Computing a Center-Transversal Line

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.

P.A. was supported by NSF under grants CCR-00-86013 EIA-98-70724, EIA-99-72879, EIA-01-31905, and CCR-02-04118. S.C. was partially supported by the European Community Sixth Framework Programme under a Marie Curie Intra-European Fellowship, and by the Slovenian Research Agency, project J1-7218-0101. J.A.S. was partially supported by grant TIN2004-08065-C02-02 of the Spanish Ministry of Education and Science (MEC). M.S. was partially supported by NSF Grants CCR-00-98246 and CCF-05-14079, by grant 155/05 of the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. P.A. and M.S. were also supported by a joint grant from the U.S.-Israeli Binational Science Foundation.