Since the work of Kolpakov and Kucherov in [5,6], it is known that ρ(n), the maximal number of runs in a string, is linear in the length n of the string. A lower bound of \(3/(1 + \sqrt{5})n \sim 0.927n\) has been given by Franek and al. [3,4], and upper bounds have been recently provided by Rytter, Puglisi and al., and Crochemore and Ilie (1.6n) [8.7.1]. However, very few properties are known for the ρ(n)/n function. We show here by a simple argument that limn ↦ ∞ ρ(n)/n exists and that this limit is never reached. Moreover, we further study the asymptotic behavior of ρp(n), the maximal number of runs with period at most p. We provide a new bound for some microruns : we show that there is no more than 0.971 n runs of period at most 9 in binary strings. Finally, this technique improves the previous best known upper bound, showing that the total number of runs in a binary string of length n is below 1.52n.


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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mathieu Giraud
    • 1
    • 2
  1. 1.CNRS, LIFL, Université Lille 1Villeneuve d’Acsq cedexFrance
  2. 2.INRIA Lille Nord-EuropeVilleneuve d’AscqFrance

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