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International Symposium on String Processing and Information Retrieval

SPIRE 2017: String Processing and Information Retrieval pp 27–37Cite as

Distinct Squares in Circular Words

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10508))

Abstract

A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length n. The famous conjecture attributed to Fraenkel and Simpson is that there are at most n such distinct squares, yet the best known upper bound is 1.84n by Deza et al. [Discr. Appl. Math. 180, 52–69 (2015)]. We consider a natural generalization of this question to circular words: how many distinct squares can there be in all cyclic rotations of a word of length n? We prove an upper bound of 3.14n. This is complemented with an infinite family of words implying a lower bound of 1.25n.

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Notes

  1. 1.

    Formally, we need to appropriately round both \(\frac{1}{4}n\) and \(\frac{1}{2}n\). We chose not to do so explicitly as to avoid cluttering the presentation.

References

  1. Blanchet-Sadri, F., Mercas, R., Scott, G.: Counting distinct squares in partial words. Acta Cybern. 19(2), 465–477 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Bland, W., Smyth, W.F.: Three overlapping squares: the general case characterized and applications. Theor. Comput. Sci. 596, 23–40 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Castiglione, G., Restivo, A., Sciortino, M.: Circular Sturmian words and Hopcroft’s algorithm. Theor. Comput. Sci. 410(43), 4372–4381 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crochemore, M., Fici, G., Mercaş, R., Pissis, S.P.: Linear-time sequence comparison using minimal absent words & applications. In: Kranakis, E., Navarro, G., Chávez, E. (eds.) LATIN 2016. LNCS, vol. 9644, pp. 334–346. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49529-2_25

  5. Crochemore, M., Iliopoulos, C.S., Kociumaka, T., Kubica, M., Radoszewski, J., Rytter, W., Tyczyński, W., Waleń, T.: The maximum number of squares in a tree. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 27–40. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31265-6_3

    Chapter  Google Scholar 

  6. Crochemore, M., Rytter, W.: Squares, cubes, and time-space efficient string searching. Algorithmica 13(5), 405–425 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Currie, J.D.: There are ternary circular square-free words of length \(n\) for \(n\ge 18\). Electron. J. Comb. 9(1), N10 (2002)

    MATH  Google Scholar 

  8. Currie, J.D., Fitzpatrick, D.S.: Circular words avoiding patterns. In: Ito, M., Toyama, M. (eds.) DLT 2002. LNCS, vol. 2450, pp. 319–325. Springer, Heidelberg (2003). doi:10.1007/3-540-45005-X_28

    Chapter  Google Scholar 

  9. Deza, A., Franek, F., Thierry, A.: How many double squares can a string contain? Discrete Appl. Math. 180, 52–69 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, K., Puglisi, S.J., Smyth, W.F., Turpin, A.: A new periodicity lemma. SIAM J. Discrete Math. 20(3), 656–668 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fine, N., Wilf, H.: Uniqueness theorems for periodic functions. Proc. Am. Math. Soc. 16, 109–114 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fraenkel, A.S., Simpson, J.: How many squares can a string contain? J. Comb. Theory Ser. A 82(1), 112–120 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Franek, F., Fuller, R.C.G., Simpson, J., Smyth, W.F.: More results on overlapping squares. J. Discrete Algorithms 17, 2–8 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gawrychowski, P., Kociumaka, T., Rytter, W., Waleń, T.: Tight bound for the number of distinct palindromes in a tree. In: Iliopoulos, C., Puglisi, S., Yilmaz, E. (eds.) SPIRE 2015. LNCS, vol. 9309, pp. 270–276. Springer, Cham (2015). doi:10.1007/978-3-319-23826-5_26

    Chapter  Google Scholar 

  15. Hegedüs, L., Nagy, B.: Representations of circular words. AFL. EPTCS 151, 261–270 (2014)

    Article  Google Scholar 

  16. Ilie, L.: A simple proof that a word of length n has at most 2n distinct squares. J. Comb. Theory 112(1), 163–164 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ilie, L.: A note on the number of squares in a word. Theor. Comput. Sci. 380(3), 373–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kopylova, E., Smyth, W.F.: The three squares lemma revisited. J. Discrete Algorithms 11, 3–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lothaire, M. (ed.): Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  20. Manea, F., Seki, S.: Square-density increasing mappings. In: Manea, F., Nowotka, D. (eds.) WORDS 2015. LNCS, vol. 9304, pp. 160–169. Springer, Cham (2015). doi:10.1007/978-3-319-23660-5_14

    Chapter  Google Scholar 

  21. Massé, A.B., Brlek, S., Garon, A., Labbé, S.: Equations on palindromes and circular words. Theor. Comput. Sci. 412(27), 2922–2930 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shur, A.M.: On ternary square-free circular words. Electron. J. Comb. 17(1), R140 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Simpson, J.: Intersecting periodic words. Theor. Comput. Sci. 374(1–3), 58–65 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Simpson, J.: Palindromes in circular words. Theor. Comput. Sci. 550, 66–78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Thue, A.: Über unendliche zeichenreihen. Norske Vid. Selsk. Skr. I Mat.-Nat. Kl. Christiania 7, 1–22 (1906)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. IBM Research, Haifa, Israel

    Mika Amit

  2. University of Haifa, Haifa, Israel

    Mika Amit & Paweł Gawrychowski

  3. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Paweł Gawrychowski

Authors
  1. Mika Amit
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  2. Paweł Gawrychowski
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Corresponding author

Correspondence to Paweł Gawrychowski .

Editor information

Editors and Affiliations

  1. Università di Palermo, Palermo, Italy

    Gabriele Fici

  2. Università di Palermo, Palermo, Italy

    Marinella Sciortino

  3. Università di Pisa, Pisa, Italy

    Rossano Venturini

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Amit, M., Gawrychowski, P. (2017). Distinct Squares in Circular Words. In: Fici, G., Sciortino, M., Venturini, R. (eds) String Processing and Information Retrieval. SPIRE 2017. Lecture Notes in Computer Science(), vol 10508. Springer, Cham. https://doi.org/10.1007/978-3-319-67428-5_3

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  • DOI: https://doi.org/10.1007/978-3-319-67428-5_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67427-8

  • Online ISBN: 978-3-319-67428-5

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