A Fast Algorithm for Approximating the Detour of a Polygonal Chain

Ebbers-Baumann, Annette; Klein, Rolf; Langetepe, Elmar; Lingas, Andrzej

doi:10.1007/3-540-44676-1_27

Cited by 14 publications

(19 citation statements)

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“…These are the first subquadratic-time algorithms for finding the exact spanning ratio or detour, and they solve open problems posed in at least two papers [10,23]. Our algorithm for the spanning ratio is worstcase optimal, as shown in [23], and we suspect that the algorithm for the detour is also optimal, although we are not aware of a published Ω(n log n) lower bound.…”

Section: New Resultsmentioning

confidence: 98%

“…We first describe an algorithm for the decision problem for the detour: "Given a parameter κ ≥ 1, determine whether δ(P ) ≤ κ." Our algorithm makes crucial use of the following properties established in [10]. The proof of property (iii) is straightforward.…”

Section: Overall Approachmentioning

confidence: 99%

“…Ebbers-Baumann et al [10] have studied the problem of computing the detour of a planar polygonal chain G with n vertices. They have established several geometric properties, the most significant of which (restated in Lemma 2.1) is that the detour of G is always attained by two mutually visible points p, q, one of which is a vertex of G. Using these properties, they develop an ε-approximation algorithm that runs in O((n/ε) log n) time.…”

Section: Introductionmentioning

confidence: 99%

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

Agarwal

Klein

Knauer

et al. 2007

Discrete Comput Geom

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The detour and spanning ratio of a graph G embedded in E d measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we 18 Discrete Comput Geom (2008) 39: 17-37 obtain O(n log 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in E 3 , and show that computing the detour in E 3 is at least as hard as Hopcroft's problem.

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Section: New Resultsmentioning

confidence: 98%

Section: Overall Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Agarwal

Klein

Knauer

et al. 2007

Discrete Comput Geom

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“…The detour [7] of a chain P on the pair of points (p i , p j ) is defined as the total length |p i . .…”

Section: Using the Detour To Bound The Distance To Segmentmentioning

confidence: 99%

A Computational-Geometry Approach to Digital Image Contour Extraction

Jiang

Tejada

2011

Transactions on Computational Science XIII

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We present a method for extracting contours from digital images, using techniques from computational geometry. Our approach is different from traditional pixel-based methods in image processing. Instead of working directly with pixels, we extract a set of oriented feature points from the input digital images, then apply classical geometric techniques, such as clustering, linking, and simplification, to find contours among these points. Experiments on synthetic and natural images show that our method can effectively extract contours, even from images with considerable noise; moreover, the extracted contours have a very compact representation.(176 pages) iv Acknowledgments Many thanks to my advisor, Dr. Minghui Jiang, for his encouragement and support, and for getting me interested in the field of computational geometry. His concern for me as a student and his advice have been a great help. He introduced me to the practice of doing research, and his many comments have helped me become a better researcher.

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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%

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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A Fast Algorithm for Approximating the Detour of a Polygonal Chain

Cited by 14 publications

References 10 publications

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

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

A Computational-Geometry Approach to Digital Image Contour Extraction

Lower Bounds on the Dilation of Plane Spanners

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