Advertisement

Discrete & Computational Geometry

, Volume 42, Issue 1, pp 3–21 | Cite as

On Approximate Range Counting and Depth

  • Peyman Afshani
  • Timothy M. Chan
Article

Abstract

We improve the previous results by Aronov and Har-Peled (SODA’05) and Kaplan and Sharir (SODA’06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in \(O(\log{n\over k})\) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo. In addition, we describe new results for several related problems, including approximate Tukey depth queries in 3D, approximate regression depth queries in 2D, and approximate linear programming with violations in low dimensions.

Keywords

Range searching Data structures Approximation algorithms Randomized algorithms Statistical depth 

References

  1. 1.
    Afshani, P., Chan, T.M.: Optimal halfspace range reporting in three dimensions. In: Proceedings of the 20th Annual Symposium on Discrete Algorithms, pp. 180–186 (2009) Google Scholar
  2. 2.
    Agarwal, P.K., Erickson, J.: Geometric range searching and its relatives. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry. Contemporary Mathematics, vol. 223, pp. 1–56. American Mathematical Society, Providence (1999) Google Scholar
  3. 3.
    Aronov, B., Har-Peled, S.: On approximating the depth and related problems. In: Proceedings of the 16th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 886–894 (2005). Updated version at http://valis.cs.uiuc.edu/~sariel/research/papers/04/depth/, downloaded in November 2006
  4. 4.
    Arya, S., Mount, D.M.: Approximate range searching. Comput. Geom. Theory Appl. 17(3-4), 135–152 (2000) MATHMathSciNetGoogle Scholar
  5. 5.
    de Berg, M., Schwarzkopf, O.: Cuttings and applications. Int. J. Comput. Geom. Appl. 5, 343–355 (1995) MATHCrossRefGoogle Scholar
  6. 6.
    Bern, M., Eppstein, D.: Multivariate regression depth. In: Proceedings of the 16th Annual Symposium on Computational Geometry, pp. 315–321 (2000) Google Scholar
  7. 7.
    Chan, T.M.: Fixed-dimensional linear programming queries made easy. In: Proceedings of the 12th Annual ACM Symposium on Computational Geometry, pp. 284–290 (1996) Google Scholar
  8. 8.
    Chan, T.M.: Output-sensitive results on convex hulls, extreme points, and related problems. Discrete Comput. Geom. 16, 369–387 (1996) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete Comput. Geom. 22, 547–567 (1999) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chan, T.M.: Low-dimensional linear programming with violations. SIAM J. Comput. 34, 879–893 (2000) CrossRefGoogle Scholar
  11. 11.
    Chan, T.M.: Random sampling, halfspace range reporting, and construction of (≤k)-levels in three dimensions. SIAM J. Comput. 30(2), 561–575 (2000) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chan, T.M.: On enumerating and selecting distances. Int. J. Comput. Geom. Appl. 11, 291–304 (2001) MATHCrossRefGoogle Scholar
  13. 13.
    Chan, T.M.: An optimal randomized algorithm for maximum Tukey depth. In: Proceedings of the 15th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 430–436 (2004) Google Scholar
  14. 14.
    Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25(1), 76–90 (1985) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cohen, E.: Size-estimation framework with applications to transitive closure and reachability. J. Comput. Syst. Sci. 55, 441–453 (1997) MATHCrossRefGoogle Scholar
  17. 17.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer, Heidelberg (1987) MATHGoogle Scholar
  18. 18.
    Edelsbrunner, H., Welzl, E.: Constructing belts in two-dimensional arrangements with applications. SIAM J. Comput. 15, 271–284 (1986) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Har-Peled, S., Sharir, M.: Relative ε-approximations in geometry. http://valis.cs.uiuc.edu/~sariel/research/papers/06/relative/ (2006). Also with B. Aronov, in Proceedings of the 23rd ACM Symposium on Computational Geometry, pp. 327–336 (2007)
  20. 20.
    Hart, S., Sharir, M.: Nonlinearity of Davenport–Schinzel sequences and of generalized path compression schemes. Combinatorica 6(2), 151–177 (1986) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kaplan, H., Sharir, M.: Randomized incremental constructions of three-dimensional convex hulls and planar Voronoi diagrams, and approximate range counting. In: Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 484–493 (2006) Google Scholar
  23. 23.
    Kaplan, H., Ramos, E., Sharir, M.: Range minima queries with respect to a random permutation, and approximate range counting. Discrete Comput. Geom. (to appear) Google Scholar
  24. 24.
    Kaplan, H., Ramos, E., Sharir, M.: The overlay of minimization diagrams in a randomized incremental construction. Manuscript (2007) Google Scholar
  25. 25.
    Langerman, S., Steiger, W.: The complexity of hyperplane depth in the plane. Discrete Comput. Geom. 30(2), 299–309 (2003) MATHMathSciNetGoogle Scholar
  26. 26.
    Matoušek, J.: Reporting points in halfspaces. Comput. Geom. Theory Appl. 2(3), 169–186 (1992) MATHGoogle Scholar
  27. 27.
    Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Matoušek, J.: On geometric optimization with few violated constraints. Discrete Comput. Geom. 14, 365–384 (1995) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995) MATHGoogle Scholar
  30. 30.
    Mulmuley, K.: Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs (1994) Google Scholar
  31. 31.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985) Google Scholar
  32. 32.
    Ramos, E.A.: On range reporting, ray shooting and k-level construction. In: Proceedings of the 14th Annual Symposium on Computational Geometry, pp. 390–399 (1999) Google Scholar
  33. 33.
    Rousseeuw, P.J., Hubert, M.: Regression depth. J. Am. Stat. Assoc. 94(446), 388–402 (1999) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Sharir, M., Smorodinsky, S., Tardos, G.: An improved bound for k-sets in three dimensions. Discrete Comput. Geom. 26, 195–204 (2001) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations