A loop-free two-close Gray-code algorithm for listing k-ary Dyck words

  • Authors:

  • Vincent Vajnovszki;Timothy Walsh

  • Affiliations:

  • LE21 FRE-CNRS 2309, Université de Bourgogne, B.P. 47 870, 21078 Dijon-Cedex, France;Department of Computer Science, University of Quebec At Montreal, P.O. 8888, Station A, Montreal, Quebec, Canada, H3C 3P8

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2006

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Abstract

P. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close-that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at least k-1 times as many 0's as 1's, we obtain k-suffixes: suffixes of k-ary Dyck words. Combinations are retrieved as special case by setting k=1 and k-ary Dyck words are retrieved as a special case by imposing the additional condition that the entire string has exactly k-1 times as many 0's as 1's. We generalize Ruskey's Gray code to the first two-close Gray code for k-suffixes and we provide a loop-free implementation for k=2. For k=1 we use a simplified version of Chase's loop-free algorithm for generating his two-close Gray code for combinations. These results are optimal in the sense that there does not always exist a Gray code, either for combinations or Dyck words, in which the 1 and the 0 that exchange positions are adjacent.