Longest α-Gapped Repeat and Palindrome

We propose an efficient algorithm finding, for a word w and an integer \(\alpha >0\), the longest word u such that w has a factor uvu, with \(|uv|\le \alpha |u|\) (i.e., the longest \(\alpha \)-gapped repeat of w). Our algorithm runs in \({\mathcal O}(\alpha n)\) time. Moreover, it can be easily adapted to find the longest u such that w has a factor \(u^Rvu\), with \(|uv|\le \alpha |u|\) (i.e., the longest \(\alpha \)-gapped palindrome), again in \({\mathcal O}(\alpha n)\) time.