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Ham-Sandwich Cuts for Abstract Order Types

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.

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Notes

  1. 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. 1.

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

  2. 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. 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. 4.

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

  5. 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)

  6. 6.

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

  7. 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)

  8. 8.

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

  9. 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. 10.

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

  11. 11.

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

  12. 12.

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

  13. 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)

  14. 14.

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

  15. 15.

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

  16. 16.

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

  17. 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. 18.

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

  19. 19.

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

  20. 20.

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

  21. 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)

  22. 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)

  23. 23.

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

  24. 24.

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

  25. 25.

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

  26. 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)

  27. 27.

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

  28. 28.

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

  29. 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)

  30. 30.

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

  31. 31.

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

  32. 32.

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

  33. 33.

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

  34. 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)

  35. 35.

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

  36. 36.

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

  37. 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. 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. 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. 40.

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

  41. 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)

  42. 42.

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

  43. 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. 44.

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

  45. 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)

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Author information

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.

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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

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Keywords

  • Ham-sandwich cut
  • Abstract order type
  • Pseudo-line arrangement