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Computing a Center-Transversal Line

  • Pankaj K. Agarwal
  • Sergio Cabello
  • J. Antoni Sellarès
  • Micha Sharir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4337)

Abstract

A center-transversal line for two finite point sets in ℝ3 is a line with the property that any closed halfspace that contains it also contains at least one third of each point set. It is known that a center-transversal line always exists [12],[24] but the best known algorithm for finding such a line takes roughly n 12 time. We propose an algorithm that finds a center-transversal line in \({\it O}({\it n}^{\rm 1+{\it \epsilon}}{\it \kappa}^{\rm 2}({\it n}))\) worst-case time, for any \({\it \epsilon}>\)0, where \({\it \kappa}({\it n})\) is the maximum complexity of a single level in an arrangement of n planes in ℝ3. With the current best upper bound \({\it \kappa}\)(n)=O(n 5/2) of [21], the running time is \({\it O}({\it n}^{\rm 6+{\it \epsilon}})\), for any \({\it \epsilon} > 0\). We also show that the problem of deciding whether there is a center-transversal line parallel to a given direction u can be solved in O(nlogn) expected time. Finally, we We also extend the concept of center-transversal line to that of bichromatic depth of lines in space, and give an algorithm that computes a deepest line exactly in time \({\it O}({\it n}^{\rm 1+{\it \epsilon}}{\it \kappa}^{\rm 2}({\it n}))\), and a linear-time approximation algorithm that computes, for any specified \({\it \delta}>0\), a line whose depth is at least \(1-{\it \delta}\) times the maximum depth.

Keywords

Maximum Complexity Closed Halfspace Candidate Line Halfspace Depth Tukey Depth 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Sergio Cabello
    • 2
  • J. Antoni Sellarès
    • 3
  • Micha Sharir
    • 4
    • 5
  1. 1.Department of Computer ScienceDuke UniversityUSA
  2. 2.Dep. of Mathematics, IMFM and FMFUniversity of LjubljanaSlovenia
  3. 3.Institut d’Informàtica i AplicacionsUniversitat de GironaSpain
  4. 4.School of Computer ScienceTel Aviv UniversityIsrael
  5. 5.Courant Institute of Mathematical SciencesNew York UniversityUSA

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