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A linear algorithm for computing convex hulls for random lines

Authors:

Daniel Berend,

Vladimir BravermanAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 5, Issue 4

Article No.: 42, Pages 1 - 21

https://doi.org/10.1145/1597036.1597046

Published: 06 November 2009 Publication History

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Abstract

Finding the convex hull of n points in the plane requires O(n log n) time in general. In Devroye and Toussaint [1993] and Golin et al. [2002] the problem of computing the convex hull of the intersection points of n lines was considered, where the lines are chosen randomly according to two various models. In both models, linear-time algorithms were developed. Here we improve the results of Devroye and Toussaint [1993] by giving a universal algorithm for a wider range of distributions.

References

[1]

Atallah, M. 1986. Computing the convex hull of line intersections. J. Algor. 7, 285--288.

Digital Library

Google Scholar

[2]

Berend, D. and Braverman, V. 2005. Convex hull for intersections of random lines. In Proceedings of the International Conference on the Analysis of Algorithms 25, 11--12.

Google Scholar

[3]

Chan, T. 1996. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete Comput. Geom. 16, 361--368.

Digital Library

Google Scholar

[4]

Ching, Y. and Lee, D. 1985. Finding the diameter of a set of lines. Pattern Recogn. 18, 249--255.

Crossref

Google Scholar

[5]

Cormen, T., Leiserson, G., and Rivest, R. 2000. Introduction to Algorithms. The MIT Press.

Digital Library

Google Scholar

[6]

Devroye, L. and Toussaint, G. 1993. Convex hulls for random lines. J. Algor. 14, 381--394.

Digital Library

Google Scholar

[7]

Golin, M., Langerman, S., and Steiger, W. 2002. The convex hull for random lines in the plane. In Proceedings of the Japan Conference on Discrete and Computational Geometry (JCDCG'02).

Google Scholar

[8]

Graham, R. 1972. An efficient algorithm for determining the convex hull of a finite planar set. Inf. Process. Lett. 1, 132--133.

Crossref

Google Scholar

[9]

Jarvis, R. 1973. On the identification of the convex hull of a finite set of points in the plane. Inf. Process. Lett. 2, 18--21.

Crossref

Google Scholar

[10]

Kirkpatrick, D. and Seidel, R. 1986. The ultimate planar convex hull algorithm? SIAM J. Comput. 15, 287--299.

Digital Library

Google Scholar

[11]

Melkman, A. 1987. On-line construction of the convex hull of a simple polyline. Inf. Process. Lett. 25, 11--12.

Digital Library

Google Scholar

[12]

Preparata, F. and Shamos, M. 1986. Computational Geometry: An Introduction. Springer-Verlag, New-York.

Digital Library

Google Scholar

Index Terms

A linear algorithm for computing convex hulls for random lines
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 5, Issue 4

October 2009

281 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/1597036

Issue’s Table of Contents

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Association for Computing Machinery

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Publication History

Published: 06 November 2009

Accepted: 01 May 2009

Revised: 01 February 2009

Received: 01 August 2005

Published in TALG Volume 5, Issue 4

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Convex hulls of spheres and convex hulls of disjoint convex polytopes

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

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