Generalized de Bruijn Words and the State Complexity of Conjugate Sets

  • Daniel Gabric
  • Štěpán Holub
  • Jeffrey ShallitEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)


We consider a certain natural generalization of de Bruijn words, and use it to compute the exact maximum state complexity for the language consisting of the conjugates of a single word. In other words, we determine the state complexity of cyclic shift on languages consisting of a single word.



We thank the anonymous referees for helpful comments and suggestions.


  1. 1.
    van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Stevin 28, 203–217 (1951)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anisiu, M., Blázsik, Z., Kása, Z.: Maximal complexity of finite words. Pure Math. Appl. 13, 39–48 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    de Bruijn, N.G.: A combinatorial problem. Proc. Konin. Neder. Akad. Wet. 49, 758–764 (1946)MathSciNetzbMATHGoogle Scholar
  4. 4.
    de Bruijn, N.G.: Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of \(2^n\) zeros and ones that show each \(n\)-letter word exactly once. Technical report 75-WSK-06, Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands (1975)Google Scholar
  5. 5.
    Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010). Scholar
  6. 6.
    Etzion, T.: An algorithm for generating shift-register cycles. Theoret. Comput. Sci. 44, 209–224 (1986). Scholar
  7. 7.
    Flaxman, A., Harrow, A.W., Sorkin, G.B.: Strings with maximally many distinct subsequences and substrings. Electron. J. Combin. 11(1), 8 (2004). Scholar
  8. 8.
    Flye Sainte-Marie, C.: Question 48. L’Intermédiaire Math. 1, 107–110 (1894)Google Scholar
  9. 9.
    Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24, 195–221 (1982). Scholar
  10. 10.
    Good, I.J.: Normal recurring decimals. J. London Math. Soc. 21, 167–169 (1946)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hemmati, F., Costello Jr., D.J.: An algebraic construction for q-ary shift register sequences. IEEE Trans. Comput. 27(12), 1192–1195 (1978). Scholar
  12. 12.
    Iványi, A.: On the \(d\)-complexity of words. Ann. Univ. Sci. Budapest. Sect. Comput. 8, 69–90 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jirásek, J., Jirásková, G.: Cyclic shift on prefix-free languages. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 246–257. Springer, Heidelberg (2013). Scholar
  14. 14.
    Jirásková, G., Okhotin, A.: State complexity of cyclic shift. RAIRO Theor. Inform. Appl. 42(2), 335–360 (2008). Scholar
  15. 15.
    Lempel, A.: \(m\)-ary closed sequences. J. Combin. Theory 10, 253–258 (1971)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Martin, M.H.: A problem in arrangements. Bull. Am. Math. Soc. 40, 859–864 (1934)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194(6), 1266–1268 (1970). In Russian. English translation in Soviet Math. Doklady 11(5), 1373–1375 (1970)MathSciNetGoogle Scholar
  18. 18.
    Ralston, A.: De Bruijn sequences — a model example of the interaction of discrete mathematics and computer science. Math. Mag. 55, 131–143 (1982). Scholar
  19. 19.
    Shallit, J.: On the maximum number of distinct factors of a binary string. Graphs Combin. 9(2–4), 197–200 (1993). Scholar
  20. 20.
    Sloane, N.J.A. et al.: The on-line encyclopedia of integer sequences (2019).

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  • Daniel Gabric
    • 1
  • Štěpán Holub
    • 2
  • Jeffrey Shallit
    • 1
    Email author
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

Personalised recommendations