometric Spanners

Contact Info

Product

Resources

About