A Faster Algorithm for Computing Maximal \alpha -gapped Repeats in a String

A string $$x = uvu$$ with both u,i¾źv being non-empty is called a gapped repeat with period$$p = |uv|$$, and is denoted by pair x,i¾źp. If $$p \le \alpha |x|-p$$ with $$\alpha > 1$$, then x,i¾źp is called an $$\alpha $$-gapped repeat. An occurrence $$[i, i+|x|-1]$$ of an $$\alpha $$-gapped repeat x,i¾źp in a string w is called a maximal$$\alpha $$-gapped repeat of w, if it cannot be extended either to the left or to the right in w with the same period p. Kolpakov et al. CPM 2014 showed that, given a string of length n over a constant alphabet, all the occurrences of maximal $$\alpha $$-gapped repeats in the string can be computed in $$O\alpha ^2 n + occ $$ time, where $$ occ $$ is the number of occurrences. In this paper, we propose a faster $$O\alpha n + occ $$-time algorithm to solve this problem, improving the result of Kolpakov et al. by a factor of $$\alpha $$.