Fault-Tolerant Geometric Spanners

Czumaj, Artur; Zhao, Hairong

doi:10.1007/s00454-004-1121-7

Cited by 39 publications

(14 citation statements)

References 27 publications

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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%

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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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Section: Robustness Versus Fault-tolerancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust Geometric Spanners

Bose¹,

Dujmović²,

Morin³

et al. 2013

SIAM J. Comput.

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“…That is, given a set S of n points in R d , their algorithm finds a sparse graph H such that for every set F ⊆ S of size f and any pair of points u, v ∈ S \ F , the distances in H satisfy δ H\F (u, v) ≤ (1 + )|uv|, where |uv| is the Euclidean distance between u and v. A fault tolerant geometric spanner of improved size was presented by Lukovszki [21]. Finally, Czumaj and Zhao [8] presented a fault tolerant geometric spanner with optimal maximum degree and total weight. In [8] they raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary weighted undirected graph.…”

Section: Background and Previous Workmentioning

confidence: 99%

“…Finally, Czumaj and Zhao [8] presented a fault tolerant geometric spanner with optimal maximum degree and total weight. In [8] they raised as an open problem the question whether it is possible to obtain a fault tolerant spanner for an arbitrary weighted undirected graph.…”

Section: Background and Previous Workmentioning

confidence: 99%

Fault-tolerant spanners for general graphs

Chechik

Langberg

Peleg

et al. 2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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The paper concerns graph spanners that are resistant to vertex or edge failures. Given a weighted undirected n-vertex graph G = (V, E) and an integer k ≥ 1, the subgraph u, v) denotes the distance between u and v in G . Graph spanners were extensively studied since their introduction over two decades ago. It is known how to efficiently construct a (2k −1)-spanner of size O(n 1+1/k ), and this sizestretch tradeoff is conjectured to be tight.

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“…Xu et al [20] provided an algorithm to construct minimal energy interference spanner. Czumaj et al [7] provided a greedy algorithm that constructs a k fault tolerant distance spanner in which every vertex is of degree O(k). Note that, to the best of our knowledge, there is no work addressing fault tolerant interference spanners.…”

Section: Related Workmentioning

confidence: 99%

Fault Tolerant Interference-Aware Topology Control for Ad Hoc Wireless Networks

Haque¹,

Rahman

2011

Ad-Hoc, Mobile, and Wireless Networks

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Abstract. Interference imposes a major challenge for efficient data communication in wireless networks. Increased level of interference may increase number of collisions, energy consumption and latency due to retransmissions of the interfered data. Interference of a link is the number of nodes interfered while a pair of nodes are communicating over a bi-directional link. Approaches have been proposed to reduce interference by dropping links that create high interference. However, dropping links make a network more susceptible to node failure/departure-a frequent phenomenon in ad hoc networks. Thus dropping high interference links while keeping the network significantly connected is an important goal to achieve. In this paper, we formulate the problem of constructing minimum interference path preserving and fault tolerant wireless ad hoc networks and then provide algorithms, both centralized and distributed with local information, to solve the problem. Moreover, for the first time in literature, we conceive the concept of fault tolerant interference spanner and provide a local algorithm to construct such spanner of a communication graph.

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Fault-Tolerant Geometric Spanners

Cited by 39 publications

References 27 publications

Robust Geometric Spanners

Robust Geometric Spanners

Fault-tolerant spanners for general graphs

Fault Tolerant Interference-Aware Topology Control for Ad Hoc Wireless Networks

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