Minimum dilation stars

Eppstein, David; Wortman, Kevin

doi:10.1145/1064092.1064142

Cited by 15 publications

(16 citation statements)

References 9 publications

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“…As mentioned in the introduction, it is NP-hard to decide whether there is a t-spanner of a point set with at most n − 1 + k edges, even in the case when the spanner is a tree [4]. However, if the spanner is restricted to be a star then the minimum dilation graph can be computed in polynomial time [10]. Are there other restrictions on the spanner or the point set that allow for efficient algorithms?…”

Section: Open Problemsmentioning

confidence: 99%

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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Section: Open Problemsmentioning

confidence: 99%

Sparse geometric graphs with small dilation

Aronov¹,

Berg²,

Cheong³

et al. 2008

Computational Geometry

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show abstract

“…A lower bound is √ 2, attained by the vertices of a square. A partial answer to Problem 8 was recently given by Eppstein and Wortman [11] who provided a randomized O(n log n) algorithm for computing an optimal star center for n points in d-space.…”

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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“…Testing a candidate edge e entails computing the stretch factor of the graph G = (V, E ∪ {e}), therefore we briefly consider the problem of computing the stretch factor of a given Euclidean graph. This problem has recently received considerable attention, see for example [11,13,21,25].…”

Section: Finding An Optimal Solutionmentioning

confidence: 99%

“…This algorithm is quite slow and we would like to be able to compute the stretch factor more efficiently, but no faster algorithm is known for any graphs except planar graphs, paths, cycles, stars and trees [13,21,25].…”

Section: Finding An Optimal Solutionmentioning

confidence: 99%

Finding the best shortcut in a geometric network

Farshi

Giannopoulos

Gudmundsson

2005

Proceedings of the Twenty-First Annual Symposium on Computational Geometry

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Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)-approximation algorithm with running time O(nm + n 2 (log n + 1/ε 3d )) using O(n 2 ) space.

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Minimum dilation stars

Cited by 15 publications

References 9 publications

Sparse geometric graphs with small dilation

Sparse geometric graphs with small dilation

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

Finding the best shortcut in a geometric network

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