A fast algorithm for constructing sparse Euclidean spanners

Das, Gautam; Narasimhan, Giri

doi:10.1145/177424.177579

Cited by 64 publications

(125 citation statements)

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“…G has weight O(wt|MST|). This follows from the following result in [11], [13], [14], and [19]. Let G be an arbitrary t-spanner for a point set S having O(n) edges, where t is a constant, and let ε > 0 be an arbitrary real constant.…”

Section: Putting It All Togethermentioning

confidence: 95%

“…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%

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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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Section: Putting It All Togethermentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Constructing Plane Spanners of Bounded Degree and Low Weight

2005

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“…Other desirable properties of a spanner are, for example, that the total weight of the edges is small or that the maximum degree is low. As it turns out, such spanners do indeed exist: it has been shown that for any set P of n points and for any fixed ε > 0, there exists a (1 + ε)-spanner with O(n) edges, bounded degree, and whose total weight is O(wt (MST(P ))), where wt (MST(P )) is the weight of a minimum spanning tree of P [7,18].…”

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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“…The following result is shown in [12] and [14] (see also Theorem 3 of [13]). The following result gives a tight upper bound for the cost of (k, t)-VFTSs generated by the k-Greedy Algorithm.…”

Section: Upper Bound For the Cost Of Spanners Generated By The K-greementioning

confidence: 81%

“…Our construction in Theorem 1 is a generalization of the greedy algorithm (which is our algorithm with k = 0) that has been used before to construct spanners [1], [2], [7], [13], [18]. Our main contribution in this context is the first, precise analysis of the fault-tolerant spanners obtained in that construction.…”

Section: Theorem 1 Let V Be a Set Of Points In R D Let T > 1 And Lmentioning

confidence: 99%

Fault-Tolerant Geometric Spanners

Czumaj

Zhao

2004

Discrete Comput Geom

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We present two new results about vertex and edge fault-tolerant spanners in Euclidean spaces.We describe the first construction of vertex and edge fault-tolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t > 1 and any non-negative integer k, constructs a k-fault-tolerant t-spanner in which every vertex is of degree O(k) and whose total cost is O(k 2 ) times the cost of the minimum spanning tree; these bounds are asymptotically optimal.Our next contribution is an efficient algorithm for constructing good fault-tolerant spanners. We present a new, sufficient condition for a graph to be a k-fault-tolerant spanner. Using this condition, we design an efficient algorithm that finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.

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A fast algorithm for constructing sparse Euclidean spanners

Cited by 64 publications

References 7 publications

Constructing Plane Spanners of Bounded Degree and Low Weight

Constructing Plane Spanners of Bounded Degree and Low Weight

Region-Fault Tolerant Geometric Spanners

Fault-Tolerant Geometric Spanners

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