A New Periodicity Lemma

• Authors: Kangmin Fan;Simon J. Puglisi;W. F. Smyth;Andrew Turpin
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2006

Given a string $\s{x}=\s{x}[1..n]$, a repetition of period $p$ in {\mbox{\boldmath $x$}} is a substring ${\mbox{\boldmath $u$}}^r = \break {\mbox{\boldmath $x$}}[i..i\+ rp\- 1]$, $p = |{\mbox{\boldmath $u$}}|$, $r \ge 2$, where neither ${\mbox{\boldmath $u$}} = {\mbox{\boldmath $x$}}[i..i\+ p\- 1]$ nor ${\mbox{\boldmath $x$}}[i..i\+ (r\+ 1)p\- 1]$ is a repetition. The maximum number of repetitions in any string {\mbox{\boldmath $x$}} is well known to be $\Theta(n\log n)$. A run or maximal periodicity of period $p$ in {\mbox{\boldmath $x$}} is a substring ${\mbox{\boldmath $u$}}^r{\mbox{\boldmath $t$}} = {\mbox{\boldmath $x$}}[i..i\+ rp\+ |{\mbox{\boldmath $t$}}|\- 1]$ of {\mbox{\boldmath $x$}}, where ${\mbox{\boldmath $u$}}^r$ is a repetition, {\mbox{\boldmath $t$}} is a proper prefix of {\mbox{\boldmath $u$}}, and no repetition of period $p$ begins at position $i\- 1$ of {\mbox{\boldmath $x$}} or ends at position $i\+ rp\+ |{\mbox{\boldmath $t$}}|$. In 2000 Kolpakov and Kucherov [J. Discrete Algorithms, 1 (2000), pp. 159-186] showed that the maximum number $\rho(n)$ of runs in any string {\mbox{\boldmath $x$}} 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 $\rho(n) J. Combin. Theory Ser. A., 82 (1998), pp. 112-120] showed that the maximum number $\sigma(n)$ of distinct squares in any string {\mbox{\boldmath $x$}} satisfies $\sigma(n)