Computing a center-transversal line
© Springer-Verlag Berlin Heidelberg 2006. 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 n12 time. We propose an algorithm that finds a center-transversal line in O(n1+εκ2(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 R3. With the current best upper bound κ(n) = O(n5/2) of , the running time is O(n6+ε), for any ε>0. 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 O(n1+εκ2(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.
Agarwal, PK; Cabello, S; Sellarès, JA; Sharir, M
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International Standard Book Number 13 (ISBN-13)