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Exhaustive generation of combinatorial objects by ECO

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Abstract.

The problem of exhaustively generating combinatorial objects can currently be applied to many disciplines, such as biology, chemistry, medicine and computer science. A well known approach to the exhaustive generation problem is given by the Gray code scheme for listing n-bit binary numbers in such a way that successive numbers differ in exactly one bit position. In this work, we introduce an exhaustive generation algorithm, which is general for the classes of succession rules considered in [1]. We also show that our algorithm is efficient in an amortized sense; it actually uses only a constant amount of computation per object.

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References

  1. Banderier, C., Denise, A., Flajolet, P., Gardy, D., Gouyou-Beauchamps, D.: Generating functions for generating trees. Discrete Math. 246 (1-3), 29-55 (2002)

    Google Scholar 

  2. Barcucci, E., Del Lungo, A., Pergola, E., Pinzani, R.: ECO: a methodology for the enumeration of combinatorial objects. Journal of Difference Equations and Applications 5, 435-490 (1999)

  3. Chase, P. J.: Combination generation and graylex ordering. Congressus Numerantium 69, 215-242 (1989)

    Google Scholar 

  4. Chung, F. R. K., Graham, R. L., Hoggatt, V. E., Kleimann, M.: The number of Baxter permutations. J. Combin. Theory Ser. A 24, 382-394 (1978)

    Google Scholar 

  5. Del Lungo, A., Frosini, A., Rinaldi, S.: ECO method and the exhaustive generation of convex polyominoes. In: Calude, C. S., Dinneen, M. J., Vajnovszki, V. (eds.) DMTCS 2003. Lecture Notes in Computer Sciences 2731, pp. 129-140 (2003)

  6. Dulucq, S.: Some combinatorial and algorithmic problems in biology. In: Rechenmann, F. (eds.) Septiémes Entretiens Jacques Cartier. (1994)

  7. Eades, P., McKay, B.: An algorithm for generating subsets of fixed size with a strong minimal change property. Information Processing Letters 19, 131-133 (1984)

    Google Scholar 

  8. Ferrari, L., Pergola, E., Pinzani, R., Rinaldi, S.: An algebraic characterization of the set of succession rules. Theoretical Computer Science 281, 351-367 (2002)

    Google Scholar 

  9. Gardner, M.: Curious properties of the Gray code and how it can be used to solve puzzles. Scientific American 227 (2), 106-109 (1972)

    Google Scholar 

  10. Gray, F.: Pulse code communications. U.S. Patent 2632058, March 1953

  11. Liu, C. N., Tang, D. T.: Algorithm 452, enumerating M out N objects. Comm. ACM 16, 485 (1973)

    Google Scholar 

  12. Lucas, J. M., Roelants van Baronaigien, D., Ruskey, F.: On rotations and the generation of binary trees. Journal of Algorithms 15 (3), 343-366 (1993)

  13. Ludman, J. E.: Gray code generation for MPSK signals. IEEE Transactions on Communications COM-29, 1519-1522 (1981)

    Google Scholar 

  14. Johnson, S. M.: Generation of permutations by adjacent transpositions. Mathematics of Computation 17, 282-285 (1963)

    Google Scholar 

  15. Robinson, J., Cohn, M.: Counting sequences. IEEE Transactions on computers C-30, 17-23 (1981)

    Google Scholar 

  16. Richards, D.: Data compression and Gray-code sorting. Information Processing Letters 22 (4), 210-205 (1986)

    Google Scholar 

  17. Ruskey, F., Sawada, J.: An efficient algorithm for generating necklaces with fixed density. SIAM J. Computing 29 (2), 671-684 (1999)

    Google Scholar 

  18. Ruskey, F., Effler, S.: A CAT algorithm for generating permutations with a fixed number of inversions. Information Processing Letters 86 (2), 107-112 (2003)

    Google Scholar 

  19. Ruskey, F.: Adjacent interchange generation of combinations. Journal of Algorithms 9, 162-180 (1988)

    Google Scholar 

  20. Savage, C.: A survey of combinatorial Gray codes. SIAM Rev. 39 (4), 605-629 (1997)

    Google Scholar 

  21. Trotter, H. F.: PERM (Algorithm 115). Comm. ACM 5 (8), 434-435 (1962)

  22. Vajnovszki, V. : Le codage des arbres binaires. Computer Science Journal of Moldova 3 (2), 194-209 (1995) (Mathematical Reviews 1485353)

  23. Vajnovszki, V.: Generating a Gray code for P-sequences. The Journal of Mathematical Modelling and Algorithms 1 (1), 31-41 (2002)

    Google Scholar 

  24. Vajnovszki, V.: Gray visiting Motzkins. Acta Informatica 38, 793-811 (2003)

    Google Scholar 

  25. West, J.: Generating trees and the Catalan and Schröder numbers. Discrete Math. 146, 247-262 (1995)

    Google Scholar 

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Authors and Affiliations

  1. University of Bologna, Via Sacchi, 3, 47023, Cesena, Italy

    Silvia Bacchelli

  2. Dipartimento di Sistemi e Informatica, University of Firenze, Via Lombroso, 6/17, 50134, Firenze, Italy

    Elena Barcucci, Elisabetta Grazzini & Elisa Pergola

Authors
  1. Silvia Bacchelli
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  2. Elena Barcucci
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  3. Elisabetta Grazzini
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  4. Elisa Pergola
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Correspondence to Silvia Bacchelli.

Additional information

Received: 10 October 2003, Revised: 10 February 2004, Published online: 21 April 2004

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Bacchelli, S., Barcucci, E., Grazzini, E. et al. Exhaustive generation of combinatorial objects by ECO. Acta Informatica 40, 585–602 (2004). https://doi.org/10.1007/s00236-004-0139-x

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  • DOI: https://doi.org/10.1007/s00236-004-0139-x

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