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On Rectilinear Partitions with Minimum Stabbing Number

  • Mark de Berg
  • Amirali Khosravi
  • Sander Verdonschot
  • Vincent van der Weele
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

Abstract

Let S be a set of n points in \({\Bbb R}^d\), and let r be a parameter with 1 ≤ r ≤ n. A rectilinear r-partition for S is a collection Ψ(S) : = {(S 1,b 1),…,(S t ,b t )}, such that the sets S i form a partition of S, each b i is the bounding box of S i , and n/2r ≤ |S i | ≤ 2n/r for all 1 ≤ i ≤ t. The (rectilinear) stabbing number of Ψ(S) is the maximum number of bounding boxes in Ψ(S) that are intersected by an axis-parallel hyperplane h. We study the problem of finding an optimal rectilinear r-partition—a rectilinear partition with minimum stabbing number—for a given set S. We obtain the following results.

  • Computing an optimal partition is np-hard already in \({\Bbb R}^2\).

  • There are point sets such that any partition with disjoint bounding boxes has stabbing number Ω(r 1 − 1/d ), while the optimal partition has stabbing number 2.

  • An exact algorithm to compute optimal partitions, running in polynomial time if r is a constant, and a faster 2-approximation algorithm.

  • An experimental investigation of various heuristics for computing rectilinear r-partitions in \({\Bbb R}^2\).

Keywords

Exact Algorithm Black Point Optimal Partition Hilbert Curve Partition Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P.K., Erickson, J.: Geometric Range Searching and its Relatives. In: Chazelle, B., Goodman, J., Pollack, R. (eds.) Advances in Discrete and Computational Geometry, pp. 1–56 (1998)Google Scholar
  2. 2.
    Ahuja, P.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)MATHGoogle Scholar
  3. 3.
    Arthur, D., Vassilvitskii, S.: K-means++: the Advantages of Careful Seeding. In: Proc. of the 18th Annual ACM-SIAM Sym. of Desc. Alg., pp. 1027–1035 (2007)Google Scholar
  4. 4.
    Chazelle, B., Welzl, E.: Quasi-optimal Range Searching in Spaces of Finite VC-dimension. Arch. Rat. Mech. Anal. 4, 467–490 (1989)MathSciNetMATHGoogle Scholar
  5. 5.
    Ahuja, P.K., Magnanti, T.L., Orlin, J.B.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill (2001)Google Scholar
  6. 6.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)CrossRefMATHGoogle Scholar
  7. 7.
    Dijkstra, E.W., Feijen, W.H.J., Sterringa, J.: A Method of Programming. Addison-Wesley, Reading (1988)Google Scholar
  8. 8.
    Fekete, S.P., Lübbecke, M.E., Meijer, H.: Minimizing the Stabbing Number of Matchings, Trees, and Triangulations. Discr. Comput. Geom. 40, 595–621 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Interactibility: A Guide to the Theory of NP-Completness. W.H. Freeman and Co., New York (1979)Google Scholar
  10. 10.
    Haverkort, H., van Walderveen, F.: Four-dimensional Hilbert Curves for R-trees. In: Proc. Workshop on Algorithms Engineering and Experiments, ALANEX (2009)Google Scholar
  11. 11.
    Manolopoulos, Y., Nanopoulos, A., Theodoridis, Y., Papadopoulos, A.: R-trees: Theory and Applications. Series in Adv. Inf. and Knowledge Processing. Springer, Heidelberg (2005)MATHGoogle Scholar
  12. 12.
    Matoušek, J., Tarantello, G.: Efficient Partition Trees. Discr. Comput. Geom. 8, 315–334 (1992)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mark de Berg
    • 1
  • Amirali Khosravi
    • 1
  • Sander Verdonschot
    • 2
  • Vincent van der Weele
    • 3
  1. 1.Department of Mathematics and Computing ScienceTU EindhovenEindhovenThe Netherlands
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

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