Delaunay graphs are almost as good as complete graphs

Dobkin, David P.; Friedman, Steven J.; Supowit, Kenneth J.

doi:10.1007/bf02187801

Cited by 181 publications

(92 citation statements)

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“…Bose and Morin [7] showed (by modifying an argument in [17]) that the length of D P (s i , s j ) is at most π/2 times |s i s j |, provided that (i) the straight-line segment between s i and s j lies outside the Voronoi region induced by u, and (ii) that the path lies on one side of the line through s i and s j .…”

Section: 1mentioning

confidence: 99%

“…G is a (21.04)-spanner of the complete visibility graph on S and C. One cannot compare CDT with the complete graph, instead we compare it with the visibility graph of S and C. The visibility graph VIS(S, C) of S and C is the undirected graph that has S as a vertex set and in which two vertices are connected by an edge whenever they "see" each other, i.e., the open line segment joining them is disjoint from all segments in C or is contained in a segment. It is straightforward to modify Dobkin et al's [17] proof to show that CDT (S, C) is a (π(1 + √ 5 )/2)-spanner of VIS(S, C), which implies that G is a (π(π + 1)(1 + √ 5 )/2)-spanner of VIS(S, C), since G is a (π + 1)-spanner of CDT (S, C). 3.…”

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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: 1mentioning

confidence: 99%

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confidence: 99%

Constructing Plane Spanners of Bounded Degree and Low Weight

2005

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“…Otherwise, if y is free, (x, y) is added to set F (x) of edges that connect x to (possibly) free vertices (lines [12][13]. When this set becomes too large, F (x) is scanned to determine whether the other endpoint of each edge is still free, and if not, whether the corresponding edge constitutes a bridge between two clusters (lines [14][15][16][17][18][19]. If, after this process, at least n 1/3 c free vertices remain, then a new cluster centered at x (including x itself if it is free), is created (lines 20-25).…”

Section: The Algorithm By Ausiello Et Al (A)mentioning

confidence: 99%

“…Graph spanners arise in many applications, including communication networks, computational biology, computational geometry, distributed computing, and robotics ( [4,7,13,14,[16][17][18][24][25][26][27][28]). Intuitively, a spanner of a given graph is a subgraph on the same vertex set which preserves approximate distances between all pairs of vertices.…”

Section: Introductionmentioning

confidence: 99%

Graph Spanners in the Streaming Model: An Experimental Study

et al. 2008

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This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (α, β)-spanner of a graph G is a subgraph S ⊆ G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and by Elkin (In: Algorithmica (2009) 55: 346-374 347 for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: ), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana (http://www.citebase.org/abstract?id=oai:arXiv.org:cs/0611023) and Elkin (In: to know in advance the number of vertices in the graph.The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources.

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“…Given a point set V , the Delaunay triangulation is obtained by all triangles T = (u, v, w) ∈ V 3 that satisfy that there exists no further node x ∈ V that is contained in the disk U(u, v, w) passing through the nodes u, v, and w. The geometric graph formed by a Delaunay triangulation is planar and is known to be a spanner with respect to the Euclidean distance metric [9,18]. It is thus an ideal candidate planar graph routing schemes might be applied on.…”

Section: Localized Delaunay Triangulationmentioning

confidence: 99%

Localized Topology Control Algorithms for Ad Hoc and Sensor Networks

Nayak

2007

Handbook of Applied Algorithms

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Delaunay graphs are almost as good as complete graphs

Cited by 181 publications

References 2 publications

Constructing Plane Spanners of Bounded Degree and Low Weight

Constructing Plane Spanners of Bounded Degree and Low Weight

Graph Spanners in the Streaming Model: An Experimental Study

Localized Topology Control Algorithms for Ad Hoc and Sensor Networks

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