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Theory of Computing Systems

, Volume 62, Issue 6, pp 1490–1524 | Cite as

Monotone Paths in Geometric Triangulations

  • Adrian Dumitrescu
  • Ritankar Mandal
  • Csaba D. Tóth
Article
  • 54 Downloads
Part of the following topical collections:
  1. Special Issue on Combinatorial Algorithms

Abstract

(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of n points in the plane is O(1.7864 n ). This improves an earlier upper bound of O(1.8393 n ); the current best lower bound is Ω(1.7003 n ). (II) Given a planar geometric graph G with n vertices, we show that the number of monotone paths in G can be computed in O(n2) time.

Keywords

Monotone path Triangulation Counting algorithm 

References

  1. 1.
    Adler, I., Papadimitriou, C., Rubinstein, A.: On simplex pivoting rules and complexity theory. In: Proceedings of the 17th IPCO, LNCS 8494, Springer (2014)Google Scholar
  2. 2.
    Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Ann. Discret. Math. 12, 9–12 (1982)MathSciNetMATHGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry, 3rd edn. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Buchin, K., Knauer, C., Kriegel, K., Schulz, A., Seidel, R.: On the number of cycles in planar graphs. In: Proceedings of the 13th COCOON, LNCS 4598, Springer (2007)Google Scholar
  5. 5.
    Dumitrescu, A., Löffler, M., Schulz, A., Tóth, C. s. D.: Counting carambolas. Graphs Combin. 32(3), 923–942 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dumitrescu, A., Rote, G., Tóth, Cs. D.: Monotone paths in planar convex subdivisions and polytopes. In: Discrete Geometry and Optimization, vol. 69 of Fields Institute of Communications, Springer, pp. 79–104 (2013)Google Scholar
  7. 7.
    Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, C. s. D.: Bounds on the maximum multiplicity of some common geometric graphs. SIAM J. Discret. Math. 27(2), 802–826 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dumitrescu, A., Tóth, C. s. D.: Computational Geometry Column 54. SIGACT News Bullet. 43(4), 90–97 (2012)CrossRefGoogle Scholar
  9. 9.
    Dumitrescu, A., Tóth, C. s. D.: Convex polygons in geometric triangulations. Combin. Probab. Comput. 26(5), 641–659 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    García, A., Noy, M., Tejel, A.: Lower bounds on the number of crossing-free subgraphs of K N. Comput. Geom. 16(4), 211–221 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gärtner, B., Kaibel, V.: Two new bounds for the random-edge simplex-algorithm. SIAM J. Discret. Math. 21(1), 178–190 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kaibel, V., Mechtel, R., Sharir, M., Ziegler, G.M.: The simplex algorithm in dimension three. SIAM J. Comput. 34(2), 475–497 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kalai, G.: Upper bounds for the diameter and height of graphs of convex polyhedra. Discret. Comput. Geom. 8(4), 363–372 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kalai, G.: Polytope skeletons and paths. In: Handbook of Discrete and Computational Geometry Goodman, J., O’Rourke, J., Tóth, C. D. (eds), Chapter 19, pp. 505–532, 3rd edn, CRC Press, Boca Raton (2017)Google Scholar
  15. 15.
    Klee, V.: Paths on polyhedra I. J. SIAM 13(4), 946–956 (1965)MathSciNetMATHGoogle Scholar
  16. 16.
    van Kreveld, M., Löffler, M., Pach, J.: How many potatoes are in a mesh?, in Proc. 23rd ISAAC, LNCS 7676, Springer, pp. 166–176 (2012)Google Scholar
  17. 17.
    Matoušek, J., Szabó, T.: RANDOM EDGE can be exponential on abstract cubes. Adv. Math. 204(1), 262–277 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46, 39–47 (2004). Corrected version: arXiv:http://arXiv.org/abs/1101.0967,2011MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Razen, A., Snoeyink, J., Welzl, E.: Number of crossing-free geometric graphs vs. triangulations. Electron. Notes Discret. Math. 31, 195–200 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Santos, F.: A counterexample to the Hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Santos, F.: Recent progress on the combinatorial diameter of polytopes and simplicial complexes. TOP 21(3), 426–460 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electron. J. Combin. 18, P70 (2011)MathSciNetMATHGoogle Scholar
  23. 23.
    Sharir, M., Sheffer, A.: Counting plane graphs: cross-graph charging schemes. Combin. Probab. Comput. 22(6), 935–954 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sharir, M., Sheffer, A., Welzl, E.: Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn’s technique. J. Combin. Theory, Ser. A 120(4), 777–794 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695–720 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Sheffer, A.: Numbers of plane graphs, https://adamsheffer.wordpress.com/numbers-of-plane-graphs/ (version of April, 2016)
  27. 27.
    Todd, M.J.: The monotonic bounded Hirsch conjecture is false for dimension at least 4. Math. Oper. Res. 5(4), 599–601 (1980)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Todd, M.J.: An improved Kalai-Kleitman bound for the diameter of a polyhedron. SIAM J. Discret. Math. 28, 1944–1947 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ziegler, G.M.: Lectures on Polytopes, vol. 152 of GTM, Springer, pp. 83–93 (1994)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Ritankar Mandal
    • 1
  • Csaba D. Tóth
    • 2
    • 3
  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.Department of MathematicsCalifornia State UniversityNorthridge, Los AngelesUSA
  3. 3.Department of Computer ScienceTufts UniversityMedfordUSA

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