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

• Published in: String Processing and Information Retrieval pp. 124 - 136
• Main Authors: Tanimura, Yuka, Fujishige, Yuta, I, Tomohiro, Inenaga, Shunsuke, Bannai, Hideo, Takeda, Masayuki
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing 01.01.2015
• Series: Lecture Notes in Computer Science
• Subjects: Gapped Repeat; Integer Array; Link Path; Link Tree; Outgoing Edge
• ISBN: 9783319238258; 3319238256
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-23826-5_13

A string $$x = uvu$$ with both u, v being non-empty is called a gapped repeat with period $$p = |uv|$$ , and is denoted by pair (x, p). If $$p \le \alpha (|x|-p)$$ with $$\alpha > 1$$ , then (x, p) is called an $$\alpha $$ -gapped repeat. An occurrence $$[i, i+|x|-1]$$ of an $$\alpha $$ -gapped repeat (x, p) 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 $$\alpha $$ .