The number of runs in a string: Improved analysis of the linear upper bound

Abstract

A run (or a maximal repetition) in a string is an inclusion-maximal periodic segment in a string. Let p(n) be the maximal number of runs in a string of length n. It has been shown in [8] that p(n) = O(n), the proof was very complicated and the constant coefficient in O(n) has not been given explicitly. We propose a new approach to the analysis of runs based on the properties of subperiods: the periods of periodic parts of the runs. We show that p(n) ≤ 5 n. Our proof is inspired by the results of [4], where the role of new periodicity lemmas has been emphasized. © Springer-Verlag Berlin Heidelberg 2006.