Fast Algorithm for Partial Covers in Words

Abstract

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 α-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least α positions in w. We develop a data structure of O(n) size (where n = |w|) that can be constructed in O(n log n) time which we apply to compute all shortest α-partial covers for a given α. We also employ it for an O(n log n)-time algorithm computing a shortest α-partial cover for each α = 1, 2, . . . , n.Keywords Cover of a word · Quasiperiodicity · Suffix tree

Efficient algorithms for shortest partial seeds in words

“…The Cover Suffix Tree, introduced in [13], is an augmented version of a suffix tree. It allows to efficiently compute C(v) for any explicit or implicit node, as shown in the following theorem: Actually, [13] provides explicit formulae for c(v) and ∆(v) in terms of Occ(v).…”

Efficient Algorithms for Shortest Partial Seeds in Words

“…It allows to efficiently compute C(v) for any explicit or implicit node, as shown in the following theorem: Actually, [13] provides explicit formulae for c(v) and ∆(v) in terms of Occ(v). Their form is not important here; the only property that we use in this contribution is the following:…”

“…One of the ideas, resulting in the notions of approximate covers [11] and approximate seeds [12], is to require that each position lies within an approximate occurrence of the corresponding quasiperiod. Another approach, introduced recently in [13], yields the notion of partial covers-factors required to cover a certain number of positions of a word. Partial covers generalize the earlier notion of enhanced covers [14], which are additionally required to be borders (both prefixes and suffixes) of the word.…”

“…The cover index C(u, w) of a factor u in a word w has been defined in [13] as the number of positions in w covered by (full) occurrences of u in w. The word u is called an α-partial cover of w if C(u, w) ≥ α.…”

“…In [13] a data structure called the Cover Suffix Tree and denoted by CST (w) was introduced. For a word w of length n the size of CST (w) is O(n) and the construction time is O(n log n).…”