Discrete & Computational Geometry

, Volume 11, Issue 4, pp 433-452

First online:

Algorithms for ham-sandwich cuts

  • Chi-Yuan LoAffiliated withAT&T Bell Laboratories
  • , J. MatoušekAffiliated withCharles University Email author 
  • , W. SteigerAffiliated withRutgers University

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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 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 inR 3 when the three sets are suitably separated.