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Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

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Abstract

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matoušek (Comput Geom 2(3):169–186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG’99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, \(({\le }k)\)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, \(\varepsilon \)-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matoušek (Discrete Comput Geom 6(1):385–406, 1991) and Chazelle (Discrete Comput Geom 9(1):145–158, 1993).

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References

  1. Afshani, P., Chan, T.M.: Optimal halfspace range reporting in three dimensions. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’09, pp. 180–186. SIAM, Philadelphia (2009)

  2. Afshani, P., Chan, T.M., Tsakalidis, K.: Deterministic rectangle enclosure and offline dominance reporting on the RAM. In: Proceedings of the the Forty-First International Colloquium on Automata, Languages and Programming, ICALP’14. LNCS, vol. 8572, pp. 77–88. Springer, Berlin (2014)

  3. Afshani, P., Tsakalidis, K.: Optimal deterministic shallow cuttings for 3d dominance ranges. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA’14, pp. 1389–1398. SIAM, Portland (2014)

  4. Agarwal, P.K.: Partitioning arrangements of lines I: an efficient deterministic algorithm. Discrete Comput. Geom. 5(1), 449–483 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal, P.K.: Intersection and Decomposition Algorithms for Planar Arrangements. Cambridge University Press, New York (1991)

    MATH  Google Scholar 

  6. Agarwal, P.K., Aronov, B., Chan, T.M., Sharir, M.: On levels in arrangements of lines, segments, planes, and triangles. Discrete Comput. Geom. 19(3), 315–331 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brodal, G.S., Jacob, R.: Dynamic planar convex hull with optimal query time. In: Proceedings of the Seventh Scandinavian Workshop on Algorithm Theory, SWAT’00, pp. 57–70 (2000)

  8. Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proceedings of the Forty-Third Symposium on Foundations of Computer Science, FOCS’02, pp. 617–626. IEEE, Vancouver (2002)

  9. Chan, T.M.: Random sampling, halfspace range reporting, and construction of \(({\le }k)\)-levels in three dimensions. SIAM J. Comput. 30(2), 561–575 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, T.M.: Low-dimensional linear programming with violations. SIAM J. Comput. 34(4), 879–893 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chan, T.M.: A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. ACM 57(3), 16:1–16:15 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, T.M.: Three problems about dynamic convex hulls. Int. J. Comput. Geom. Appl. 22(04), 341–364 (2012)

    Article  MathSciNet  Google Scholar 

  13. Chan, T.M., Larsen, K.G., Pǎtraşcu, M.: Orthogonal range searching on the RAM, revisited. In: Proceedings of the Twenty-Seventh Symposium on Computational Geometry, SoCG’11, pp. 1–10. ACM, New York (2011)

  14. Chan, T.M., Pǎtraşcu, M.: Transdichotomous results in computational geometry, I: point location in sublogarithmic time. SIAM J. Comput. 39(2), 703–729 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chan, T.M., Tsakalidis, K.: Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. In: Proceedings of the Thirty-First Annual Symposium on Computational Geometry, SoCG’15, pp. 719–732. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2015)

  16. Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(1), 145–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chazelle, B., Friedman, J.: A deterministic view of random sampling and its use in geometry. Combinatorica 10(3), 229–249 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Dyer, M.E.: Linear time algorithms for two- and three-variable linear programs. SIAM J. Comput. 13(1), 31–45 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goodrich, M.T.: Planar separators and parallel polygon triangulation. J. Comput. Syst. Sci. 51(3), 374–389 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Har-Peled, S., Kaplan, H., Sharir, M., Smorodinsky, S.: Epsilon-nets for halfspaces revisited. CoRR http://arxiv.org/abs/1410.3154 (2014)

  23. Haussler, D., Welzl, E.: \(\varepsilon \)-nets and simplex range queries. Discrete Comput. Geom. 2(1), 127–151 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. Matoušek, J.: Construction of \(\varepsilon \)-nets. Discrete Comput. Geom. 5(1), 427–448 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Matoušek, J.: Cutting hyperplane arrangements. Discrete Comput. Geom. 6(1), 385–406 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matoušek, J.: Reporting points in halfspaces. Comput. Geom. 2(3), 169–186 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Matoušek, J.: On constants for cuttings in the plane. Discrete Comput. Geom. 20(4), 427–448 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Megiddo, N.: Linear-time algorithms for linear programming in \({R}^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  30. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31(1), 114–127 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ramos, E.A.: On range reporting, ray shooting and \(k\)-level construction. In: Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG’99, pp. 390–399. ACM, New York (1999)

  32. Ramos, E.A.: Deterministic algorithms for 3-d diameter and some 2-d lower envelopes. In: Proceedings of the Sixteenth Annual Symposium on Computational Geometry, SoCG’00, pp. 290–299. ACM, New York (2000)

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Authors and Affiliations

  1. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada

    Timothy M. Chan

  2. Computer Engineering and Informatics Department, University of Patras, Patras, Greece

    Konstantinos Tsakalidis

Authors
  1. Timothy M. Chan
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  2. Konstantinos Tsakalidis
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Correspondence to Timothy M. Chan.

Additional information

Editor in Charge: János Pach

A preliminary version of this work appeared in the Proceedings of the 31st International Symposium on Computational Geometry (SoCG’15) [15]. Part of this work was done during the authors’ visit to the Hong Kong University of Science and Technology.

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Chan, T.M., Tsakalidis, K. Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings. Discrete Comput Geom 56, 866–881 (2016). https://doi.org/10.1007/s00454-016-9784-4

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  • DOI: https://doi.org/10.1007/s00454-016-9784-4

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