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Efficient Enumeration of Ordered Trees with k Leaves (Extended Abstract)

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WALCOM: Algorithms and Computation (WALCOM 2009)

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Abstract

In this paper, we give a simple algorithm to generate all ordered trees with exactly n vertices including exactly k leaves. The best known algorithm generates such trees in O(n − k) time for each, while our algorithm generates such trees in O(1) time for each in worst case.

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References

  1. Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1-3), 21–46 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beyer, T., Hedetniemi, S.M.: Constant time generation of rooted trees. SIAM J. Comput. 9(4), 706–712 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fenner, T.I., Loizou, G.: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332–337 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Goldberg, L.: Efficient algorithms for listing combinatorial structures. Cambridge University Press, New York (1993)

    Book  MATH  Google Scholar 

  5. Kikuchi, Y., Tanaka, H., Nakano, S., Shibata, Y.: How to obtain the complete list of caterpillars. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 329–338. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  6. Knuth, D.: The art of computer programming. Generating all tuples and permutations, vol. 4, fascicle 2. Addison-Wesley, Reading (2005)

    MATH  Google Scholar 

  7. Knuth, D.E.: The art of computer programming. Generating all trees, history of combinatorial generation, vol. 4, fascicle 4. Addison-Wesley, Reading (2006)

    Google Scholar 

  8. Korsh, J.F., LaFollette, P.: Multiset permutations and loopless generation of ordered trees with specified degree sequence. Journal of Algorithms 34(2), 309–336 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Korsh, J.F., LaFollette, P.: Loopless generation of trees with specified degrees. The Computer Journal 45(3), 364–372 (2002)

    Article  MATH  Google Scholar 

  10. Kreher, D.L., Stinson, D.R.: Combinatorial algorithms. CRC Press, Boca Raton (1998)

    MATH  Google Scholar 

  11. Li, G., Ruskey, F.: The advantages of forward thinking in generating rooted and free trees. In: Proc. 10th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA 1999), pp. 939–940 (1999)

    Google Scholar 

  12. 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)

    Chapter  Google Scholar 

  13. McKay, B.D.: Isomorph-free exhaustive generation. J. Algorithms 26(2), 306–324 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Muramatsu, T., Nakano, S.: A random generation of plane trees with exactly k leaves. IEICE Transaction on Fundamentals J90-A(12), 940–947 (2007) (in Japanese)

    Google Scholar 

  15. Nakano, S.: Efficient generation of plane trees. Inf. Process. Lett. 84(3), 167–172 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nakano, S.: Efficient generation of triconnected plane triangulations. Comput. Geom. Theory and Appl. 27(2), 109–122 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nakano, S., Uno, T.: Constant time generation of trees with specified diameter. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 33–45. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  18. Nakano, S., Uno, T.: Generating colored trees. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 249–260. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  19. Pallo, J.: Generating trees with n nodes and m leaves. International Journal of Computer Mathematics 21(2), 133–144 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Read, R.C.: Every one a winner or how to avoid isomorphism search. Annuals of Discrete Mathematics 2, 107–120 (1978)

    Article  MATH  Google Scholar 

  21. Reingold, E.M., Nievergelt, J., Deo, N.: Combinatorial Algorithms. Prentice-Hall, Englewood Cliffs (1977)

    MATH  Google Scholar 

  22. Ruskey, F., van Baronaigien, D.R.: Fast recursive algorithms for generating combinatorial objects. Congressus Numerantium 41, 53–62 (1984)

    MathSciNet  MATH  Google Scholar 

  23. Sawada, J.: Generating rooted and free plane trees. ACM Transactions on Algorithms 2(1), 1–13 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stanley, R.P.: Enumerative combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  25. Wilf, H.S.: Combinatorial algorithms: An update. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

  26. Wright, R.A., Richmond, B., Odlyzko, A., McKay, B.D.: Constant time generation of free trees. SIAM J. Comput. 15(2), 540–548 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yamanaka, K., Kawano, S., Kikuchi, Y., Nakano, S.: Constant time generation of integer partitions. IEICE Trans. Fundamentals E90-A(5), 888–895 (2007)

    Article  Google Scholar 

  28. Yamanaka, K., Nakano, S.: Listing all plane graphs. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 210–221. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  29. Zaks, S., Richards, D.: Generating trees and other combinatorial objects lexicographically. SIAM J. Comput. 8(1), 73–81 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zoghbi, A., Stojmenović, I.: Fast algorithms for generating integer partitions. Int. J. Comput. Math. 70, 319–332 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Information Systems, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan

    Katsuhisa Yamanaka

  2. Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, 376-8515, Japan

    Yota Otachi & Shin-ichi Nakano

Authors
  1. Katsuhisa Yamanaka
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  2. Yota Otachi
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  3. Shin-ichi Nakano
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Editor information

Editors and Affiliations

  1. Indian Statistical Institute, 700 108, Kolkata, India

    Sandip Das

  2. Japan Advanced Institute of Science and Technology, 923-1292, Ishikawa, Japan

    Ryuhei Uehara

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Yamanaka, K., Otachi, Y., Nakano, Si. (2009). Efficient Enumeration of Ordered Trees with k Leaves (Extended Abstract). In: Das, S., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2009. Lecture Notes in Computer Science, vol 5431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00202-1_13

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  • DOI: https://doi.org/10.1007/978-3-642-00202-1_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00201-4

  • Online ISBN: 978-3-642-00202-1

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