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) : = {(S1,b1),…,(St,bt)}, such that the sets Si form a partition of S, each bi is the bounding box of Si, and n/2r ≤ |Si| ≤ 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(r1 − 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 \({\Bbb R}^2\).

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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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