Almost optimal searching of maximal subrepetitions in a word

Abstract

For $0<\delta <1$ a $\delta$-subrepetition in a word is a factor which exponent is less than~2 but is not less than $1+\delta$ (the exponent of the factor is the ratio of the factor length to its minimal period). The $\delta$-subrepetition is maximal if it cannot be extended to the left or to the right by at least one letter with preserving its minimal period. In the paper we propose an algorithm for searching all maximal $\delta$-subrepetitions in a word of length~$n$ in $O(\frac{n}{\delta}\log\frac{1}{\delta})$ time (the lower bound for this time is $\Omega (\frac{n}{\delta})$).