Fault-tolerant geometric spanners

Czumaj, Artur; Zhao, Hairong

doi:10.1145/777792.777794

Cited by 30 publications

(45 citation statements)

References 17 publications

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“…This was improved by Lukovszki [17], who presented a fault-tolerant spanner with O(nk) edges, which is optimal. Later Czumaj and Zhao [6] showed that a greedy approach produces a k-vertex (or k-edge) fault-tolerant geometric (1 + ε)-spanner with degree O(k) and total weight O(k 2 · wt (MST(P ))); these bounds are asymptotically optimal.…”

Section: Introductionmentioning

confidence: 99%

Region-Fault Tolerant Geometric Spanners

Abam

Berg

Farshi

et al. 2009

Discrete Comput Geom

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We introduce the concept of region-fault tolerant spanners for planar point sets and prove the existence of region-fault tolerant spanners of small size. For a geometric graph G on a point set P and a region F , we define G F to be what Communicated by Joseph S.B. Mitchell. M.A. Discrete Comput Geom (2009) 41: 556-582 557remains of G after the vertices and edges of G intersecting F have been removed. A C-fault tolerant t-spanner is a geometric graph G on P such that for any convex region F , the graph G F is a t-spanner for G c (P ) F , where G c (P ) is the complete geometric graph on P . We prove that any set P of n points admits a C-fault tolerant (1 + ε)-spanner of size O(n log n) for any constant ε > 0; if adding Steiner points is allowed, then the size of the spanner reduces to O(n), and for several special cases, we show how to obtain region-fault tolerant spanners of O(n) size without using Steiner points. We also consider fault-tolerant geodesic t-spanners: this is a variant where, for any disk D, the distance in G D between any two points u, v ∈ P \ D is at most t times the geodesic distance between u and v in R 2 \ D. We prove that for any P , we can add O(n) Steiner points to obtain a fault-tolerant geodesic (1 + ε)-spanner of size O(n).

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Section: Introductionmentioning

confidence: 99%

Region-Fault Tolerant Geometric Spanners

Abam

Berg

Farshi

et al. 2009

Discrete Comput Geom

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“…Many algorithms are known that compute t-spanners with O(n) edges [1], [2], [8], [20], [22], [24], [26] that have additional properties such as bounded degree [4], [6], small spanner diameter [4] (i.e., any two points are connected by a t-spanner path consisting of only a small number of edges), low weight [11], [13], [19] (i.e., the total length of all edges is proportional to the weight of a minimum spanning tree of S), planarity [3], [21], and fault-tolerance [10], [23]; see also the surveys [18] and [25]. All these algorithms compute t-spanners for any given constant t > 1.…”

Section: Introductionmentioning

confidence: 99%

Constructing Plane Spanners of Bounded Degree and Low Weight

2005

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Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree. Introduction.Givena set S of n points in the plane and a real number t > 1, a t-spanner for S is a graph G with vertex set S such that any two vertices u and v are connected by a path in G whose length is at most t · |uv|, where |uv| is the Euclidean distance between u and v. If this graph has O(n) edges, then it is a sparse approximation of the (dense) complete Euclidean graph on S.Many algorithms are known that compute t-spanners with O(n) edges [1], [2], [8], [20], [22],[24], [26] that have additional properties such as bounded degree [4], [6], small spanner diameter [4] (i.e., any two points are connected by a t-spanner path consisting of only a small number of edges), low weight [11], [13], [19] (i.e., the total length of all edges is proportional to the weight of a minimum spanning tree of S), planarity [3], [21], and fault-tolerance [10], [23]; see also the surveys [18] and [25]. All these algorithms compute t-spanners for any given constant t > 1. Observe that some properties are conflicting. For instance a spanner with constant degree cannot have spanner diameter o(log n).In this paper we consider the construction of plane t-spanners. A graph, whose vertices are points in R 2 with straight-line edges, is said to be plane, if no two edges intersect, except possibly at their endpoints. Obviously, in order for a t-spanner to be plane, t must be at least √ 2. It is known that the Delaunay triangulation is a t-spanner for t = 2π/(3 cos(π/6)), see [21]. Furthermore, Das and Joseph [12] showed that other plane graphs such as the minimum weight triangulation and the greedy triangulation are t-spanners, for some constant t. In 1992 Levcopoulos and Lingas [22] showed that, for

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“…Sparse geometrical spanners with low stretch avoiding a given region in the plane were introduced in [1]. The closest work concerns fault-tolerant geometrical spanners [17], [18], [7]. In that setting, the input is a set of nodes in an euclidean space.…”

Section: Related Workmentioning

confidence: 99%

“…• Remote-spanners allow to extend the notion of stretch to multi-connected graphs in a novel manner: by considering sum of lengths of disjoint paths. Similar properties were only studied in the context of (fault-tolerant) geometrical spanners where the graph is given by all pair distances in an euclidean space [17], [18], [7]. This setting cannot be extended to graphs in general.…”

Section: Introductionmentioning

confidence: 99%

Remote-spanners: What to know beyond neighbors

Jacquet

Viennot

2009

2009 IEEE International Symposium on Parallel &Amp; Distributed Processing

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Motivated by the fact that neighbors are generally known in practical routing algorithms, we introduce the notion of remote-spanner. Given an unweighted graph G, a sub-graph H with vertex set V (H) = V (G) is an (α, β)-remote-spanner if for each pair of points u and v the distance between u and v in H u , the graph H augmented by all the edges between u and its neighbors in G, is at most α times the distance between u and v in G plus β. We extend this definition to k-connected graphs by considering the minimum length sum over k disjoint paths as a distance. We then say that an (α, β)-remote-spanner is k-connecting.In this paper, we give distributed algorithms for computing (1 + ε, 1 − 2ε)-remote-spanners for any ε > 0, k-connecting (1, 0)-remote-spanners for any k ≥ 1 (yielding (1, 0)-remote-spanners for k = 1) and 2-connecting (2, −1)-remote-spanners. All these algorithms run in constant time for any unweighted input graph. The number of edges obtained for k-connecting (1, 0)-remote-spanner is within a logarithmic factor from optimal (compared to the best k-connecting (1, 0)-remote-spanner of the input graph). Interestingly, sparse (1, 0)-remote-spanners (i.e. preserving exact distances) with O(n 4/3 ) edges exist in random unit disk graphs. The number of edges obtained for (1+ε, 1−2ε)-remotespanners and 2-connecting (2, −1)-remote-spanners is linear if the input graph is the unit ball graph of a doubling metric (even if distances between nodes are unknown). Our methodology consists in characterizing remote-spanners as sub-graphs containing the union of small depth tree sub-graphs dominating nearby nodes. This leads to simple local distributed algorithms.

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Fault-tolerant geometric spanners

Cited by 30 publications

References 17 publications

Region-Fault Tolerant Geometric Spanners

Region-Fault Tolerant Geometric Spanners

Constructing Plane Spanners of Bounded Degree and Low Weight

Remote-spanners: What to know beyond neighbors

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