# 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

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## 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