# Computing a center-transversal line

Published

Conference Paper

© 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 [21], 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.

### Duke Authors

### Cited Authors

- Agarwal, PK; Cabello, S; Sellarès, JA; Sharir, M

### Published Date

- January 1, 2006

### Published In

### Volume / Issue

- 4337 LNCS /

### Start / End Page

- 93 - 104

### Electronic International Standard Serial Number (EISSN)

- 1611-3349

### International Standard Serial Number (ISSN)

- 0302-9743

### International Standard Book Number 13 (ISBN-13)

- 9783540499947

### Citation Source

- Scopus