Discrete & Computational Geometry

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

Algorithms for ham-sandwich cuts

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.

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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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