The number of runs in a string

A run in a string is a nonextendable (with the same minimal period) periodic segment in a string. The set of runs corresponds to the structure of internal periodicities in a string. Periodicities in strings were extensively studied and are important both in theory and practice (combinatorics of words, pattern-matching, computational biology). Let rho(n) be the maximal number of runs in a string of length n. It has been shown that rho(n) = O(n), the proof was very complicated and the constant coefficient in O(n) has not been given explicitly. We demystify the proof of the linear upper bound for p(n) and 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) <= 3.44 n and there are at most 0.67n runs with periods larger than 87. This supports the conjecture that the number of all runs is smaller than n. We also give a completely new proof of the linear bound and discover several new interesting "periodicity lemmas". (c) 2007 Elsevier Inc. All rights reserved.