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)

Abstract

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.

Keywords

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

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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

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