Embedding Point Sets into Plane Graphs of Small Dilation

Ebbers-Baumann, Annette; Grüne, Ansgar; Karpiński, Marek; Klein, Rolf; Knauer, Christian; Lingas, Andrzej

doi:10.1007/11602613_3

Cited by 4 publications

(7 citation statements)

References 8 publications

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“…For recent results, see, e.g., [4,18]. If Steiner vertices are allowed, their number should also be minimized, and different versions of the problem arise if we include the Steiner points in measuring the dilation, see [16].…”

Section: Introductionmentioning

confidence: 99%

Self-approaching Graphs

et al. 2013

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Abstract. In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner.We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3) constructing a self-approaching Steiner network connecting a given set of points.We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in R 2 and R 3 , but it is NP-hard to test if a given graph drawing in R 3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.

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Section: Introductionmentioning

confidence: 99%

Self-approaching Graphs

et al. 2013

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“…Ebbers-Baumann et al [34] have shown that if P is the (infinite) set of points on a closed convex curve, then its dilation is larger than 1.00157. They have also shown that the dilation of every finite point set is less than 1.1247.…”

Section: Dilationmentioning

confidence: 99%

On plane geometric spanners: A survey and open problems

Bose

Smid

2013

Computational Geometry

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“…In this paper we address this relaxation of the exact problem above, which has been brought up recently by Ebbers-Baumann et al [8].…”

Section: Introductionmentioning

confidence: 97%

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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Embedding Point Sets into Plane Graphs of Small Dilation

Cited by 4 publications

References 8 publications

Self-approaching Graphs

Self-approaching Graphs

On plane geometric spanners: A survey and open problems

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

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