, Volume 36, Issue 3, pp 437–462 | Cite as

Bottom-Up: a New Algorithm to Generate Random Linear Extensions of a Poset

  • P. García-Segador
  • P. MirandaEmail author


In this paper we present a new method for deriving a random linear extension of a poset. This new strategy combines Probability with Combinatorics and obtains a procedure where each minimal element of a sequence of subposets is selected via a probability distribution. The method consists in obtaining a weight vector on the elements of P, so that an element is selected with a probability proportional to its weight. From some properties on the graph of adjacent linear extensions, it is shown that the probability distribution can be obtained by solving a linear system. The number of equations involved in this system relies on the number of what we have called positioned antichains, that allows a reduced number of equations. Finally, we give some examples of the applicability of the algorithm. This procedure cannot be applied to every poset, but it is exact when it can be used. Moreover, the method is quick and easy to implement. Besides, it allows a simple way to derive the number of linear extensions of a given poset.


Poset Linear extension Random generation Probability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This paper has been supported by the Spanish Grant MTM-2015-67057.


  1. 1.
    Ayyer, A., Klee, S., Shilling, A.: Combinatorial Markov chains on linear extensions. J. Algebr. Comb. 39(4), 853–881 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bollobás, B., Brightwell, G., Sidorenko, A.: Geometrical techniques for estimating numbers of linear extensions. Eur. J. Comb. 20, 329–335 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brightwell, G: The number of linear extensions of ranked posets. CDAM Research Report (2003)Google Scholar
  4. 4.
    Brightwell, G., Tetali, P.: The number of linear extensions of the Boolean Lattice. Order 20(3), 333–345 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brightwell, G., Winkler, P.: Counting linear extensions. Order 8(3), 225–242 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bubley, R., Dyer, M.: Faster random generation of linear extensions. Discret. Math. 20, 81–88 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Choquet, G: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  9. 9.
    Denneberg, D.: Non-additive Measures and Integral. Kluwer Academic, Dordrecht (1994)CrossRefGoogle Scholar
  10. 10.
    Devroye, L.: Non-uniform Random Variate Generation. Springer, New York (1986)CrossRefGoogle Scholar
  11. 11.
    Edelman, P., Hibi, T., Stanley, R.: A recurrence for linear extensions. Order 6(1), 15–18 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eriksson, K., Jonsson, M., Sjöstrand, J.: Markov chains on graded posets: compatibility of up-directed and down-directed transition probabilities. Order, Online Open Access. (2016)CrossRefGoogle Scholar
  13. 13.
    Greene, C., Nijenhuis, A., Wilf, H.: A probabilistic proof of a formula for the number of Young Tableaux of a given shape. Adv. Math. 31, 104–109 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals- Theory and Applications. Number 40 in Studies in Fuzziness and Soft Computing. Physica–Verlag, Heidelberg (2000)Google Scholar
  15. 15.
    Huber, M.: Fast perfect sampling from linear extensions. Discret. Math. 306, 420–428 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huber, M.: Near-linear time simulation of linear extensions of a height-2 poset with bounded interaction. Chic. J. Theor. Comput. Sci. 03, 1–16 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kalvin, A.D., Varol, Y.L.: On the generation of all topological sortings. J. Algorithms 4(2), 150–162 (1983)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karzanov, A., Khachiyan, L.: On the conductance of order Markov chains. Order 8(1), 7–15 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Knuth, D.E., Szwarcfiter, J.: A structured program to generate all topological sorting arrangements. Inform. Process. Lett. 2(6), 153–157 (1974)CrossRefGoogle Scholar
  20. 20.
    Korsh, J.F., Lafollette, P.S.: Loopless generation of linear extensions of a poset. Order 19, 115–126 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Levin, D., Peres, Y., Wilmer, E.: Markov Mixing and Mixing Times. American Mathematical Society (2008)Google Scholar
  22. 22.
    Leydold, J., Hörmann, W.: A sweep-plane algorithm for generating random tuples in simple polytopes. J. Math. Comput. 67(224), 1617–1635 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Matousek, J: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefGoogle Scholar
  24. 24.
    Nakada, K., Okamura, S.: An algorithm which generates linear extensions for a generalized Young diagram with uniform probability. DMTCS, proc. AN, pp. 801–808 (2010)Google Scholar
  25. 25.
    Neggers, J., Kim, H. S.: Basic Posets. World Scientific, Singapore (1998)CrossRefGoogle Scholar
  26. 26.
    Pruesse, G., Ruskey, F.: Generating linear extensions fast. SIAM J. Comput. 23(2), 373–386 (1994)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ruskey, F.: Generating linear extensions of posets by transpositions. J. Comb. Theory, Ser. B 54, 77–101 (1992)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Stanley, R.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Stanley, R.: Enumerative Combinatorics. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  30. 30.
    Sugeno, M.: Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology (1974)Google Scholar
  31. 31.
    Varol, Y.L., Rotem, D.: An algorithm to generate all topological sorting arrangements. Comput. J. 24(1), 83–84 (1981)CrossRefGoogle Scholar
  32. 32.
    Vose, M.D.: A linear algorithm for generating random numbers with a given distribution. IEEE Trans. Softw. Eng. 17(9), 972–975 (1991)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wilf, H.S.: Generating Functionology. Academic, New York (1994)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.National Statistics InstituteMadridSpain
  2. 2.Complutense University of MadridMadridSpain

Personalised recommendations