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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(nā€‰āˆ’ā€‰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.Ā 65(1-3), 21ā€“46 (1996)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    BookĀ  MATHĀ  Google ScholarĀ 

  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.Ā 2697, pp. 329ā€“338. Springer, Heidelberg (2003)

    ChapterĀ  Google ScholarĀ 

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

    MATHĀ  Google ScholarĀ 

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

    Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MATHĀ  Google ScholarĀ 

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

    MATHĀ  Google ScholarĀ 

  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. 939ā€“940 (1999)

    Google ScholarĀ 

  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.Ā 2076, pp. 433ā€“443. Springer, Heidelberg (2001)

    ChapterĀ  Google ScholarĀ 

  13. McKay, B.D.: Isomorph-free exhaustive generation. J. AlgorithmsĀ 26(2), 306ā€“324 (1998)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  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.Ā 3353, pp. 33ā€“45. Springer, Heidelberg (2004)

    ChapterĀ  Google ScholarĀ 

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

    ChapterĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MATHĀ  Google ScholarĀ 

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

    MATHĀ  Google ScholarĀ 

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

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  23. Sawada, J.: Generating rooted and free plane trees. ACM Transactions on AlgorithmsĀ 2(1), 1ā€“13 (2006)

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    BookĀ  MATHĀ  Google ScholarĀ 

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

    BookĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  Google ScholarĀ 

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

    ChapterĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

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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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