Abstract
Givenn hyperplanes inE d, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverE d 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(r d) size inO(nr d−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.
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References
Agarwal, P. K. Partitioning arrangements of lines, I: An efficient deterministic algorithm,Discrete Comput. Geom. 5 (1990), 449–483.
Agarwal, P. K. Partitioning arrangements of lines, II: Applications,Discrete Comput. Geom. 5 (1990), 533–573.
Chazelle, B. An optimal convex hull algorithm in any fixed dimension,Discrete Comput. Geom., to appear.
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.
Chazelle, B., Friedman, J. A deterministic view of random sampling and its use in geometry,Combinatorica 10 (1990), 229–249.
Chazelle, B., Friedman, J. Point location among hyperplanes and unidirectional ray-shooting, Technical Report CS-TR-333-91, Princeton University, June 1991.
Clarkson, K. L. Linear programming inO(n3d 2) time,Inform. Process. Lett. 22 (1986), 21–24.
Clarkson, K. L. New applications of random sampling in computational geometry,Discrete Comput. Geom. 2 (1987), 195–222.
Clarkson, K. L. A randomized algorithm for closest-point queries,SIAM J. Comput. 17 (1988), 830–847.
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.
Clarkson, K. L., Shor, P. W. Applications of random sampling in computational geometry, II,Discrete Comput. Geom. 4 (1989), 387–421.
Dyer, M. E. Linear time algorithms for two- and three-variable linear programs,SIAM J. Comput. 13 (1984), 31–45.
Edelsbrunner, H.Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.
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.
Edelsbrunner, H., Mücke, P. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms,ACM Trans. Comput. Graphics 9 (1990), 66–104.
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.
Haussler, D., Welzl, E. Epsilon-nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.
Matoušek, J. Construction ofɛ-nets,Discrete Comput. Geom. 5 (1990), 427–448.
Matoušek, J. Cutting hyperplane arrangements,Discrete Comput. Geom. 6 (1991), 385–406.
Matoušek, J. Approximations and optimal geometric divide-and-conquer,Proc. 23rd Annu. ACM Symp. on Theory of Computing, 1991, pp. 505–511.
Matoušek, J. Efficient partition trees,Proc. 7th Annu. ACM Symp. on Computational Geometry, 1991, pp. 1–9.
Matoušek, J. Private communication, 1991.
Matoušek, J. Range searching with efficient hierarchical cuttings, Technical Report KAM Series, Dept. Applied Mathematics, Charles University, 1992.
Megiddo, N. Linear programming in linear time when the dimension is fixed,J. Assoc. Comput. Mach. 31 (1984), 114–127.
Raghavan, P. Probabilistic construction of deterministic algorithms: approximating packing integer programs,J. Comput. System Sci. 37 (1988), 130–143.
Seidel, R. Linear programming and convex hulls made easy,Proc. 6th Annu. ACM Symp. on Computational Geometry, 1990, pp. 211–215.
Spencer, J.Ten Lectures on the Probabilistic Method, CBMS-NSF, SIAM, Philadelphia, PA, 1987.
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.
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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This research was supported in part by the National Science Foundation under Grant CCR-9002352.
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Chazelle, B. Cutting hyperplanes for divide-and-conquer. Discrete Comput Geom 9, 145–158 (1993). https://doi.org/10.1007/BF02189314
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DOI: https://doi.org/10.1007/BF02189314