Algorithms for ham-sandwich cuts Article

First Online: 01 April 1994 Received: 21 September 1992 Revised: 28 June 1993 DOI :
10.1007/BF02574017

Cite this article as: Lo, CY., Matoušek, J. & Steiger, W. Discrete Comput Geom (1994) 11: 433. doi:10.1007/BF02574017
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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 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.

A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.

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Authors and Affiliations 1. AT&T Bell Laboratories Murray Hill USA 2. Charles University Praha 1 Czech Republic 3. Free University Berlin Berlin Germany 4. Rutgers University Piscataway USA