A New Periodicity Lemma

Abstract

Given a string x = x[1..n], a repetition of period p in x is a substring u ^r = x[i..i + rp − 1], p = |u|, r ≥ 2, where neither u = x[i..i + p − 1] nor x[i..i + (r + 1)p − 1] is a repetition. The maximum number of repetitions in any string x is well known to be Θ(nlog n). A run or maximal periodicity of period p in x is a substring u ^r t = x[i..i + rp + |t| − 1] of x, where u ^r is a repetition, t a proper prefix of x, and no repetition of period p begins at position i – 1 of x or ends at position i + rp + |t|. In 2000 Kolpakov and Kucherov showed that the maximum number ρ(n) of runs in any string x is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data strongly suggesting that ρ(n) < n. In this paper, as a first step toward proving this conjecture, we present a periodicity lemma that establishes limitations on the number of squares, and their periods, that can occur over a specified range of positions in x. We then apply this result to specify corresponding limitations on the occurrence of runs.

The work of the first and second authors was supported in part by grants from the Natural Sciences & Engineering Research Council of Canada.