Asymptotic Analysis and Random Sampling of Digitally Convex Polyominoes

  • O. Bodini
  • Ph. Duchon
  • A. Jacquot
  • L. Mutafchiev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)


Recent work of Brlek et al. gives a characterization of digitally convex polyominoes using combinatorics on words. From this work, we derive a combinatorial symbolic description of digitally convex polyominoes and use it to analyze their limit properties and build a uniform sampler. Experimentally, our sampler shows a limit shape for large digitally convex polyominoes.


Boltzmann Distribution Limit Shape Combinatorial Class Coprime Integer Standard Decomposition 
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.


  1. 1.
    Bodini, O., Gardy, D., Roussel, O.: Boys-and-girls birthdays and hadamard products. Fundam. Inform. 117(1-4), 85–101 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bodini, O., Jacquot, A.: Boltzmann Samplers for Colored Combinatorial Objects. In: Proceedings of Gascom 2008 (2008)Google Scholar
  3. 3.
    Bodini, O., Lumbroso, J.: Dirichlet random samplers for multiplicative combinatorial structures. In: 2012 Proceedings of the Eigth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), Kyoto, Japan, pp. 92–106 (2012)Google Scholar
  4. 4.
    Bodini, O., Ponty, Y.: Multi-dimensional boltzmann sampling of languages. DMTCS Proceedings (01), 49–64 (2010)Google Scholar
  5. 5.
    Bodini, O., Roussel, O., Soria, M.: Boltzmann samplers for first-order differential specifications. Discrete Applied Mathematics 160(18), 2563–2572 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bodirsky, M., Fusy, E., Kang, M., Vigerske, S.: An unbiased pointing operator for unlabeled structures, with applications to counting and sampling. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 356–365. Society for Industrial and Applied Mathematics, Philadelphia (2007)Google Scholar
  7. 7.
    Brlek, S., Lachaud, J.-O., Provencal, X., Reutenauer, C.: Lyndon christoffel digitally convex. Pattern Recognition 42(10), 2239–2246 (2009)zbMATHCrossRefGoogle Scholar
  8. 8.
    Duchon, P., Flajolet, P., Louchard, G., Schaeffer, G.: Boltzmann samplers for the random generation of combinatorial structures. Combinatorics, Probablity, and Computing 13(4-5), 577–625 (2004); Special issue on Analysis of AlgorithmsGoogle Scholar
  9. 9.
    Flajolet, P., Fusy, E., Pivoteau, C.: Boltzmann sampling of unlabelled structures. In: SIAM Press (ed.) Proceedings of ANALCO 2007 (Analytic Combinatorics and Algorithms) Conference, vol. 126, pp. 201–211 (2007)Google Scholar
  10. 10.
    Flajolet, P., Salvy, B., Zimmermann, P.: Automatic average-case analysis of algorithm. Theoretical Computer Science 79(1), 37–109 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009)Google Scholar
  12. 12.
    Ivic, A., Koplowitz, J., Zunic, J.D.: On the number of digital convex polygons inscribed into an (m, m)-grid. IEEE Transactions on Information Theory 40(5), 1681–1686 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pivoteau, C., Salvy, B., Soria, M.: Boltzmann oracle for combinatorial systems. In: Proceedings of the Fifth Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Blaubeuren, Germany, September 22-26. Discrete Mathematics and Theoretical Computer Science, pp. 475–488 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • O. Bodini
    • 1
  • Ph. Duchon
    • 2
  • A. Jacquot
    • 1
  • L. Mutafchiev
    • 3
    • 4
  1. 1.LIPNUniversité Paris 13VilletaneuseFrance
  2. 2.LaBRITalence cedexFrance
  3. 3.American University in BulgariaBlagoevgradBulgaria
  4. 4.Institute of Mathematics and InformaticsSofiaBulgaria

Personalised recommendations