Advertisement

Efficient Lattice Width Computation in Arbitrary Dimension

  • Émilie Charrier
  • Lilian Buzer
  • Fabien Feschet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)

Abstract

We provide an algorithm for the exact computation of the lattice width of an integral polygon K in linear-time with respect to the size of K. Moreover, we describe how this new algorithm can be extended to an arbitrary dimension thanks to a greedy approach avoiding complex geometric processings.

Keywords

Convex Hull Convex Body Arbitrary Dimension Integer Point Supporting Line 
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.

References

  1. 1.
    Arnold, V.I.: Higher dimensional continued fractions. Regular and chaotic dynamics 3, 10–17 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity. Graduates Studies in Mathematics, vol. 54. Amer. Math. Soc, Providence (2002)zbMATHGoogle Scholar
  3. 3.
    Boyce, J.E., Dobkin, D.P., Drysdale, R.L., Guibas, L.J.: Finding extremal polygons. In: STOC, pp. 282–289 (1982)Google Scholar
  4. 4.
    Cook, W., Hartman, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12, 27–37 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Berg, M., Schwarzkopf, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Debled-Rennesson, I., Reveillès, J.-P.: A linear algorithm for segmentation of digital curves. IJPRAI 9(4), 635–662 (1995)Google Scholar
  7. 7.
    Dobkin, D.P., Snyder, L.: On a general method for maximizing and minimizing among certain geometric problems. In: SFCS 1979: Proceedings of the 20th Annual Symposium on Foundations of Computer Science, Washington, DC, USA, pp. 9–17. IEEE Computer Society, Los Alamitos (1979)Google Scholar
  8. 8.
    Eisenbrand, F., Laue, S.: A linear algorithm for integer programming in the plane. Math. Program. Ser. A 102, 249–259 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Feschet, F.: The exact lattice width of planar sets and minimal arithmetical thickness. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 25–33. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Feschet, F.: The lattice width and quasi-straightness in digital spaces. In: 19th International Conference on Pattern Recognition (ICPR), pp. 1–4. IEEE, Los Alamitos (2008)Google Scholar
  12. 12.
    Fleischer, R., Mehlhorn, K., Rote, G., Welzl, E., Yap, C.-K.: Simultaneous inner and outer approximation of shapes. Algorithmica 8(5&6), 365–389 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Harvey, W.: Computing two-dimensional Integer Hulls. SIAM Journal on Computing 28(6), 2285–2299 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hirschberg, D., Wong, C.K.: A polynomial-time algorithm for the knapsack problem with two variables. J. Assoc. Comput. Mach. 23, 147–154 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Houle, M.E., Toussaint, G.T.: Computing the width of a set. IEEE Trans. on Pattern Analysis and Machine Intelligence 10(5), 761–765 (1988)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hübler, A., Klette, R., Voss, K.: Determination of the convex hull of a finite set of planar points within linear time. Elektronische Informationsverarbeitung und Kybernetik 17(2/3), 121–139 (1981)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kaib, M., Schnörr, C.-P.: The Generalized Gauss Reduction Algorithm. Journal of Algorithms 21(3), 565–578 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kannan, R.: A polynomial algorithm for the two variable integer programming problem. J. Assoc. Comput. Mach. 27, 118–122 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lachaud, G.: Klein polygons and geometric diagrams. Contemporary Math. 210, 365–372 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lachaud, G.: Sails and klein polyhedra. Contemporary Math. 210, 373–385 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lenstra, H.W.: Integer Programming with a Fixed Number of Variables. Math. Oper. Research 8, 535–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lyashko, S.I., Rublev, B.V.: Minimal ellipsoids and maximal simplexes in 3D euclidean space. Cybernetics and Systems Analysis 39(6), 831–834 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reveillès, J.-P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’etat, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  24. 24.
    Rote, G.: Finding a shortest vector in a two-dimensional lattice modulo m. Theoretical Computer Science 172(1-2), 303–308 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Scarf, H.E.: Production sets with indivisibilities part i and part ii. Econometrica 49, 1–32, 395–423 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, Chichester (1998)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Émilie Charrier
    • 1
    • 2
    • 3
  • Lilian Buzer
    • 1
    • 2
  • Fabien Feschet
    • 4
  1. 1.Université Paris-Est, LABINFO-IGM, CNRS, UMR 8049France
  2. 2.ESIEE, 2, boulevard Blaise Pascal, Cité DESCARTESNoisy le Grand CedexFrance
  3. 3.DGA/D4S/MRISFrance
  4. 4.LAICUniv. Clermont 1Aubière CedexFrance

Personalised recommendations