article

Partitioning arrangements of lines I: An efficient deterministic algorithm

Author:

Pankaj K. AgarwalAuthors Info & Claims

Discrete & Computational Geometry, Volume 5, Issue 5

Pages 449 - 483

https://doi.org/10.1007/BF02187805

Published: 01 December 1990 Publication History

Abstract

In this paper we consider the following problem: Given a set ofn lines in the plane, partition the plane intoO(r2) triangles so that no triangle meets more thanO(n/r) lines of . We present a deterministic algorithm for this problem withO(nr logn/r) running time, where is a constant <3.33.

References

[1]

P. K. Agarwal, Partitioning arrangements of lines, II: Applications, Discrete and Computational Geometry 5 (1990), 533-574.

Digital Library

[2]

P. K. Agarwal and M. Sharir, Red-blue intersection detection algorithms with applications to motion planning and collision detection, SIAM Journal on Computing 19 (1990), 297-322.

Digital Library

[3]

M. Ajtai, J. Komlos, and E. Szemerédi, Sorting in c log n parallel steps, Proceedings of the 15th Annual ACM Symposium on Theory of Computing , 1983, pp. 1-9.

[4]

K. E. Batcher, Sorting networks and their applications, Proceedings of the AFIPS Spring Joint Summer Computer Conference , vol. 32 (1968), pp. 307-314.

Digital Library

[5]

B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry, Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science , 1988, pp. 539-549.

Digital Library

[6]

K. Clarkson, A probabilistic algorithm for the post office problem, Proceedings of the 17th Annual ACM Symposium on Theory of Computing , 1985, pp. 75-84.

Digital Library

[7]

K. Clarkson, New applications of random sampling in computational geometry, Discrete and Computational Geometry 2 (1987), 195-222.

Digital Library

[8]

K. Clarkson, Applications of random sampling in computational geometry, II, Proceedings of the 4th Annual Symposium on Computational Geometry , 1988, pp. 1-11.

Digital Library

[9]

K. Clarkson and P. Shor, Algorithms for diametric pairs and convex hulls that are optimal, randomized and incremental, Proceedings of the 4th Annual Symposium on Computational Geometry , 1988, pp. 12-17.

Digital Library

[10]

K. Clarkson, R. E. Tarjan, and C. J. Van Wyk, A fast Las Vegas algorithm for triangulating a simple polygon, Discrete and Computational Geometry 4 (1989), 423-432.

Digital Library

[11]

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, Journal of the Association for Computing Machinery 31 (1984), 200-208.

Digital Library

[12]

R. Cole, J. Salowe, W. Steiger, and E. Szemerédi, An optimal-time algorithm for slope selection, SIAM Journal on Computing 16 (1989), 792-810.

Digital Library

[13]

R. Cole, M. Sharir, and C. K. Yap, On k -hulls and related problems, SIAM Journal on Computing 16 (1987), 61-77.

Digital Library

[14]

H. Edelsbrunner, Algorithms in Combinatorial Geometry , Springer-Verlag, Heidelberg, 1987.

Digital Library

[15]

H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir, J. Snoeyink, and E. Welzl, Implicitly representing arrangements of lines or segments, Discrete and Computational Geometry 4 (1989), 433-466.

Digital Library

[16]

H. Edelsbrunner, L. Guibas, and M. Sharir, The complexity and construction of many faces in arrangements of lines or of segments, Discrete and Computational Geometry 5 (1990), 161-196.

Digital Library

[17]

H. Edelsbrunner and E. Welzl, Constructing belts in two-dimensional arrangements with applications, SIAM Journal on Computing 15 (1986), 271-284.

Digital Library

[18]

L. Guibas, M. Overmars, and M. Sharir, Ray shooting, implicit point location, and related queries in arrangements of segments, Technical Report 433, Dept. Computer Science, New York University, March 1989.

[19]

D. Haussler and E. Welzl, ε-nets and simplex range queries, Discrete and Computational Geometry 2 (1987), 127-151.

Digital Library

[20]

J. Matoušek, Construction of ε-nets, Discrete and Computational Geometry , this issue, 427-448.

[21]

N. Megiddo, Applying parallel computation algorithms in design of serial algorithms, Journal of the Association of Computing Machinery 30 (1983), 852-865.

Digital Library

[22]

J. Reif and S. Sen, Optimal randomized parallel algorithms for computational geometry, Proceedings of the 16th International Conference on Parallel Processing , 1987, pp. 270-277.

[23]

J. Reif and S. Sen, Polling: A new randomized sampling technique for computational geometry, Proceedings of the 21st Annual ACM Symposium on Theory of Computing , 1989, pp. 394-404.

Digital Library

[24]

S. Suri, A linear algorithm for minimum link paths inside a simple polygon, Computer Vision, Graphics and Image Processing 35 (1986), 99-110.

Digital Library

[25]

E. Welzl, More on k -sets of finite sets in the plane, Discrete and Computational Geometry 1 (1986), 83-94.

Digital Library

[26]

G. Woeginger, Epsilon-nets for half planes, Technical Report B-88-02, Dept. of Mathematics, Free University, Berlin, March 1988.

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Biniaz ABose PMaheshwari ASmid M(2018)Plane Bichromatic Trees of Low DegreeDiscrete & Computational Geometry10.1007/s00454-017-9881-z59:4(864-885)Online publication date: 1-Jun-2018
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Partitioning arrangements of lines I: An efficient deterministic algorithm
1. Mathematics of computing
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 5, Issue 5

October 1990

101 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1990 Springer-Verlag New York Inc.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1990

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Cited By

Biniaz ABose PMaheshwari ASmid M(2018)Plane Bichromatic Trees of Low DegreeDiscrete & Computational Geometry10.1007/s00454-017-9881-z59:4(864-885)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00454-017-9881-z
Chan TTsakalidis K(2016)Optimal Deterministic Algorithms for 2-d and 3-d Shallow CuttingsDiscrete & Computational Geometry10.1007/s00454-016-9784-456:4(866-881)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s00454-016-9784-4
Alon N(2012)A Non-linear Lower Bound for Planar Epsilon-netsDiscrete & Computational Geometry10.5555/3115956.311602847:2(235-244)Online publication date: 1-Mar-2012
https://dl.acm.org/doi/10.5555/3115956.3116028
Rafalin ESouvaine DTóth C(2010)Cuttings for Disks and Axis-Aligned Rectangles in Three-SpaceDiscrete & Computational Geometry10.5555/3116275.311653143:2(221-241)Online publication date: 1-Mar-2010
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Cheng SVigneron A(2007)Motorcycle Graphs and Straight SkeletonsAlgorithmica10.5555/3118786.311925347:2(159-182)Online publication date: 1-Feb-2007
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Trajcevski GWolfson OHinrichs KChamberlain S(2004)Managing uncertainty in moving objects databasesACM Transactions on Database Systems10.1145/1016028.101603029:3(463-507)Online publication date: 1-Sep-2004
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Matoušek J(1996)Derandomization in Computational GeometryJournal of Algorithms10.1006/jagm.1996.002720:3(545-580)Online publication date: 1-May-1996
https://dl.acm.org/doi/10.1006/jagm.1996.0027
Thurimella RLopez M(1995)On Computing Connected Components of Line SegmentsIEEE Transactions on Computers10.1109/12.37617444:4(597-601)Online publication date: 1-Apr-1995
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Gupta RSmolka SBhaskar S(1994)On randomization in sequential and distributed algorithmsACM Computing Surveys10.1145/174666.17466726:1(7-86)Online publication date: 1-Mar-1994
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