Optimal searching of gapped repeats in a word

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the number of {\em maximal} $\alpha$-gapped repeats in a string of length $n$, previously proved to be $O(\alpha^2 n)$ in (Kolpakov et al., 2013). For a closely related notion of maximal $\delta$-subrepetition (maximal factors of exponent between $1+\delta$ and $2$), our result implies the $O(n/\delta)$ bound on their number, which improves the bound of (Kolpakov et al., 2010) by a $\log n$ factor. We also prove an algorithmic time bound $O(\alpha n+S)$ ($S$ size of the output) for computing all maximal $\alpha$-gapped repeats. Our solution, inspired by (Gawrychowski and Manea, 2015), is different from the recently published proof by (Tanimura et al., 2015) of the same bound. Together with our bound on $S$, this implies an $O(\alpha n)$-time algorithm for computing all maximal $\alpha$-gapped repeats.