Normal, Abby Normal, Prefix Normal

  • Péter Burcsi
  • Gabriele Fici
  • Zsuzsanna Lipták
  • Frank Ruskey
  • Joe Sawada
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8496)


A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number \(\textit{pnw}(n)\) of prefix normal words of length n, showing that \(\textit{pnw}(n) =\Omega\left(2^{n - c\sqrt{n\ln n}}\right)\) for some c and \(\textit{pnw}(n) = O \left(\frac{2^n (\ln n)^2}{n}\right)\). We introduce efficient algorithms for testing the prefix normal property and a “mechanical algorithm” for computing prefix normal forms. We also include games which can be played with prefix normal words. In these games Alice wishes to stay normal but Bob wants to drive her “abnormal” – we discuss which parameter settings allow Alice to succeed.


prefix normal words binary jumbled pattern matching normal forms enumeration membership testing binary languages 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amir, A., Apostolico, A., Landau, G.M., Satta, G.: Efficient text fingerprinting via Parikh mapping. J. Discrete Algorithms 1(5-6), 409–421 (2003)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Badkobeh, G., Fici, G., Kroon, S., Lipták, Zs.: Binary jumbled string matching for highly run-length compressible texts. Inf. Process. Lett. 113(17), 604–608 (2013)Google Scholar
  3. 3.
    Benson, G.: Composition alignment. In: Benson, G., Page, R.D.M. (eds.) WABI 2003. LNCS (LNBI), vol. 2812, pp. 447–461. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Böcker, S.: Simulating multiplexed SNP discovery rates using base-specific cleavage and mass spectrometry. Bioinformatics 23(2), 5–12 (2007)CrossRefGoogle Scholar
  5. 5.
    Böcker, S., Jahn, K., Mixtacki, J., Stoye, J.: Computation of median gene clusters. In: Vingron, M., Wong, L. (eds.) RECOMB 2008. LNCS (LNBI), vol. 4955, pp. 331–345. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
  7. 7.
    Burcsi, P., Cicalese, F., Fici, G., Lipták, Zs.: On table arrangements, scrabble freaks, and jumbled pattern matching. In: Boldi, P. (ed.) FUN 2010. LNCS, vol. 6099, pp. 89–101. Springer, Heidelberg (2010)Google Scholar
  8. 8.
    Burcsi, P., Fici, G., Lipták, Zs., Ruskey, F., Sawada, J.: On combinatorial generation of prefix normal words. In: Kulikov, A. (ed.) CPM 2014. LNCS, vol. 8486, pp. 60–69. Springer, Heidelberg (2014)Google Scholar
  9. 9.
    Butman, A., Eres, R., Landau, G.M.: Scaled and permuted string matching. Inf. Process. Lett. 92(6), 293–297 (2004)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cicalese, F., Fici, G., Lipták, Zs.: Searching for jumbled patterns in strings. In: Proc. of the Prague Stringology Conference 2009 (PSC 2009), pp. 105–117. Czech Technical University in Prague (2009)Google Scholar
  11. 11.
    Cicalese, F., Gagie, T., Giaquinta, E., Laber, E.S., Lipták, Zs., Rizzi, R., Tomescu, A.I.: Indexes for jumbled pattern matching in strings, trees and graphs. In: Kurland, O., Lewenstein, M., Porat, E. (eds.) SPIRE 2013. LNCS, vol. 8214, pp. 56–63. Springer, Heidelberg (2013)Google Scholar
  12. 12.
    Cicalese, F., Laber, E.S., Weimann, O., Yuster, R.: Near linear time construction of an approximate index for all maximum consecutive sub-sums of a sequence. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 149–158. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Dührkop, K., Ludwig, M., Meusel, M., Böcker, S.: Faster mass decomposition. In: Darling, A., Stoye, J. (eds.) WABI 2013. LNCS, vol. 8126, pp. 45–58. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Fici, G., Lipták, Zs.: On prefix normal words. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 228–238. Springer, Heidelberg (2011)Google Scholar
  15. 15.
    Gagie, T., Hermelin, D., Landau, G.M., Weimann, O.: Binary jumbled pattern matching on trees and tree-like structures. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 517–528. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Giaquinta, E., Grabowski, Sz.: New algorithms for binary jumbled pattern matching. Inf. Process. Lett. 113(14-16), 538–542 (2013)Google Scholar
  17. 17.
    Hermelin, D., Landau, G.M., Rabinovich, Y., Weimann, O.: Binary jumbled pattern matching via all-pairs shortest paths. Arxiv: 1401.2065v3 (2014)Google Scholar
  18. 18.
    Kociumaka, T., Radoszewski, J., Rytter, W.: Efficient indexes for jumbled pattern matching with constant-sized alphabet. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 625–636. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Moosa, T.M., Rahman, M.S.: Indexing permutations for binary strings. Inf. Process. Lett. 110, 795–798 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Moosa, T.M., Rahman, M.S.: Sub-quadratic time and linear space data structures for permutation matching in binary strings. J. Discrete Algorithms 10, 5–9 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Ian Munro, J.: Tables. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 37–42. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  23. 23.
    Parida, L.: Gapped permutation patterns for comparative genomics. In: Bücher, P., Moret, B.M.E. (eds.) WABI 2006. LNCS (LNBI), vol. 4175, pp. 376–387. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Ruskey, F., Sawada, J., Williams, A.: Binary bubble languages and cool-lex order. J. Comb. Theory, Ser. A 119(1), 155–169 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, Sequence A194850
  26. 26.
    Williams, A.M.: Shift Gray Codes. PhD thesis, University of Victoria, Canada (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Péter Burcsi
    • 1
  • Gabriele Fici
    • 2
  • Zsuzsanna Lipták
    • 3
  • Frank Ruskey
    • 4
  • Joe Sawada
    • 5
  1. 1.Dept. of Computer AlgebraEötvös Loránd Univ.BudapestHungary
  2. 2.Dip. di Matematica e InformaticaUniversity of PalermoItaly
  3. 3.Dip. di InformaticaUniversity of VeronaItaly
  4. 4.Dept. of Computer ScienceUniversity of VictoriaCanada
  5. 5.School of Computer ScienceUniversity of GuelphCanada

Personalised recommendations