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.
Article PDF
Similar content being viewed by others
References
P.K. Agarwal and J. Matoušek. Dynamic half-space range reporting and its applications. Technical Report CS-91-43, Department of Computer Science, Duke University, 1991. The results combined with results of D. Eppstein appear inProc. 33rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 80–89.
M. Ajtai, J. Komlós, and E. Szemerédi. Sorting inc Logn parallel steps.Combinatorica 3, 1–19, 1983.
J. Akiyama and N. Alon. Disjoint simplices and geometric hypergraphs.Ann. N.Y. Acad. Sci. 555, 1–3, 1989.
N. Alon, I. Bárány, Z. Füredi, and D. Kleitman. Point selections and weak ε-nets for convex hulls. Manuscript, 1991.
B. Aronov, B. Chazelle, H. Edelsbrunner, L. Guibas, M. Sharir, and R. Wenger. Points and triangles in the plane and halving planes in the space.Discrete Comput. Geom. 6, 435–442, 1991.
M. Atallah. A matching problem in the plane.J. Comput. System. Sci. 3, 63–70, 1985.
I. Bárány and W. Steiger. On the expected number ofk-sets. Technical Report, Rutgers University, 1992. AlsoDiscrete Comput. Geom. 11 243–263, 1994.
R. Cole, J. S. Salowe, W. L. Steiger and E. Szemerédi. An optimal-time algorithm for slope selection.SIAM J. Comput. 18, 792–810, 1989.
R. Cole, M. Sharir, and C. Yap. Onk-hulls and related topics.SIAM J. Comput. 16, 61–77, 1987.
T. Dey and H. Edelsbrunner. Counting simplex crossings and halving hyperplanes.Proc. 9th Annual ACM Symposium on Computational Geometry, 1993, pp. 270–273.
M. B. Dillencourt, D. M. Mount, and N. S. Netanyahu. A randomized algorithm for slope selection.Internat. J. Comput. Geom. Appl. 2, 1–27, 1992.
H. Edelsbrunner.Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, 1987.
H. Edelsbrunner. Edge-skeletons in arrangements with applications.Algorithmica 1, 93–109, 1986.
H. Edelsbrunner and R. Waupotitsch. Computing a ham sandwich cut in two dimensions.J. Symbolic Comput. 2, 171–178, 1986.
H. Edelsbrunner and E. Welzl. Constructing belts in two-dimensional arrangements with applications.SIAM J. Comput. 15, 271–284, 1986.
C.-Y. Lo, J. Matoušek, and W. L. Steiger. Ham-sandwich cuts inR d.Proc. 24th ACM Symposium on Theory of Computing, 1992, pp. 539–545.
C.-Y. Lo and W. L. Steiger. An optimal time algorithm for ham-sandwich cuts in the plane.Proc. 2nd Canadian Conference on Computational Geometry, 1990, pp. 5–9.
J. Matoušek. Construction of ε-nets.Discrete Comput. Geom. 5, 427–448, 1990.
J. Matoušek. Approximations and optimal geometric divide-and-conquer.Proc. 23rd Annual ACM Symposium on Theory of Computing 1991, pp. 505–511.
J. Matoušek. Randomized optimal algorithm for slope selection.Inform. Process. Lett., 183–187, 1991.
N. Megiddo. Partitioning with two lines in the plane,J. Algorithms 6, 430–433, 1985.
J. Pach, W. Steiger, and E. Szemerédi. An upper bound on the number of planark-sets.Discrete Comput. Geom. 7, 109–123, 1992.
L. Shafer and W. Steiger. Randomizing optimal geometric algorithms.Proc. 5th Canadian Conference on Computational Geometry, 1993.
D. Willard. Polygon retrieval.SIAM J. Comput. 11, 149–165, 1982.
R. T. Živaljević and S. T. Vrećica. The colored Tverberg problem and complexes of injective functions.J. Combin. Theory Ser. A 61, 309–318, 1992.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Lo, CY., Matoušek, J. & Steiger, W. Algorithms for ham-sandwich cuts. Discrete Comput Geom 11, 433–452 (1994). https://doi.org/10.1007/BF02574017
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574017