Algorithms for hamsandwich cuts
 ChiYuan Lo,
 J. Matoušek,
 W. Steiger
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Abstract
Given disjoint setsP _{1},P _{2}, ...,P _{ d } inR ^{ d } withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _{ i }. We present algorithms for finding hamsandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worstcase time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofksets inR ^{ d−1 }. With the current estimates, we get complexity close toO(n ^{ 3/2 }) ford=3, roughlyO(n ^{ 8/3 }) ford=4, andO(n ^{ d−1−a(d) }) for somea(d)>0 (going to zero asd increases) for largerd. We also give a lineartime algorithm for hamsandwich cuts inR ^{3} when the three sets are suitably separated.
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 Title
 Algorithms for hamsandwich cuts
 Journal

Discrete & Computational Geometry
Volume 11, Issue 1 , pp 433452
 Cover Date
 19941201
 DOI
 10.1007/BF02574017
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 ChiYuan Lo ^{(1)}
 J. Matoušek ^{(2)}
 W. Steiger ^{(4)}
 Author Affiliations

 1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA
 2. Charles University, Malostranské nám, 25, 118 00, Praha 1, Czech Republic
 4. Rutgers University, 08903, Piscataway, NJ, USA