On rectilinear partitions with minimum stabbing number

  • Authors:

  • Mark De Berg;Amirali Khosravi;Sander Verdonschot;Vincent Van Der Weele

  • Affiliations:

  • Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands;Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands;School of Computer Science, Carleton University, Ottawa, Canada;Max-Planck-Institut für Informatik, Saarbrücken, Germany

  • Venue:
  • WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
  • Year:
  • 2011

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Abstract

Let S be a set of n points in Rd, 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 R2. - 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 R2.