Computing a Center-Transversal Line

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 [14, 29], but the best known algorithm for finding such a line takes roughly n12 time. We propose an algorithm that finds a center-transversal line in O(n1+ 2 (n)) worst-case time, for any > 0, where (n) is the maximum complexity of a single level in an arrangement of n planes in R3 . With the current best upper bound (n) = O(n5/2 ) of [26], the running time is O(n6+ ), for any > 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 log n) 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(n1+ 2 (n)), and a linear-time appr...

• Type: Conference
• Year: 2006
• Where: FSTTCS
• Authors: Pankaj K. Agarwal, Sergio Cabello, Joan Antoni Sellarès, Micha Sharir
• Keywords: Center-transversal Line, Center-transversal Line Parallel, FSTTCS 2006, Point Set, Software Engineering