The Exact Lattice Width of Planar Sets and Minimal Arithmetical Thickness

  • F. Feschet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)


We provide in this paper an algorithm for the exact computation of the lattice width of an integral polygon K with n vertices in O(n log s) arithmetic operations where s is a bound on all integers defining vertices and edges. We also provide an incremental version of the algorithm whose update complexity is shown to be O(log n + log s). We apply this algorithm to construct the arithmetical line with minimal thickness, which contains a given set of integer points.


Convex Hull Integer Linear Program Arithmetic Operation Supporting Point Integer Point 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kannan, R.: A polynomial algorithm for the two variable integer programming problem. J. Assoc. Comput. Mach. 27, 118–122 (1980)MATHMathSciNetGoogle Scholar
  2. 2.
    Scarf, H.E.: Production sets with indivisibilities part i and part ii. Econometrica 49, 1–32, 395–423 (1981)Google Scholar
  3. 3.
    Hirschberg, D., Wong, C.: A polynomial-time algorithm for the knapsack problem with two variables. J. Assoc. Comput. Mach. 23, 147–154 (1976)MATHMathSciNetGoogle Scholar
  4. 4.
    Lenstra, H.: Integer Programming with a Fixed Number of Variables. Math. Oper. Research 8, 535–548 (1983)Google Scholar
  5. 5.
    Eisenbrand, F., Rote, G.: Fast 2-variable integer programming. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 78–89. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Eisenbrand, F., Laue, S.: A linear algorithm for integer programming in the plane. Math. Program. Ser. A 102, 249–259 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, Chichester (1998)MATHGoogle Scholar
  8. 8.
    Barvinok, A.: A Course in Convexity. Graduates Studies in Mathematics, vol. 54. Amer. Math. Soc. (2002)Google Scholar
  9. 9.
    de Berg, M., Schwarzkopf, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2000)MATHGoogle Scholar
  10. 10.
    Kaib, M., Schnörr, C.P.: The Generalized Gauss Reduction Algorithm. Journal of Algorithms 21(3), 565–578 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rote, G.: Finding a shortest vector in a two-dimensional lattice modulo m. Theoretical Computer Science 172(1-2), 303–308 (1997)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Reveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’etat, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  13. 13.
    Debled-Rennesson, I., Reveillès, J.P.: A linear algorithm for segmentation of digital curves. International Journal on Pattern Recognition and Artificial Intelligence 9, 635–662 (1995)CrossRefGoogle Scholar
  14. 14.
    Françon, J., Schramm, J.M., Tajine, M.: Recognizing artimethic straight lines and planes. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 141–150. Springer, Heidelberg (1996)Google Scholar
  15. 15.
    Gérard, Y., Debled-Rennesson, I., Zimmermann, P.: An elementary digital plane recognition algorithm. Discrete Applied Mathematics 151, 169–183 (2005)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Houle, M., Toussaint, G.: Computing the width of a set. IEEE Trans. on Pattern Analysis and Machine Intelligence 10(5), 761–765 (1988)MATHCrossRefGoogle Scholar
  17. 17.
    Lachaud, G.: Klein polygons and geometric diagrams. Contemporary Math. 210, 365–372 (1998)MathSciNetGoogle Scholar
  18. 18.
    Lachaud, G.: Sails and klein polyhedra. Contemporary Math. 210, 373–385 (1998)MathSciNetGoogle Scholar
  19. 19.
    Arnold, V.: Higher dimensional continued fractions. Regular and chaotic dynamics 3, 10–17 (1998)MATHCrossRefGoogle Scholar
  20. 20.
    Harvey, W.: Computing two-dimensional Integer Hulls. SIAM Journal on Computing 28(6), 2285–2299 (1999)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Cook, W., Hartman, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12, 27–37 (1992)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1996)Google Scholar
  23. 23.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • F. Feschet
    • 1
  1. 1.LLAIC1 – IUT Clermont-FerrandAubièreFrance

Personalised recommendations