Skip to main content
SpringerLink
Log in

Ham-Sandwich Cuts for Abstract Order Types

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The linear-time ham-sandwich cut algorithm of Lo, Matoušek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen vertical lines. We consider the setting where we are not given the coordinates of the point set, but only the orientation of each point triple (the order type) and give a deterministic linear-time algorithm for the mentioned sub-algorithm. This yields a linear-time ham-sandwich cut algorithm even in our restricted setting. We also show that our methods are applicable to abstract order types.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Lo et al. [30] refer to [32], where [27, Lemma 4.5] (Lemma 12 herein) is used, and also refer to [33] in this context, where a general algorithm for constructing \(\varepsilon \)-approximations is given.

References

  1. Agarwal, P.K., Sharir, M.: Pseudo-line arrangements: duality, algorithms, and applications. SIAM J. Comput. 34(3), 526–552 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aichholzer, O., Hackl, T., Korman, M., Pilz, A., Vogtenhuber, B.: Geodesic-preserving polygon simplification. In: Cai, L., Cheng, S.-W., Lam, T.W. (eds.) ISAAC. LNCS, vol. 8283, pp. 11–21. Springer, Berlin (2013)

  3. Aichholzer, O., Korman, M., Pilz, A., Vogtenhuber, B.: Geodesic order types. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON. LNCS, vol. 7434. Springer, Berlin (2012)

  4. Aichholzer, O., Miltzow, T., Pilz, A.: Extreme point and halving edge search in abstract order types. Comput. Geom. 46(8), 970–978 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avnaim, F., Boissonnat, J.D., Devillers, O., Preparata, F.P., Yvinec, M.: Evaluating signs of determinants using single-precision arithmetic. Algorithmica 17(2), 111–132 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bland, R.: A combinatorial abstraction of linear programming. J. Comb. Theory B 23, 33–57 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., Tarjan, R.E.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), 448–461 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boissonnat, J.D., Snoeyink, J.: Efficient algorithms for line and curve segment intersection using restricted predicates. Comput. Geom. 16(1), 35–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bose, P., Demaine, E.D., Hurtado, F., Iacono, J., Langerman, S., Morin, P.: Geodesic ham-sandwich cuts. In: SoCG, pp. 1–9. ACM (2004)

  10. Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38(1), 165–194 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Edelsbrunner, H., Guibas, L.J.: Corrigendum: topologically sweeping an arrangement. J. Comput. Syst. Sci. 42(2), 249–251 (1991)

    Article  MATH  Google Scholar 

  13. Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9(1), 66–104 (1990)

    Article  MATH  Google Scholar 

  14. Edelsbrunner, H., Waupotitsch, R.: Computing a ham-sandwich cut in two dimensions. J. Symb. Comput. 2(2), 171–178 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Erickson, J., Hurtado, F., Morin, P.: Centerpoint theorems for wedges. Discrete Math. Theor. Comput. Sci. 11(1), 45–54 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Erickson, J.G.: Lower bounds for fundamental geometric problems. Ph.D. Thesis, University of California at Berkeley (1996)

  17. Felsner, S., Pilz, A.: Ham-Sandwich cuts for abstract order types. In: Algorithms and Computation—25th International Symposium, ISAAC 2014, Jeonju, Korea, December 15–17, 2014, Proceedings, LNCS, vol. 8889, pp. 726–737. Springer, Berlin (2014)

  18. Fukuda, K.: Oriented matroid programming. Ph.D. thesis, University of Waterloo, Canada (1981)

  19. Gärtner, B., Welzl, E.: Vapnik–Chervonenkis dimension and (pseudo-)hyperplane arrangements. Discrete Comput. Geom. 12, 399–432 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goodman, J.E.: Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Math. 32(1), 27–35 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goodman, J.E., Pollack, R.: On the combinatorial classification of nondegenerate configurations in the plane. J. Comb. Theory Ser. A 29(2), 220–235 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goodman, J.E., Pollack, R.: Proof of Grünbaum’s conjecture on the stretchability of certain arrangements of pseudolines. J. Comb. Theory Ser. A 29(3), 385–390 (1980)

    Article  MATH  Google Scholar 

  23. Goodman, J.E., Pollack, R.: Multidimensional sorting. SIAM J. Comput. 12(3), 484–507 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goodman, J.E., Pollack, R.: Semispaces of configurations, cell complexes of arrangements. J. Comb. Theory Ser. A 37(3), 257–293 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in \(R^{{d}}\). Discrete Comput. Geom. 1, 219–227 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Goodman, J.E., Pollack, R.: Allowable sequences and order types in discrete and computational geometry. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, pp. 103–134. Springer, Berlin (1993)

    Chapter  Google Scholar 

  27. Haussler, D., Welzl, E.: \(\varepsilon \)-Nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Knuth, D.E.: Axioms and Hulls, LNCS, vol. 606. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  29. Levi, F.: Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade. Ber. Math. Phys. Kl. Sächs. Akad. Wiss. Leipzig 78, 256–267 (1926). (In German)

    MATH  Google Scholar 

  30. Lo, C.Y., Matoušek, J., Steiger, W.: Algorithms for ham-sandwich cuts. Discrete Comput. Geom. 11, 433–452 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lo, C.Y., Steiger, W.: An optimal time algorithm for ham-sandwich cuts in the plane. In: CCCG, pp. 5–9 (1990)

  32. Matoušek, J.: Construction of \(\varepsilon \)-nets. Discrete Comput. Geom. 5, 427–448 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Matoušek, J.: Approximations and optimal geometric divide-and-conquer. In: STOC, pp. 505–511. ACM (1991)

  34. Matoušek, J.: Epsilon-nets and computational geometry. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, pp. 69–89. Springer, Berlin (1993)

    Chapter  Google Scholar 

  35. Matoušek, J.: Approximations and optimal geometric divide-and-conquer. J. Comput. Syst. Sci. 50(2), 203–208 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Megiddo, N.: Partitioning with two lines in the plane. J. Algorithms 6(3), 430–433 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  37. Meikle, L.I., Fleuriot, J.D.: Mechanical theorem proving in computational geometry. In: Hong, H., Wang, D. (eds.) Automated Deduction in Geometry. LNCS, vol. 3763, pp. 1–18. Springer, Berlin (2004)

  38. Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytope varieties. In: Viro, O.Y. (ed.) Topology and Geometry—Rohlin Seminar. Lecture Notes Math., vol. 1346, pp. 527–544. Springer, Berlin (1988)

  39. Pichardie, D., Bertot, Y.: Formalizing convex hull algorithms. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs, LNCS, vol. 2152, pp. 346–361. Springer, Berlin (2001)

  40. Pilz, A.: On the complexity of problems on order types and geometric graphs. Ph.D. Thesis, Graz University of Technology (2014)

  41. Richter-Gebert, J., Ziegler, G.M.: Oriented matroids. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn, pp. 129–151. Chapman and Hall/CRC, Boca Raton (2004)

    Google Scholar 

  42. Roy, S., Steiger, W.: Some combinatorial and algorithmic applications of the Borsuk–Ulam theorem. Graphs Comb. 23(1), 331–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) Graph Drawing, LNCS, vol. 5849, pp. 334–344. Springer, Berlin (2009)

  44. Snoeyink, J., Hershberger, J.: Sweeping arrangements of curves. In: SoCG, pp. 354–363 (1989)

  45. Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Stefan Felsner

  2. Department of Computer Science, ETH Zürich, Zurich, Switzerland

    Alexander Pilz

Authors
  1. Stefan Felsner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Pilz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Pilz.

Additional information

This work has been supported by the ESF EUROCORES programme EuroGIGA-ComPoSe. A.P. is supported by an Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J-3847-N35. A preliminary version of this paper appeared in the proceedings of ISAAC 2014 [17]. Part of this work was presented in the PhD thesis [40] of the second author.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Felsner, S., Pilz, A. Ham-Sandwich Cuts for Abstract Order Types. Algorithmica 80, 234–257 (2018). https://doi.org/10.1007/s00453-016-0246-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0246-4

Keywords

Search

Navigation