Exhaustive generation of combinatorial objects by ECO

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.

[1]  Frank Ruskey,et al.  Adjacent Interchange Generation of Combinations , 1988, J. Algorithms.

[2]  Selmer M. Johnson.  Generation of permutations by adjacent transposition , 1963 .

[3]  Mireille Bousquet-Mélou,et al.  Generating functions for generating trees , 2002, Discret. Math..

[4]  Frank Ruskey,et al.  A CAT algorithm for generating permutations with a fixed number of inversions , 2003, Inf. Process. Lett..

[5]  Andrea Frosini,et al.  ECO Method and the Exhaustive Generation of Convex Polyominoes , 2003, DMTCS.

[6]  Frank Ruskey,et al.  On Rotations and the Generation of Binary Trees , 1993, J. Algorithms.

[7]  Alberto Del Lungo,et al.  ECO:a methodology for the enumeration of combinatorial objects , 1999 .

[8]  J. Ludman,et al.  Gray Code Generation for MPSK Signals , 1981, IEEE Trans. Commun..

[9]  Vincent Vajnovszki,et al.  Gray visiting Motzkins , 2002, Acta Informatica.

[10]  Vincent Vajnovszki,et al.  Le codage des arbres binaires , 1995, Comput. Sci. J. Moldova.

[11]  Frank Ruskey,et al.  An efficient algorithm for generating necklaces with fixed density , 1999, SODA '99.

[12]  Dana S. Richards.  Data Compression and Gray-Code Sorting , 1986, Inf. Process. Lett..

[13]  Donald T. Tang,et al.  Enumerating Combinations of m Out of n Objects [G6] (Algorithm 452) , 1973, Commun. ACM.

[14]  Julian West,et al.  Generating trees and the Catalan and Schröder numbers , 1995, Discret. Math..

[15]  Vincent Vajnovszki.  Generating a Gray Code for P-Sequences , 2002, J. Math. Model. Algorithms.

[16]  Renzo Pinzani,et al.  An algebraic characterization of the set of succession rules , 2002, Theor. Comput. Sci..

[17]  John P. Robinson,et al.  Counting sequences , 1981, IEEE Transactions on Computers.

[18]  Fan Chung Graham,et al.  The Number of Baxter Permutations , 1978, J. Comb. Theory, Ser. A.

[19]  Carla Savage,et al.  A Survey of Combinatorial Gray Codes , 1997, SIAM Rev..

[20]  Brendan D. McKay,et al.  An Algorithm for Generating Subsets of Fixed Size With a Strong Minimal Change Property , 1984, Inf. Process. Lett..

[21]  D. Tang,et al.  Algorithm 452: enumerating combinations of m out of n objects [G6] , 1973, CACM.