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)


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.


Maximum Complexity Closed Halfspace Candidate Line Halfspace Depth Tukey Depth 
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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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