Elsevier

Theoretical Computer Science

Volume 506, 30 September 2013, Pages 102-114
Theoretical Computer Science

Enhanced string covering

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Abstract

A factor u of a string y is a cover of y if every letter of y lies within some occurrence of u in y; thus every cover u is also a border—both prefix and suffix—of y. If u is a cover of a superstring of y then u is a seed of y. Covers and seeds are two formalisations of quasiperiodicity, and there exist linear-time algorithms for computing all the covers and seeds of y. A string y covered by u thus generalises the idea of a repetition; that is, a string composed of exact concatenations of u. Even though a string is coverable somewhat more frequently than it is a repetition, still a string that can be covered by a single u is rare. As a result, seeking to find a more generally applicable and descriptive notion of cover, many articles were written on the computation of a minimum k-cover of y; that is, the minimum cardinality set of strings of length k that collectively cover y. Unfortunately, this computation turns out to be NP-hard. Therefore, in this article, we propose new, simple, easily-computed, and widely applicable notions of string covering that provide an intuitive and useful characterisation of a string: the enhanced cover; the enhanced left cover; and the enhanced left seed.

Keywords

Periodicity
Quasiperiodicity
Covers
Seeds

Cited by (0)

A preliminary version of this article appeared in the Proceedings of the Prague Stringology Conference 2012 (PSC 2012), pp. 75–88, 2012.

1

Supported by the DFG grant STA 860/4.

2

Supported by the NSF-funded iPlant Collaborative (NSF grant #DBI-0735191).

3

Supported by a Newton Fellowship.

4

Supported by a grant from the Natural Sciences & Engineering Research Council (NSERC) of Canada.