A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(nlog n)$ time which we apply to compute all shortest $alpha$-partial covers for a given $alpha$. We also employ it for an $O(nlog n)$-time algorithm computing a shortest $alpha$-partial cover for each $alpha=1,2,ldots,n$.
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