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Random Generation and Approximate Counting of Ambiguously Described Combinatorial Structures

  • Alberto Bertoni
  • Massimiliano Goldwurm
  • Massimo Santini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)

Abstract

This paper concerns the uniform random generation and the approximate counting of combinatorial structures admitting an ambiguous description. We propose a general framework to study the complexity of these problems and present some applications to specific classes of languages. In particular, we give a uniform random generation algorithm for finitely ambiguous context-free languages of the same time complexity of the best known algorithm for the unambiguous case. Other applications include a polynomial time uniform random generator and approximation scheme for the census function of (i) languages recognized in polynomial time by one-way nondeterministic auxiliary pushdown automata of polynomial ambiguity and (ii) polynomially ambiguous rational trace languages.

Keywords

uniform random generation approximate counting context-free languages auxiliary pushdown automata rational trace languages inherent ambiguity 

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References

  1. [1]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA, 1974.zbMATHGoogle Scholar
  2. [2]
    A. V. Aho and J. D. Ullman. The Theory of Parsing, Translation and Compiling-Vol.I: Parsing. Prentice Hall, Englewood Cliffs, NJ, 1972.Google Scholar
  3. [3]
    E. Allender, D. Bruschi, and G. Pighizzini. The complexity of computing maximal word functions. Computational Complexity, 3:368–391, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Avellone and M. Goldwurm. Analysis of algorithms for the recognition of rational and context-free trace languages. RAIRO Informatique théorique et Applications/Theoretical Informatics and Applications, 32(4-5-6):141–152, 1998.MathSciNetGoogle Scholar
  5. [5]
    A. Bertoni, M. Goldwurm, G. Mauri, and N. Sabadini. Counting techniques for inclusion, equivalence and membership problems. In V. Diekert and G. Rozenberg, editors, The Book of Traces, chapter 5, pages 131–164. World Scientific, Singapore, 1995.Google Scholar
  6. [6]
    A. Bertoni, M. Goldwurm, and N. Sabadini. The complexity of computing the number of strings of given length in context-free languages. Theoretical Computer Science, 86(2):325–342, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    F.-J. Brandenburg. On one-way auxiliary pushdown automata. In H. Waldschmidt H. Tzschach and H. K.-G. Walter, editors, Proceedings of the 3rd GI Conference on Theoretical Computer Science, volume 48 of Lecture Notes in Computer Science, pages 132–144, Darmstadt, FRG, March 1977. Springer.Google Scholar
  8. [8]
    C. Choffrut and M. Goldwurm. Rational transductions and complexity of counting problems. Mathematical Systems Theory, 28(5):437–450, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    S. A. Cook. Characterizations of pushdown machines in terms of time-bounded computers. Journal of the ACM, 18(1):4–18, January 1971.zbMATHCrossRefGoogle Scholar
  10. [10]
    V. Diekert and Y. Métivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III, pages 457–527. Springer, Berlin-Heidelberg, 1997.Google Scholar
  11. [11]
    V. Diekert and G. Rozenberg. The Book of Traces. World Scientific, Singapore, 1995.Google Scholar
  12. [12]
    J. Earley. An efficient context-free parsing algorithm. Communications of the ACM, 13(2):94–102, February 1970.zbMATHCrossRefGoogle Scholar
  13. [13]
    P. Flajolet. Mathematical methods in the analysis of algorithms and data structures. In Egon Börger, editor, Trends in Theoretical Computer Science, chapter 6, pages 225–304. Computer Science Press, Rockville, Maryland, 1988.Google Scholar
  14. [14]
    P. Flajolet, P. Zimmerman, and B. Van Cutsem. A calculus for the random generation of labelled combinatorial structures. Theoretical Computer Science, 132(1–2):1–35, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. Goldwurm. Random generation of words in an algebraic language in linear binary space. Information Processing Letters, 54(4):229–233, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    V. Gore, M. Jerrum, S. Kannan, Z. Sweedyk, and S. Mahaney. A quasi-polynomial-time algorithm for sampling words from a context-free language. Information and Computation, 134(1):59–74, 10 April 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. A. Harrison. Introduction to Formal Language Theory. Addison-Wesley, Reading, MA, 1978.zbMATHGoogle Scholar
  18. [18]
    T. Hickey and J. Cohen. Uniform random generation of strings in a context-free language. SIAM Journal on Computing, 12(4):645–655, nov 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13–30, 1963.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Language, and Computation. Addison-Wesley, Reading, MA, 1979.Google Scholar
  21. [21]
    M. R. Jerrum, L. G. Valiant, and V. V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43(2–3):169–188, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    R. M. Karp, M. Luby, and N. Madras. Monte-carlo approximation algorithms for enumeration problems. Journal of Algorithms, 10:429–448, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    D. E. Knuth and A. C. Yao. The complexity of nonuniform random number generation. In J. F. Traub, editor, Algorithms and Complexity: New Directions and Recent Results, pages 357–428. Academic Press, 1976.Google Scholar
  24. [24]
    C. Lautemann. On pushdown and small tape. In K. Wagener, editor, Dirk-Siefkes, zum 50. Geburststag (proceedings of a meeting honoring Dirk Siefkes on his fiftieth birthday), pages 42–47. Technische Universität Berlin and Universität Ausgburg, 1988.Google Scholar
  25. [25]
    H. G. Mairson. Generating words in a context-free language uniformly at random. Information Processing Letters, 49(2):95–99, January 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    M. Santini. Random Uniform Generation and Approximate Counting of Combinatorial Structures. PhD thesis, Dipartimento di Scienze dell’Informazione — Università degli Studi di Milano, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Alberto Bertoni
    • 1
  • Massimiliano Goldwurm
    • 1
  • Massimo Santini
    • 1
  1. 1.Dipartimento di Scienze dell’InformazioneUniversità degli Studi di MilanoMilanoItalia

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