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Abelian periods, partial words, and an extension of a theorem of Fine and Wilf∗

Published online by Cambridge University Press:  25 April 2013

Francine Blanchet-Sadri ,
Sean Simmons ,
Amelia Tebbe  and
Amy Veprauskas
Francine Blanchet-Sadri
Affiliation:
Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402–6170, USA.. blanchet@uncg.edu
Sean Simmons
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Building 2, Room 236, 77 Massachusetts Avenue, Cambridge, MA 02139–4307, USA
Amelia Tebbe
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA
Amy Veprauskas
Affiliation:
Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089 Tucson, AZ 85721–0089, USA
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Abstract

Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a result for the case of two non-relatively prime abelian periods. In this paper, we answer some open problems they suggested. We show that their conjecture is false but we give bounds, that depend on the two abelian periods, such that the conjecture is true for all words having length at least those bounds and show that some of them are optimal. We also extend their study to the context of partial words, giving optimal lengths and describing an algorithm for constructing optimal words.

Keywords

Combinatorics on wordsFine and Wilf’s theorempartial wordsabelian periodsperiodsoptimal lengths
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 3 , July 2013 , pp. 215 - 234
Copyright
© EDP Sciences 2013

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References

Avgustinovich, S.V., Glen, A., Halldórsson, B.V. and Kitaev, S., On shortest crucial words avoiding abelian powers. Discrete Appl. Math. 158 (2010) 605–607. Google Scholar
S.V. Avgustinovich, J. Karhumäki and S. Puzynina, On abelian versions of the critical factorization theorem. In JM 2010, 13ièmes Journées Montoises d’Informatique Théorique, Amiens, France (2010).
Berstel, J. and Boasson, L., Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135–141. Google Scholar
F. Blanchet-Sadri, Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008).
Blanchet-Sadri, F., Kim, J.I., Mercaş, R., Severa, W., Simmons, S. and Xu, D., Avoiding abelian squares in partial words. J. Combin. Theory Ser. A 119 (2012) 257–270. Google Scholar
Blanchet-Sadri, F., Mandel, T. and Sisodia, G., Periods in partial words: An algorithm. J. Discrete Algorithms 16 (2012) 113–128. Google Scholar
Blanchet-Sadri, F., Oey, T. and Rankin, T., Fine and Wilf’s theorem for partial words with arbitrarily many weak periods. Internat. J. Foundations Comput. Sci. 21 (2010) 705–722. Google Scholar
Blanchet-Sadri, F., Simmons, S. and Xu, D., Abelian repetitions in partial words. Adv. Appl. Math. 48 (2012) 194–214. Google Scholar
F. Blanchet-Sadri, A. Tebbe and A. Veprauskas, Fine and Wilf’s theorem for abelian periods in partial words. In JM 2010, 13ièmes Journées Montoises d’Informatique Théorique, Amiens, France (2010).
Castelli, M.G., Mignosi, F. and Restivo, A., Fine and Wilf’s theorem for three periods and a generalization of Sturmian words. Theoret. Comput. Sci. 218 (1999) 83–94. Google Scholar
C. Choffrut and J. Karhumäki, Combinatorics of Words. In Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa, Springer-Verlag, Berlin Vol. 1 (1997) 329–438.
Constantinescu, S. and Ilie, L., Generalised Fine and Wilf’s theorem for arbitrary number of periods. Theor. Comput. Sci. 339 (2005) 49–60. Google Scholar
Constantinescu, S. and Ilie, L., Fine and Wilf’s theorem for abelian periods. Bull. Eur. Assoc. Theor. Comput. Sci. 89 (2006) 167–170. Google Scholar
Cummings, L.J. and Smyth, W.F., Weak repetitions in strings. J. Combin. Math. Combin. Comput. 24 (1997) 33–48. Google Scholar
Currie, J. and Aberkane, A., A cyclic binary morphism avoiding abelian fourth powers. Theoret. Comput. Sci. 410 (2009) 44–52. Google Scholar
M. Domaratzki and N. Rampersad, Abelian primitive words. In DLT 2011, 15th International Conference on Developments in Language Theory, Milano, Italy, Lect. Notes Comput. Sci. Vol. 6795 edited by G. Mauri and A. Leporati. Springer-Verlag, Berlin, Heidelberg (2011) 204–215.
G. Fici, T. Lecroq, A. Lefebvre and E. Prieur-Gaston, Computing abelian periods in words. PSC 2011, Prague Stringology Conference, Prague, Czech Republic, (2011) 184–196.
Fine, N.J. and Wilf, H.S., Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109–114. Google Scholar
Halava, V., Harju, T. and Kärki, T., Interaction properties of relational periods. Discrete Math. Theoret. Comput. Sci. 10 (2008) 87–112. Google Scholar
Justin, J., On a paper by Castelli, Mignosi, Restivo. Theoret. Inform. Appl. 34 (2000) 373–377. Google Scholar
V. Keränen, Abelian squares are avoidable on 4 letters. In ICALP 1992, 19th International Colloquium on Automata, Languages and Programming, Lect. Notes Comput. Sci. Vol. 623 edited by W. Kuich. Springer-Verlag, Berlin (1992) 241–52.
A.V. Samsonov and A.M. Shur, On abelian repetition threshold. In JM 2010, 13ièmes Journées Montoises d’Informatique Théorique, Amiens, France (2010).
Shur, A.M. and Gamzova, Y.V., Partial words and the interaction property of periods. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 68 (2004) 191–214. Google Scholar
A.M. Shur and Y.V. Konovalova, On the periods of partial words. In MFCS 2001, 26th International Symposium on Mathematical Foundations of Computer Science, Lect. Notes Comput. Sci. Vol. 2136 edited by J. Sgall, A. Pultr and P. Kolman. London, UK, Springer-Verlag. (2001) 657–665.
Smyth, W. F. and Wang, S., A new approach to the periodicity lemma on strings with holes. Theoret. Comput. Sci. 410 (2009) 4295–4302. Google Scholar
Tijdeman, R. and Zamboni, L., Fine and Wilf words for any periods. Indagationes Math. 14 (2003) 135–147. Google Scholar