On Periodicity Lemma for Partial Words

  • Tomasz Kociumaka
  • Jakub Radoszewski
  • Wojciech Rytter
  • Tomasz Waleń
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10792)


We investigate the function L(hpq), called here the threshold function, related to periodicity of partial words (words with holes). The value L(hpq) is defined as the minimum length threshold which guarantees that a natural extension of the periodicity lemma is valid for partial words with h holes and (strong) periods pq. We show how to evaluate the threshold function in \(\mathcal {O}(\log p + \log q)\) time, which is an improvement upon the best previously known \(\mathcal {O}(p+q)\)-time algorithm. In a series of papers, the formulae for the threshold function, in terms of p and q, were provided for each fixed \(h \le 7\). We demystify the generic structure of such formulae, and for each value h we express the threshold function in terms of a piecewise-linear function with \(\mathcal {O}(h)\) pieces.


Partial words Words with don’t cares Periodicity lemma 


  1. 1.
    Bai, H., Franek, F., Smyth, W.F.: The new periodicity lemma revisited. Discrete Appl. Math. 212, 30–36 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berstel, J., Boasson, L.: Partial words and a theorem of Fine and Wilf. Theor. Comput. Sci. 218(1), 135–141 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blanchet-Sadri, F., Bal, D., Sisodia, G.: Graph connectivity, partial words, and a theorem of Fine and Wilf. Inf. Comput. 206(5), 676–693 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blanchet-Sadri, F., Hegstrom, R.A.: Partial words and a theorem of Fine and Wilf revisited. Theor. Comput. Sci. 270(1–2), 401–419 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blanchet-Sadri, F., Mandel, T., Sisodia, G.: Periods in partial words: an algorithm. J. Discrete Algorithms 16, 113–128 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchet-Sadri, F., Oey, T., Rankin, T.D.: Fine and Wilf’s theorem for partial words with arbitrarily many weak periods. Int. J. Found. Comput. Sci. 21(5), 705–722 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blanchet-Sadri, F., Simmons, S., Tebbe, A., Veprauskas, A.: Abelian periods, partial words, and an extension of a theorem of Fine and Wilf. RAIRO - Theor. Inform. Appl. 47(3), 215–234 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castelli, M.G., Mignosi, F., Restivo, A.: Fine and Wilf’s theorem for three periods and a generalization of Sturmian words. Theor. Comput. Sci. 218(1), 83–94 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for abelian periods. Bull. EATCS 89, 167–170 (2006). MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fan, K., Puglisi, S.J., Smyth, W.F., Turpin, A.: A new periodicity lemma. SIAM J. Discrete Math. 20(3), 656–668 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fine, N.J., Wilf, H.S.: Uniqueness theorems for periodic functions. Proc. Am. Math. Soc. 16(1), 109–114 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giancarlo, R., Mignosi, F.: Generalizations of the periodicity theorem of Fine and Wilf. In: Tison, S. (ed.) CAAP 1994. LNCS, vol. 787, pp. 130–141. Springer, Heidelberg (1994). CrossRefGoogle Scholar
  13. 13.
    Justin, J.: On a paper by Castelli, Mignosi, Restivo. RAIRO - Theor. Inform. Appl. 34(5), 373–377 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Karhumäki, J., Puzynina, S., Saarela, A.: Fine and Wilf’s theorem for \(k\)-abelian periods. Int. J. Found. Comput. Sci. 24(7), 1135–1152 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Khinchin, A.Y.: Continued Fractions. Dover Publications, New York (1997)zbMATHGoogle Scholar
  16. 16.
    Kociumaka, T., Radoszewski, J., Rytter, W., Waleń, T.: On periodicity lemma for partial words. ArXiv preprint.
  17. 17.
    Lothaire, M.: Algebraic Combinatorics on Words: Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Manea, F., Mercaş, R., Nowotka, D.: Fine and Wilf’s theorem and pseudo-repetitions. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 668–680. Springer, Heidelberg (2012). CrossRefGoogle Scholar
  19. 19.
    Mignosi, F., Restivo, A., Silva, P.V.: On Fine and Wilf’s theorem for bidimensional words. Theor. Comput. Sci. 292(1), 245–262 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mignosi, F., Shallit, J., Wang, M.: Variations on a theorem of Fine & Wilf. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136. Springer, Heidelberg (2001). CrossRefGoogle Scholar
  21. 21.
    van Ravenstein, T.: The three gap theorem (Steinhaus conjecture). J. Aust. Math. Soc. 45(3), 360–370 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Richards, I.: Continued fractions without tears. Math. Mag. 54(4), 163–171 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shur, A.M., Gamzova, Y.V.: Partial words and the interaction property of periods. Izv. Math. 68, 405–428 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Shur, A.M., Konovalova, Y.V.: On the periods of partial words. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 657–665. Springer, Heidelberg (2001). CrossRefGoogle Scholar
  25. 25.
    Smyth, W.F., Wang, S.: A new approach to the periodicity lemma on strings with holes. Theor. Comput. Sci. 410(43), 4295–4302 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tijdeman, R., Zamboni, L.Q.: Fine and Wilf words for any periods II. Theor. Comput. Sci. 410(30–32), 3027–3034 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Tomasz Kociumaka
    • 1
  • Jakub Radoszewski
    • 1
  • Wojciech Rytter
    • 1
  • Tomasz Waleń
    • 1
  1. 1.Faculty of Mathematics, Informatics, and MechanicsUniversity of WarsawWarsawPoland

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