Computing a Center-Transversal Line

A center-transversal line for two finite point sets in R 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,but the best known algorithm for finding such a line takes roughly n time. We propose an algorithm that finds a center-transversal line in O(nκ(n)) worst-case time, for any ε > 0, where κ(n) is the maximum complexity of a single level in an arrangement of n planes in R. With the current best upper bound κ(n) = O(n), the running time is O(n), for any ε > 0. We also show that the problem of deciding whether there is a center-transversal line parallel to a given direction can be solved in O(n logn) expected time. Finally, we 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 O(nκ(n)), and a linear-time approximation algorithm that computes, for any specified δ > 0, a line whose depth is at least 1− δ 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 CCR02-04118. 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. 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. A preliminary version of the paper had appeared in the Proceedings of the Twenty-Sixth International Conference on Foundations of Software Technology and Theoretical Computer Science, 2006, pp. 93–104.