Computing a center-transversal line

Conference Paper

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 [12,24], 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(n κ (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 ) of [21], the running time is O(n ), 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(n κ (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. 3 12 1+ε 2 3 5/2 6+ε 1+ε 2

Full Text

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

Digital Object Identifier (DOI)

  • 10.1007/11944836_11

Citation Source

  • Scopus