Algorithms for ham-sandwich cuts

Given disjoint setsP1,P2, ...,Pd inRd withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects thePi. 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 inRd−1. With the current estimates, we get complexity close toO(n3/2) ford=3, roughlyO(n8/3) ford=4, andO(nd−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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