On the Geometric Dilation of Finite Point Sets

Ebbers-Baumann, Annette; Grüne, Ansgar; Klein, Rolf

doi:10.1007/978-3-540-24587-2_27

Cited by 6 publications

(8 citation statements)

References 9 publications

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“…As proved in [9], the two-dimensional version of Cauchy's surface area formula cited below, leads to the general lower dilation bound of π/2.…”

Section: Lemmamentioning

confidence: 93%

“…An easier approach dealing only with the two-dimensional case is described on the first pages of [22]. Lemma 7 immediately implies a first lower bound to the dilation of arbitrary closed curves, see [9].…”

Section: Lemma 7 (Cauchy) Let C Be a Closed Convex Curve Then Its mentioning

confidence: 99%

“…Ebbers et al [9] studied the question of embedding a finite point set into a planar network of low geometric dilation. Among other things it was shown, using Cauchy's surface area formula, that each closed curve has a geometric dilation of at least π/2, the dilation of the circle.…”

Section: Introductionmentioning

confidence: 99%

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Geometric dilation of closed planar curves: New lower bounds

Ebbers-Baumann

Grüne

Klein

2007

Computational Geometry

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Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.

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“…As proved in [9], the two-dimensional version of Cauchy's surface area formula cited below, leads to the general lower dilation bound of π/2.…”

Section: Lemmamentioning

confidence: 93%

Section: Lemma 7 (Cauchy) Let C Be a Closed Convex Curve Then Its mentioning

confidence: 99%

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Geometric dilation of closed planar curves: New lower bounds

Ebbers-Baumann

Grüne

Klein

2007

Computational Geometry

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“…, p n spaced equally on the unit circle, as shown in Fig. 1(a), and is based on similar arguments as in Ebbers-Baumann et al [7] (Lemma 3). Theorem 1.…”

Section: Steiner Treesmentioning

confidence: 93%

Sparse geometric graphs with small dilation

Aronov¹,

Berg²,

Cheong³

et al. 2008

Computational Geometry

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Given a set S of n points in R D , and an integer k such that 0 k < n, we show that a geometric graph with vertex set S, at most n − 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n − 1 + k edges has dilation (n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position.

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“…Let us recall the formal definition of geometric dilation (see [13,15]). Let G be a plane graph whose edges are curves.…”

Section: Introductionmentioning

confidence: 99%

Light orthogonal networks with constant geometric dilation

Dumitrescu

Tóth

2009

Journal of Discrete Algorithms

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An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axis-aligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small having O (n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the Euclidean distance between u and v. Such a network can be constructed in O (n log n) time.

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On the Geometric Dilation of Finite Point Sets

Cited by 6 publications

References 9 publications

Geometric dilation of closed planar curves: New lower bounds

Geometric dilation of closed planar curves: New lower bounds

Sparse geometric graphs with small dilation

Light orthogonal networks with constant geometric dilation

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