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Discrete & Computational Geometry

, Volume 11, Issue 4, pp 433–452 | Cite as

Algorithms for ham-sandwich cuts

  • Chi-Yuan Lo
  • J. MatoušekEmail author
  • W. Steiger
Article

Abstract

Given disjoint setsP1,P2, ...,P d inR d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP i . We present algorithms for finding ham-sandwich 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 worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets 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 linear-time algorithm for ham-sandwich cuts inR3 when the three sets are suitably separated.

Keywords

Linear Time Median Level Separation Condition Discrete Comput Geom Planar Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Charles UniversityPraha 1Czech Republic
  3. 3.Rutgers UniversityPiscatawayUSA

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