article

Cutting hyperplanes for divide-and-conquer

Author:

Bernard ChazelleAuthors Info & Claims

Discrete & Computational Geometry, Volume 9, Issue 2

Pages 145 - 158

https://doi.org/10.1007/BF02189314

Published: 01 December 1993 Publication History

Abstract

Givenn hyperplanes inEd, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverEd and such that the interior of each simplex intersects at mostn/r hyperplanes. We present a deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd 1) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.

References

[1]

Agarwal, P. K. Partitioning arrangements of lines, I: An efficient deterministic algorithm,Discrete Comput. Geom.5 (1990), 449---483.

Digital Library

[2]

Agarwal, P. K. Partitioning arrangements of lines, II: Applications,Discrete Comput. Geom.5 (1990), 533---573.

Digital Library

[3]

Chazelle, B. An optimal convex hull algorithm in any fixed dimension,Discrete Comput. Geom., to appear.

[4]

Chazelle, B., Edelsbrunner, H., Guibas, J. J., Sharir, M. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications,Theoret. Comput. Sci.84 (1991), 77---105.

Digital Library

[5]

Chazelle, B., Friedman, J. A deterministic view of random sampling and its use in geometry,Combinatorica10 (1990), 229---249.

[6]

Chazelle, B., Friedman, J. Point location among hyperplanes and unidirectional ray-shooting, Technical Report CS-TR-333-91, Princeton University, June 1991.

[7]

Clarkson, K. L. Linear programming inO(n3d2) time,Inform. Process. Lett.22 (1986), 21---24.

Digital Library

[8]

Clarkson, K. L. New applications of random sampling in computational geometry,Discrete Comput. Geom.2 (1987), 195---222.

Digital Library

[9]

Clarkson, K. L. A randomized algorithm for closest-point queries,SIAM J. Comput.17 (1988), 830---847.

Digital Library

[10]

Clarkson, K. L. A Las Vegas algorithm for linear programming when the dimension is small,Proc. 29th Annu. Symp. on Foundations of Computer Science, 1988, pp. 452---456.

[11]

Clarkson, K. L., Shor, P. W. Applications of random sampling in computational geometry, II,Discrete Comput. Geom.4 (1989), 387---421.

Digital Library

[12]

Dyer, M. E. Linear time algorithms for two- and three-variable linear programs,SIAM J. Comput.13 (1984), 31---45.

[13]

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

Digital Library

[14]

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

Digital Library

[15]

Edelsbrunner, H., Mücke, P. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms,ACM Trans. Comput. Graphics9 (1990), 66---104.

[16]

Guibas, L. J., Overmars, M., Sharir, M. Counting and reporting intersections in arrangements of line segments, Technical Report 434, Dept. Computer Science, New York University, 1989.

[17]

Haussler, D., Welzl, E. Epsilon-nets and simplex range queries,Discrete Comput. Geom.2 (1987), 127---151.

Digital Library

[18]

Matoušek, J. Construction of¿-nets,Discrete Comput. Geom.5 (1990), 427---448.

Digital Library

[19]

Matoušek, J. Cutting hyperplane arrangements,Discrete Comput. Geom.6 (1991), 385---406.

Digital Library

[20]

Matoušek, J. Approximations and optimal geometric divide-and-conquer,Proc. 23rd Annu. ACM Symp. on Theory of Computing, 1991, pp. 505---511.

[21]

Matoušek, J. Efficient partition trees,Proc. 7th Annu. ACM Symp. on Computational Geometry, 1991, pp. 1---9.

[22]

Matoušek, J. Private communication, 1991.

[23]

Matoušek, J. Range searching with efficient hierarchical cuttings, Technical Report KAM Series, Dept. Applied Mathematics, Charles University, 1992.

Digital Library

[24]

Megiddo, N. Linear programming in linear time when the dimension is fixed,J. Assoc. Comput. Mach.31 (1984), 114---127.

Digital Library

[25]

Raghavan, P. Probabilistic construction of deterministic algorithms: approximating packing integer programs,J. Comput. System Sci.37 (1988), 130---143.

Digital Library

[26]

Seidel, R. Linear programming and convex hulls made easy,Proc. 6th Annu. ACM Symp. on Computational Geometry, 1990, pp. 211---215.

[27]

Spencer, J.Ten Lectures on the Probabilistic Method, CBMS-NSF, SIAM, Philadelphia, PA, 1987.

[28]

Vapnik, V. N., Chervonenkis, A. Ya. On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl.16 (1971), 264---280.

[29]

Yap, C. K. A geometric consistency theorem for a symbolic perturbation scheme,Proc. 4th Annu. ACM Symp. on Computational Geometry, 1988, pp. 134---142.

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Cutting hyperplanes for divide-and-conquer
1. Mathematics of computing
2. Theory of computation
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Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 9, Issue 2

February 1993

102 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1993 Springer-Verlag New York Inc.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1993

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