Fast Algorithm for Partial Covers in Words

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