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Weighted Ham-Sandwich Cuts

  • Prosenjit Bose
  • Stefan Langerman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

Abstract

Let R and B be two sets of n points. A ham-sandwich cut is a line that simultaneously bisects R and B, and is known to always exist. This notion can be generalized to the case where each point pRB is associated with a weight w p . A ham-sandwich cut can still be proved to exist, even if weights are allowed to be negative. In this paper, we present a O(n log n) algorithm to find a weighted ham-sandwich cut, but we show that deciding whether that cut is unique is 3-SUM hard.

Keywords

Blue Point Blue Curf Weighted Point Incident Point Weighted Pair 
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.

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References

  1. 1.
    Abbott, T., Demaine, E.D., Demaine, M.L., Kane, D., Langerman, S., Nelson, J., Yeung, V.: Dynamic ham-sandwich cuts of convex polygons in the plane. In: Proceedings of the 17th Canadian Conference on Computational Geometry (CCCG 2005), Windsor, Ontario, Canada (2005) (to appear)Google Scholar
  2. 2.
    Akiyama, J., Nakamura, G., Rivera-Campo, E., Urrutia, J.: Perfect divisions of a cake. In: Proc. Canad. Conf. Comput. Geom (CCCG 1998), pp. 114–115 (1998)Google Scholar
  3. 3.
    Bereg, S., Bose, P., Kirkpatrick, D.: Equitable subdivisions of polygonal regions. Comput. Geom. Theory and Appl. (to appear)Google Scholar
  4. 4.
    Bespamyatnikh, S., Kirkpatrick, D., Snoeyink, J.: Generalizing ham sandwich cuts to equitable subdivisions. Discrete Comput. Geom. 24, 605–622 (2000)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bose, P., Demaine, E.D., Hurtado, F., Iacono, J., Langerman, S., Morin, P.: Geodesic ham-sandwich cuts. In: Proc. of the 2004 ACM Symp. on Computational Geometry, pp. 1–9 (2004)Google Scholar
  6. 6.
    Chan, T.: Remarks on k-level algorithms in the plane (1999) ManuscriptGoogle Scholar
  7. 7.
    Chien, H., Steiger, W.: Some geometric lower bounds. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 72–81. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  8. 8.
    Díaz, M., O’Rourke, J.: Ham-sandwich sectioning of polygons. In: Proc. Canad. Conf. Comput. Geom (CCCG 1990), pp. 282–286 (1990)Google Scholar
  9. 9.
    Edelsbrunner, H., Mücke, E.P.: Simulation of Simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics 9(1), 66–104 (1990)zbMATHCrossRefGoogle Scholar
  10. 10.
    Edelsbrunner, H., Waupotitisch, R.: Computing a ham sandwich cut in two dimensions. J. Symbolic Comput. 2, 171–178 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Erickson, J.: Lower bounds for linear satisfiability problems. Chicago J. Theoret. Comp. Sci. (8) (1999)Google Scholar
  12. 12.
    Erickson, J.: New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput. 28, 1198–1214 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Erickson, J., Seidel, R.: Better lower bounds on detecting affine and spherical degeneracies. Discrete Comput. Geom. 13, 41–57 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gajentaan, A., Overmars, M.: On a class of O(n 2) problems in computational geometry. Comput. Geom. Theory and Appl. 5, 165–185 (1995)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Ito, H., Uehara, H., Yokoyama, M.: 2-dimension ham sandwich theorem for partitioning into three convex pieces. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 129–157. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  16. 16.
    Ito, H., Uehara, H., Yokoyama, M.: A generalization of 2-dimension ham sandwich theorem. TIEICE: IEICE Trans. Comm./Elec./Inf./Sys (2001)Google Scholar
  17. 17.
    Langerman, S., Steiger, W.: Optimization in arrangements. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 50–61. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Lo, C.-Y., Matoušek, J., Steiger, W.L.: Algorithms for ham-sandwich cuts. Discrete Comput. Geom. 11, 433–452 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lo, C.-Y., Steiger, W.L.: An optimal time algorithm for ham-sandwich cuts in the plane. In: Proc. Canad. Conf. Comput. Geom (CCCG 1990), pp. 5–9 (1990)Google Scholar
  20. 20.
    Matousek, J.: Using the Borsuk-Ulam theorem. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  21. 21.
    Megiddo, N.: Partitioning with two lines in the plane. J. Algorithms, 430–433 (1985)Google Scholar
  22. 22.
    Ramos, E.A.: Equipartition of mass distributions by hyperplanes. Discrete Comput. Geom. 15(2), 147–167 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Stojmenovic, I.: Bisections and ham-sandwich cuts of convex polygons and polyhedra. Inf. Process. Lett. 38(1), 15–21 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Stone, A.H., Tukey, J.W.: Generalized ‘sandwich’ theorems. Duke Math. J. 9, 356–359 (1942)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Stefan Langerman
    • 2
  1. 1.School of Computer ScienceCarleton UniversityCanada
  2. 2.Département d’InformatiqueUniversité Libre de BruxellesBelgium

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