On the geometric dilation of closed curves, graphs, and point sets

Dumitrescu, Adrian; Ebbers-Baumann, Annette; Grüne, Ansgar; Klein, Rolf; Rote, Günter

doi:10.1016/j.comgeo.2005.07.004

Cited by 28 publications

(20 citation statements)

References 16 publications

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“…It is known [13] that there are point sets for which no plane t-spanner can exist if t < (1+10 −11 )π/2 ≈ 1.570. On the upper bound side, Chew introduced in [7] the triangular distanceDelaunay graphs, TD-Delaunay graphs for short, whose convex distance function is defined from an equilateral triangle.…”

Section: Delaunay-graphsmentioning

confidence: 99%

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

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Abstract. Θ k -graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations, introduced by Chew, have been shown to be plane 2-spanners of the 2D Euclidean complete graph, i.e., the distance in the TD-Delaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6-graph defined in the 2D Euclidean space, namely the half-Θ6-graph, composed of the evencone edges of the Θ6-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish:-Every Θ6-graph is the union of two spanning TD-Delaunay graphs.In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6-graphs are t-spanners for some constant t, and Θ7-graphs were only known to be t-spanners for t ≈ 7.562. -Every plane triangulation is TD-Delaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TD-Delaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.

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Section: Delaunay-graphsmentioning

confidence: 99%

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Bonichon

Gavoille

Hanusse

et al. 2010

Lecture Notes in Computer Science

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“…Yet, it is interesting to observe that in this model non-trivial upper and lower bounds could be established. Each finite point set can be embedded into a plane graph of geometric dilation ≤ 1.678, and there are point sets for which each plane graph containing them has a geometric dilation ≥ (1 + 10 −11 )π/2; see [3,7]. These results are based on methods from convex geometry that do not apply here.…”

Section: Related Workmentioning

confidence: 99%

“…There are two infinite families and one exceptional case. Quite recently, a related measure called geometric dilation has been introduced, see [1,3,4,5,6,7], where all points of the graph, vertices and interior edge points alike, are considered in computing the dilation. The small difference in definition leads to rather different results.…”

Section: Related Workmentioning

confidence: 99%

The density of iterated crossing points and a gap result for triangulations of finite point sets

Klein

Kutz

2006

Proceedings of the Twenty-Second Annual Symposium on Computational Geometry

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Consider a plane graph G, drawn with straight lines. For every pair a, b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the dilation of G. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set P ⊆ R 2 , we connect every pair of points in P by a line segment and then add to P all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set P ∞ ⊇ P . This limit set P ∞ is finite if and only if P is contained in the vertex set of a triangulation of dilation 1.The main result of this paper is the following gap theorem: For any finite point set P in the plane for which P ∞ is infinite, there exists a threshold λ > 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most λ. As a first ingredient to our proof, we show that such an infinite P ∞ must lie dense in a certain region of the plane. In the second, more difficult part, we then construct a concrete point set P 0 such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.

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“…8 (right). In this case, the primary pair is s 6 , s 18 and the secondary pair is s 8 , s 20 . As in the Case A, we assume that s 0 ∈ π(s 6 , s 18 ).…”

Section: Case Cmentioning

confidence: 99%

“…When the stretch factor (or dilation) is measured over all pairs of points on edges or vertices of a plane graph G (rather than only over pairs of vertices) one arrives at the concept of geometric dilation of G; see [20,22]. (III) For every n ≥ 6, there exists a set S of n points such that the stretch factor of the greedy triangulation of S is at least 2.0268.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds on the Dilation of Plane Spanners

Dumitrescu

Ghosh

2016

Int. J. Comput. Geom. Appl.

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Abstract(I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previous best lower bound of 1.4161 for the worst-case dilation of plane spanners.(II) For every n ≥ 13, there exists an n-element point set S such that the degree 3 dilation of S equals 1 + √ 3 = 2.7321 . . . in the domain of plane geometric spanners. In the same domain, we show that for every n ≥ 6, there exists a an n-element point set S such that the degree 4 dilation of S equals 1 + (5 − √ 5)/2 = 2.1755 . . . The previous best lower bound of 1.4161 holds for any degree.(III) For every n ≥ 6, there exists an n-element point set S such that the stretch factor of the greedy triangulation of S is at least 2.0268.

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On the geometric dilation of closed curves, graphs, and point sets

Cited by 28 publications

References 16 publications

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

The density of iterated crossing points and a gap result for triangulations of finite point sets

Lower Bounds on the Dilation of Plane Spanners

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