Abstract
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 ρ p (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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References
Crochemore, M., Ilie, L.: Maximal repetitions in strings. J. Comput. Systems Sci. 74(5), 796–807 (2008)
Fekete, M.: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift 17, 228–249 (1923)
Franek, F., Simpson, R.J., Smyth, W.F.: The maximum number of runs in a string. In: Proceedings of the 2003 Australasian Workshop on Combinatorial Algorithms (AWOCA 2003), pp. 26–35 (2003)
Franek, F., Yang, Q.: An asymptotic lower bound for the maximal-number-of-runs function. In: Prague Stringology Conference 2006, pp. 3–8 (2006)
Kolpakov, R., Kucherov, G.: Maximal repetitions in words or how to find all squares in linear time. Technical Report 98-R-227, LORIA (1998)
Kolpakov, R., Kucherov, G.: On maximal repetitions in words. Journal on Discrete Algorithms 1(1), 159–186 (2000)
Puglisi, S.J., Simpson, J., Smyth, B.: How many runs can a string contain? Theoretical Computer Science 401(1-3), 165–171 (2008)
Rytter, W.: The number of runs in a string: improved analysis of the linear upper bound. Information and Computation 205(9), 1459–1469 (2007)
Smyth, B.: The maximum number of runs in a string. In: International Workshop on Combinatorial Algorithms (IWOCA 2007), Problems Session (2007)
van Lint, J.L., Wilson, R.M.: A course in combinatorics. Cambridge University Press, Cambridge (1992)
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Giraud, M. (2008). Not So Many Runs in Strings. In: Martín-Vide, C., Otto, F., Fernau, H. (eds) Language and Automata Theory and Applications. LATA 2008. Lecture Notes in Computer Science, vol 5196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88282-4_22
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DOI: https://doi.org/10.1007/978-3-540-88282-4_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88281-7
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