Discrete & Computational Geometry

, Volume 11, Issue 4, pp 433–452 | Cite as

Algorithms for ham-sandwich cuts

  • Chi-Yuan Lo
  • J. MatoušekEmail author
  • W. Steiger


Given disjoint setsP1,P2, ...,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 inR3 when the three sets are suitably separated.


Linear Time Median Level Separation Condition Discrete Comput Geom Planar Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    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.Google Scholar
  2. 2.
    M. Ajtai, J. Komlós, and E. Szemerédi. Sorting inc Logn parallel steps.Combinatorica 3, 1–19, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Akiyama and N. Alon. Disjoint simplices and geometric hypergraphs.Ann. N.Y. Acad. Sci. 555, 1–3, 1989.MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Alon, I. Bárány, Z. Füredi, and D. Kleitman. Point selections and weak ε-nets for convex hulls. Manuscript, 1991.Google Scholar
  5. 5.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Atallah. A matching problem in the plane.J. Comput. System. Sci. 3, 63–70, 1985.MathSciNetCrossRefGoogle Scholar
  7. 7.
    I. Bárány and W. Steiger. On the expected number ofk-sets. Technical Report, Rutgers University, 1992. AlsoDiscrete Comput. Geom. 11 243–263, 1994.Google Scholar
  8. 8.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Cole, M. Sharir, and C. Yap. Onk-hulls and related topics.SIAM J. Comput. 16, 61–77, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Dey and H. Edelsbrunner. Counting simplex crossings and halving hyperplanes.Proc. 9th Annual ACM Symposium on Computational Geometry, 1993, pp. 270–273.Google Scholar
  11. 11.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    H. Edelsbrunner.Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, 1987.zbMATHCrossRefGoogle Scholar
  13. 13.
    H. Edelsbrunner. Edge-skeletons in arrangements with applications.Algorithmica 1, 93–109, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Edelsbrunner and R. Waupotitsch. Computing a ham sandwich cut in two dimensions.J. Symbolic Comput. 2, 171–178, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Edelsbrunner and E. Welzl. Constructing belts in two-dimensional arrangements with applications.SIAM J. Comput. 15, 271–284, 1986.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    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.Google Scholar
  17. 17.
    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.Google Scholar
  18. 18.
    J. Matoušek. Construction of ε-nets.Discrete Comput. Geom. 5, 427–448, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Matoušek. Approximations and optimal geometric divide-and-conquer.Proc. 23rd Annual ACM Symposium on Theory of Computing 1991, pp. 505–511.Google Scholar
  20. 20.
    J. Matoušek. Randomized optimal algorithm for slope selection.Inform. Process. Lett., 183–187, 1991.Google Scholar
  21. 21.
    N. Megiddo. Partitioning with two lines in the plane,J. Algorithms 6, 430–433, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Pach, W. Steiger, and E. Szemerédi. An upper bound on the number of planark-sets.Discrete Comput. Geom. 7, 109–123, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    L. Shafer and W. Steiger. Randomizing optimal geometric algorithms.Proc. 5th Canadian Conference on Computational Geometry, 1993.Google Scholar
  24. 24.
    D. Willard. Polygon retrieval.SIAM J. Comput. 11, 149–165, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    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.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Charles UniversityPraha 1Czech Republic
  3. 3.Rutgers UniversityPiscatawayUSA

Personalised recommendations