Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Agarwal, Pankaj K.; Klein, Rolf; Knauer, Christian; Langerman, Stefan; Morin, Pat; Sharir, Micha; Soss, Michael

doi:10.1007/s00454-007-9019-9

Cited by 33 publications

(41 citation statements)

References 28 publications

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“…Thus, path π is contained in the ellipse of diameter δ|pq|, whose foci are p and q; see Figure 4. 1 The width of this ellipse is equal to w = |pq| √ δ 2 − 1; it tends to zero as the dilation tends to 1. This has a useful consequence.…”

Section: Approximating Exact Intersectionsmentioning

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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Section: Approximating Exact Intersectionsmentioning

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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“…Geometric spanners find their applications in the areas of robotics, computer networks, distributed systems and many others. Refer to [1,2,4,13,23,32] for various algorithmic results.…”

Section: Introductionmentioning

confidence: 99%

“…Knauer and Mulzer [31] showed that for each edge e of a minimum dilation triangulation of a point set, at least one of the two half-disks of diameter about 0.2|e| on each side of e and centered at the midpoint of e must be empty of points 1 .…”

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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“…Narasimhan and Smid [7] show that the problem of approximating the stretch factor of any geometric graph on n vertices can be reduced to performing approximate shortest-path queries for O(n) pairs of points. Agarwal et al [1] show that the exact stretch factor of a geometric path, tree, and cycle on n points in the plane can be computed in O(n log n), O(n log 2 n), and O(n √ n log n) expected time, respectively. They also present algorithms for the three-dimensional versions of these problems.…”

Section: Introductionmentioning

confidence: 99%

Approximating the Average Stretch Factor of Geometric Graphs

Cheng¹,

Knauer

Langerman

et al. 2010

Algorithms and Computation

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Let G be a geometric graph whose vertex set S is a set of n points in R d . The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the sum of all n 2 stretch factors determined by all pairs of points in S. We show that for paths, cycles, and trees, this sum can be approximated, within a factor of 1 + , in O(npolylog(n)) time. For plane graphs, we present a (2 + )-approximation algorithm with running time O(n 5/3 polylog(n)), and a (4 + )-approximation algorithm with running time O(n 3/2 polylog(n)).

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Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Cited by 33 publications

References 28 publications

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

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

Lower Bounds on the Dilation of Plane Spanners

Approximating the Average Stretch Factor of Geometric Graphs

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