Combinatorics of Periods in Strings

We consider the set Γ(n) of all period sets of strings of length n over a finite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that Γ(n) is a lattice under set inclusion and does not satisfy the Jordan-Dedekind condition.We propose the first enumeration algorithm for Γ(n) and improve upon the previously known asymptotic lower bounds on the cardinality of Γ(n). Finally, we provide a new recurrence to compute the number of strings sharing a given period set.

[1]  G. Marsaglia,et al.  Monkey tests for random number generators , 1993 .

[2]  Esko Ukkonen,et al.  A Comparison of Approximate String Matching Algorithms , 1996 .

[3]  J. A. Fill.  An interruptible algorithm for perfect sampling via Markov chains , 1998 .

[4]  de Ng Dick Bruijn.  On Mahler's partition problem , 1948 .

[5]  Wojciech Rytter,et al.  Text Algorithms , 1994 .

[6]  Esko Ukkonen,et al.  Approximate String Matching with q-grams and Maximal Matches , 1992, Theor. Comput. Sci..

[7]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[8]  D. R. McGregor,et al.  Fast approximate string matching , 1988, Softw. Pract. Exp..

[9]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[10]  Christian Choffrut,et al.  Combinatorics of Words , 1997, Handbook of Formal Languages.

[11]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[12]  Martin Vingron,et al.  q-gram based database searching using a suffix array (QUASAR) , 1999, RECOMB.

[13]  P. Billingsley,et al.  Probability and Measure , 1980 .

[14]  Ora E. Percus,et al.  Theory and application of Marsaglia's monkey test for pseudorandom number generators , 1995, TOMC.

[15]  P. Pevzner,et al.  Computational Molecular Biology , 2000 .

[16]  Sven Rahmann,et al.  The Expected Number of Missing Words in a Random Text , 2000 .

[17]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[18]  Leonidas J. Guibas,et al.  Periods in Strings , 1981, J. Comb. Theory, Ser. A.

[19]  Philippe Flajolet,et al.  Analysis of algorithms , 2000, Random Struct. Algorithms.

[20]  Carl-Erik Fröberg,et al.  Accurate estimation of the number of binary partitions , 1977 .