How many runs can a string contain? - Puglisi, S.; Simpson, Jamie; Smyth, William (Elsevier, 2008)
In 2000 Kolpakov and Kucherov showed that the maximum number ρ(n) of runs in any string x[1..n] is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data to prompt the conjecture: ρ(n)<n. Recently, Rytter [Wojciech Rytter, The number of runs in a string: Improved analysis of the linear upper bound, in: B. Durand, W. Thomas (Eds.), STACS 2006, in: Lecture Notes in Computer Science, vol. 3884, Springer-Verlag, Berlin, 2006, pp. 184–195] made a significant step toward proving this conjecture by showing that ρ(n)<5n. In this paper we improve Rytter’s approach and press the bound on ρ(n) further, proving ρ(n)≤3.48n.