Not So Many Runs in Strings

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 lim_{n ↦ ∞} 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.