Advertisement

On Collapsing Prefix Normal Words

  • Pamela FleischmannEmail author
  • Mitja Kulczynski
  • Dirk Nowotka
  • Danny Bøgsted Poulsen
Conference paper
  • 88 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12038)

Abstract

Prefix normal words are binary words in which each prefix has at least the same number of \(\mathsf {1}\)s as any factor of the same length. Firstly introduced in 2011, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems.

Notes

Acknowledgments

We would like to thank Florin Manea for helpful discussions and advice.

References

  1. 1.
    Balister, P., Gerke, S.: The asymptotic number of prefix normal words. J. Comb. Theory 784, 75–80 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: Algorithms for jumbled pattern matching in strings. Int. J. Found. CS 23(2), 357–374 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Burcsi, P., Fici, G., Lipták, Z., Ruskey, F., Sawada, J.: On combinatorial generation of prefix normal words. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds.) CPM 2014. LNCS, vol. 8486, pp. 60–69. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-07566-2_7CrossRefzbMATHGoogle Scholar
  4. 4.
    Burcsi, P., Fici, G., Lipták, Z., Ruskey, F., Sawada, J.: Normal, abby normal, prefix normal. In: Ferro, A., Luccio, F., Widmayer, P. (eds.) FUN 2014. LNCS, vol. 8496, pp. 74–88. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-07890-8_7CrossRefGoogle Scholar
  5. 5.
    Burcsi, P., Fici, G., Lipták, Z., Ruskey, F., Sawada, J.: On prefix normal words and prefix normal forms. TCS 659, 1–13 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cassaigne, J., Richomme, G., Saari, K., Zamboni, L.Q.: Avoiding Abelian powers in binary words with bounded Abelian complexity. Int. J. Found. CS 22(04), 905–920 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chan, T.M., Lewenstein, M.: Clustered integer 3SUM via additive combinatorics. In: 47th ACM Symposium on TOC, pp. 31–40. ACM (2015)Google Scholar
  8. 8.
    Cicalese, F., Lipták, Z., Rossi, M.: Bubble-flip—a new generation algorithm for prefix normal words. In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 207–219. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-77313-1_16CrossRefGoogle Scholar
  9. 9.
    Cicalese, F., Lipták, Z., Rossi, M.: On infinite prefix normal words. In: Proceedings of the SOFSEM, pp. 122–135 (2019)Google Scholar
  10. 10.
    Coven, E.M., Hedlund, G.A.: Sequences with minimal block growth. TCS 7(2), 138–153 (1973)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Currie, J., Rampersad, N.: Recurrent words with constant Abelian complexity. Adv. Appl. Math. 47(1), 116–124 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dassow, J.: Parikh mapping and iteration. In: Calude, C.S., PĂun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2000. LNCS, vol. 2235, pp. 85–101. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45523-X_5CrossRefGoogle Scholar
  13. 13.
    Ehlers, T., Manea, F., Mercas, R., Nowotka, D.: k-Abelian pattern matching. J. Discrete Algorithms 34, 37–48 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fici, G., Lipták, Z.: On prefix normal words. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 228–238. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22321-1_20CrossRefGoogle Scholar
  15. 15.
    OEIS Foundation Inc.: The on-line encyclopedia of integer sequencess (2019). http://oeis.org/
  16. 16.
    Karhumäki, J.: Generalized Parikh mappings and homomorphisms. Inf. Control 47(3), 155–165 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Keränen, V.: Abelian squares are avoidable on 4 letters. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 41–52. Springer, Heidelberg (1992).  https://doi.org/10.1007/3-540-55719-9_62CrossRefGoogle Scholar
  18. 18.
    Lee, L.-K., Lewenstein, M., Zhang, Q.: Parikh matching in the streaming model. In: Calderón-Benavides, L., González-Caro, C., Chávez, E., Ziviani, N. (eds.) SPIRE 2012. LNCS, vol. 7608, pp. 336–341. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-34109-0_35CrossRefGoogle Scholar
  19. 19.
    Mateescu, A., Salomaa, A., Salomaa, K., Yu, S.: On an extension of the Parikh mapping, 06 September 2000. http://citeseer.ist.psu.edu/440186.html
  20. 20.
    Mateescu, A., Salomaa, A., Yu, S.: Subword histories and Parikh matrices. J. Comput. Syst. Sci. 68(1), 1–21 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Parikh, R.J.: On context-free languages. J. ACM 13, 570–581 (1966)CrossRefGoogle Scholar
  22. 22.
    Puzynina, S., Zamboni, L.Q.: Abelian returns in Sturmian words. J. Comb. Theory 120(2), 390–408 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Richomme, G., Saari, K., Zamboni, L.Q.: Abelian complexity of minimal subshifts. J. Lond. Math. Soc. 83(1), 79–95 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Richomme, G., Saari, K., Zamboni, L.Q.: Balance and Abelian complexity of the Tribonacci word. Adv. Appl. Math. 45(2), 212–231 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Salomaa, A.: Connections between subwords and certain matrix mappings. TCS 340(2), 188–203 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Pamela Fleischmann
    • 1
    Email author
  • Mitja Kulczynski
    • 1
  • Dirk Nowotka
    • 1
  • Danny Bøgsted Poulsen
    • 2
  1. 1.Department of Computer ScienceKiel UniversityKielGermany
  2. 2.Department of Computer ScienceAalborg UniversityAalborgDenmark

Personalised recommendations