Algorithms for hamsandwich cuts
 ChiYuan Lo,
 J. Matoušek,
 W. Steiger
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 hamsandwich 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 worstcase time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofksets 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 lineartime algorithm for hamsandwich cuts inR ^{3} when the three sets are suitably separated.
 P.K. Agarwal and J. Matoušek. Dynamic halfspace range reporting and its applications. Technical Report CS9143, 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. CrossRef
 N. Alon, I. Bárány, Z. Füredi, and D. Kleitman. Point selections and weak εnets 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. CrossRef
 I. Bárány and W. Steiger. On the expected number ofksets. Technical Report, Rutgers University, 1992. AlsoDiscrete Comput. Geom. 11 243–263, 1994.
 R. Cole, J. S. Salowe, W. L. Steiger and E. Szemerédi. An optimaltime algorithm for slope selection.SIAM J. Comput. 18, 792–810, 1989. CrossRef
 R. Cole, M. Sharir, and C. Yap. Onkhulls and related topics.SIAM J. Comput. 16, 61–77, 1987. CrossRef
 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. CrossRef
 H. Edelsbrunner.Algorithms in Combinatorial Geometry. SpringerVerlag, Berlin, 1987.
 H. Edelsbrunner. Edgeskeletons in arrangements with applications.Algorithmica 1, 93–109, 1986. CrossRef
 H. Edelsbrunner and R. Waupotitsch. Computing a ham sandwich cut in two dimensions.J. Symbolic Comput. 2, 171–178, 1986. CrossRef
 H. Edelsbrunner and E. Welzl. Constructing belts in twodimensional arrangements with applications.SIAM J. Comput. 15, 271–284, 1986. CrossRef
 C.Y. Lo, J. Matoušek, and W. L. Steiger. Hamsandwich 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 hamsandwich cuts in the plane.Proc. 2nd Canadian Conference on Computational Geometry, 1990, pp. 5–9.
 J. Matoušek. Construction of εnets.Discrete Comput. Geom. 5, 427–448, 1990. CrossRef
 J. Matoušek. Approximations and optimal geometric divideandconquer.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. CrossRef
 J. Pach, W. Steiger, and E. Szemerédi. An upper bound on the number of planarksets.Discrete Comput. Geom. 7, 109–123, 1992. CrossRef
 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. CrossRef
 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. CrossRef
 Title
 Algorithms for hamsandwich cuts
 Journal

Discrete & Computational Geometry
Volume 11, Issue 1 , pp 433452
 Cover Date
 19941201
 DOI
 10.1007/BF02574017
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 ChiYuan Lo ^{(1)}
 J. Matoušek ^{(2)}
 W. Steiger ^{(4)}
 Author Affiliations

 1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA
 2. Charles University, Malostranské nám, 25, 118 00, Praha 1, Czech Republic
 4. Rutgers University, 08903, Piscataway, NJ, USA