Analysis of Algorithms for Listing Equivalence Classes of k-ary Strings

Andrzej Proskurowski, Department of Computer and Information Science, University of Oregon, USA.
Frank Ruskey, Department of Computer Science, University of Victoria, Canada.
Malcolm Smith, Department of Computer Science, University of Victoria, Canada.

Abstract:

We give efficient algorithms for listing equivalence classes of k-ary strings under reversal and permutation of alphabet symbols. As representative of each equivalence class we choose that string which is lexicographically smallest. These algorithms use space O(n) and time O(\sqrt{k} N), where N is the total number of strings generated and n is the length of each string. For k = 2, we obtain a recursive decomposition of the set of binary strings that allows the strings to be generated without rejecting any strings. For k >= 3, some strings must be rejected. The algorithm is simple but its exact analysis is rather complicated. In the analysis we determine a quantity of independent interest: the average length of the common prefix of two randomly chosen infinite length restricted-growth'' strings.

Keywords: Combinatorial generation, k-ary strings, Stirling numbers, restricted growth function, k-paths.
AMS Classification: 05A15, 05C30, 60J10, 68Q25, 68R05, 68R15.

• The postscript file is 301,862 bytes, the dvi file is 76,832 bytes.
• Presented at 2?th S.E. Conference on Combinatorics, Graph Theory, and Computing.
• Appears in SIAM J. Discrete Mathematics, 11 (1998) 94-109.
• Typo: In the Pascal program of Figure 2.1, the Mr should be M. It is the M set being generated, not the MR set, in agreement with (8). [Thanks to Tom Spreen for noticing this typo.]
• Malcolm Smith is now at Camosun College, Victoria, B.C.
• Malcolm Smith is now at the Dominion Royal Observatory, Saanich, B.C.
• MR 99b:68100 (68Q25 (05A15 05C30 68R05 68R15)).

Selected Citations

• Stephen Swift, Allan Tucker, Jason Crampton, and David Garway-Heath, An improved restricted growth function genetic algorithm for the consensus clustering of retinal nerve fibre data, Proceedings of the 9th annual conference on Genetic and evolutionary computation, London, England SESSION: Real-world applications: Pages: 2174 - 2181 Year of Publication: 2007 ISBN:978-1-59593-697-4.
• Allan Tucker, Jason Crampton, and Stephen Swift, RGFGA: An Efficient Representation and Crossover for Grouping Genetic Algorithms, Evolutionary Computing, 13 (2006) 477-499.
• Arvind Gupta, Naomi Nishimura, Andrzej Proskurowski, and Prabhakar Ragde, Embeddings of k-connected graphs of pathwidth k, Discrete Applied Mathematics Volume 145, Issue 2, 15 January 2005, Pages 242-265.
• A. Proskurowski and J.A. Telle, Classes of graphs with restricted interval models, Discrete Mathematics and Theoretical Computer Science, 3, 1999, 167-176.

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