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Asymptotic behavior of the numbers of runs and microruns

Mathieu Giraud 1, 2, *
* Corresponding author
2 SEQUOIA - Sequential Learning
LIFL - Laboratoire d'Informatique Fondamentale de Lille, Inria Lille - Nord Europe
Abstract : The notion of run (also called maximal repetition) allows a compact representation of the set of all tandem periodicities, even fractional, in a string. Since the work of Kolpakov and Kucherov (1998), it is known that rho(n), the maximum number of runs in a string, is linear in the length n of the string. Between 2003 and 2008, lower bounds haven been provided by Franek et al. and Matsubara et al. (0.9445... n) and upper bounds have been provided by Rytter, Puglisi et al., and Crochemore and Ilie (1.048n). However, very few properties are known for the rho(n)/n function. We show here by a simple argument that limit rho(n)/n exists and that this limit is never reached. We further study the asymptotic behavior of rho_p(n), the maximal number of runs with period at most p. Finally, we provide the first exact limits for some microruns.
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Mathieu Giraud. Asymptotic behavior of the numbers of runs and microruns. Information and Computation, Elsevier, 2009, 207 (11), pp.1221-1228. ⟨10.1016/j.ic.2009.02.007⟩. ⟨hal-00438214⟩

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