Improved Algorithms for Constructing Fault-Tolerant Spanners

Levcopoulos, Christos; Narasimhan, Giri; Smid, Michiel

doi:10.1007/s00453-001-0075-x

Cited by 55 publications

(55 citation statements)

References 18 publications

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“…In [12], Callahan and Kosaraju showed that for any point set in a Euclidean space and for any positive constant s, there always exists an s-well-separated pair decomposition with linearly many pairs. This fact has been very useful in obtaining nearly linear time algorithms for many problems such as computing k-nearest neighbors, n-body potential fields, geometric spanners and approximate minimum spanning trees [9,10,12,11,6,2,30,27,19,15].…”

Section: Specific Contributionsmentioning

confidence: 99%

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Deformable spanners and applications

Gao

Guibas

Nguyẽ̂n

2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

152

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For a set S of points in R d , an s-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d ) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers.

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Section: Specific Contributionsmentioning

confidence: 99%

“…In fact, many of the spanner constructions for points in Euclidean space use the well-separated pair decomposition as a tool [6,2,30,27]. The basic idea is this: the graph defined by taking an arbitrary edge connecting each s-well-separated pair must be a spanner [9].…”

Section: Specific Contributionsmentioning

confidence: 99%

Deformable spanners and applications

Gao

Guibas

Nguyẽ̂n

2004

Proceedings of the Twentieth Annual Symposium on Computational Geometry

152

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“…Similarly, Czumaj and Lingas [7] showed approximation schemes for minimum-cost multiconnectivity problems in geometric graphs. The problem of constructing geometric spanners has received considerable attention from a theoretical perspective; see [1,3,4,5,8,9,10,17,20,21,23,24,33,36], the surveys [12,16,34], and the book by Narasimhan and Smid [28]. Note that considerable research has also been done in the construction of spanners for general graphs; see, for example, the book by Peleg [31] or the recent work by Elkin and Peleg [11] and Thorup and Zwick [35].…”

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confidence: 99%

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Farshi¹,

Giannopoulos²,

Gudmundsson³

2008

SIAM J. Comput.

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Abstract. Given a Euclidean graph G in R d with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm + n 2 log n) time, resulting in a trivial O(n 3 m + n 4 log n)-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running time we consider several approximation algorithms.Key words. computational geometry, approximation algorithms, geometric networks AMS subject classifications. 65D18, 68U05, 68Q25 DOI. 10.1137/050635675 1. Introduction. Consider a set V of n points in R d . A network on V can be modeled as an undirected graph G with vertex set V of size n and an edge set E of size m where every edge (u, v) has a positive weight w (u, v). A Euclidean network is a geometric network where the weight of the edge (u, v) is equal to the Euclidean distance |uv| between its two endpoints u and v.For two vertices u, v in a weighted graph G we use δ G (u, v) to denote a shortest path between u and v in G, and the length of the path is denoted by d G (u, v). Consider a weighted graph G = (V, E) and a graph G = (V, E ) on the same vertex set but with edge set E ⊆ E. We say that G is a t-spanner of G if for each pair of vertices (u, v). The minimum t such that G is a t-spanner for V is called the stretch factor, or dilation, of G.We say that a Euclidean network G = (V, E) is a t-spanner if G = (V, E) is a t-spanner of the complete network on V . In other words, for any two points p, q ∈ V the graph distance in G is at most t times the Euclidean distance between the two points.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas. Recently spanners found interesting practical applications

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“…This immediately implies that any r-fault-tolerant spanner with n > r vertices has at least (r + 1)n/2 edges, since every vertex must have degree at least r + 1. Several constructions of r-fault-tolerant spanners with O(rn) edges exist [22,31,32].…”

Section: Robustness Versus Fault-tolerancementioning

confidence: 99%

“…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.

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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

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Improved Algorithms for Constructing Fault-Tolerant Spanners

Cited by 55 publications

References 18 publications

Deformable spanners and applications

Deformable spanners and applications

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Robust Geometric Spanners

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