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Not so many runs in strings

Mathieu Giraud 1, 2, * 
* Corresponding author
2 SEQUOIA - Sequential Learning
LIFL - Laboratoire d'Informatique Fondamentale de Lille, Inria Lille - Nord Europe
Abstract : Since the work of Kolpakov and Kucherov in 1998, it is known that \rho(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 = 0.927 n has been given by Franek and al., and upper bounds have been recently provided by Rytter, Puglisi and al., and Crochemore and Ilie (1.6n). However, very few properties are known for the \rho(n)/n function. We show here by a simple argument that this function converges and that its limit is never reached. We further study the asymptotic behavior of the number of runs with short periods. We provide a new bound for some microruns : we show that there is no more than 0.971n 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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Submitted on : Wednesday, April 9, 2008 - 5:32:22 PM
Last modification on : Friday, February 4, 2022 - 3:18:24 AM
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  • HAL Id : inria-00271630, version 1



Mathieu Giraud. Not so many runs in strings. 2nd International Conference on Language and Automata Theory and Applications (LATA 2008), Mar 2008, Tarragona, Spain. ⟨inria-00271630⟩



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