Constant Time Generation of Linear Extensions

  • Akimitsu Ono
  • Shin-ichi Nakano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3623)


Given a poset \(\mathcal{P}\), several algorithms have been proposed for generating all linear extensions of \(\mathcal{P}\). The fastest known algorithm generates each linear extension in constant time “on average”. In this paper we give a simple algorithm which generates each linear extension in constant time “in worst case”. The known algorithm generates each linear extension exactly twice and output one of them, while our algorithm generates each linear extension exactly once.


Constant Time Schedule Problem Linear Extension Hamiltonian Path Recursive Call 
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. [A74]
    Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. [C01]
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  3. [G93]
    Goldberg, L.A.: Efficient Algorithms for Listing Combinatorial Structures. Cambridge University Press, New York (1993)zbMATHCrossRefGoogle Scholar
  4. [J80]
    Joichi, J.T., White, D.E., Williamson, S.G.: Combinatorial Gray Codes. SIAM J. on Computing 9, 130–141 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [KS98]
    Kreher, D.L., Stinson, D.R.: Combinatorial Algorithms. CRC Press, Boca Raton (1998)Google Scholar
  6. [KV83]
    Kalvin, A.D., Varol, Y.L.: On the Generation of All Topological Sortings. J. of Algorithms 4, 150–162 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [KN05]
    Kawano, S., Nakano, S.: Constant Time Generation of Set Partitions. IEICE Trans. Fundamentals (2005) (accepted, to appear)Google Scholar
  8. [LN01]
    Li, Z., Nakano, S.: Efficient Generation of Plane Triangulations without Repetitions. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 433–443. Springer, Heidelberg (2001)Google Scholar
  9. [LR99]
    Li, G., Ruskey, F.: The Advantage of Forward Thinking in Generating Rooted and Free Trees. In: Proc. 10th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 939–940 (1999)Google Scholar
  10. [M98]
    McKay, B.D.: Isomorph-free Exhaustive Generation. J. of Algorithms 26, 306–324 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [N02]
    Nakano, S.: Efficient Generation of Plane Trees. Information Processing Letters 84, 167–172 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [NU03]
    Nakano, S., Uno, T.: A Simple Constant Time Enumeration Algorithm for Free Trees, IPSJ Technical Report, 2003-AL-91-2 (2003),
  13. [PR94]
    Pruesse, G., Ruskey, F.: Generating Linear Extensions Fast. SIAM Journal on Computing 23, 373–386 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [R78]
    Read, R.C.: How to Avoid Isomorphism Search When Cataloguing Combinatorial Configurations. Annals of Discrete Mathematics 2, 107–120 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [R93]
    Ruskey, F.: Simple Combinatorial Gray Codes Constructed by Reversing Sublists. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 201–208. Springer, Heidelberg (1993)Google Scholar
  16. [R00]
    Rosen, K.H. (ed.): Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton (2000)zbMATHGoogle Scholar
  17. [S97]
    Savage, C.: A Survey of Combinatorial Gray Codes. SIAM Review 39, 605–629 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [St97]
    Stanley, R.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  19. [W89]
    Wilf, H.S.: Combinatorial Algorithms: An Update. SIAM, Philadelphia (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Akimitsu Ono
    • 1
  • Shin-ichi Nakano
    • 1
  1. 1.Gunma UniversityKiryu-ShiJapan

Personalised recommendations