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# Efficient Computation of 2-Covers of a String

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LIPIcs.ESA.2020.77.pdf
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## Acknowledgements

The authors thank Patryk Czajka for helpful discussions on the initial version of the algorithm.

## Cite As

Jakub Radoszewski and Juliusz Straszyński. Efficient Computation of 2-Covers of a String. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.77

## Abstract

Quasiperiodicity is a generalization of periodicity that has been researched for almost 30 years. The notion of cover is the classic variant of quasiperiodicity. A cover of a text T is a string whose occurrences in T cover all positions of T. There are several algorithms computing covers of a text in linear time. In this paper we consider a natural extension of cover. For a text T, we call a pair of strings a 2-cover if they have the same length and their occurrences cover the text T. We give an algorithm that computes all 2-covers of a string of length n in 𝒪(n log n log log n + output) expected time or 𝒪(n log n log² log n / log log log n + output) worst-case time, where output is the size of output. If (X,Y) is a 2-cover of T, then either X is a prefix and Y is a suffix of T, in which case we call (X,Y) a ps-cover, or one of X, Y is a border (that is, both a prefix and a suffix) of T, and then we call (X,Y) a b-cover. A string of length n has up to n ps-covers; we show an algorithm that computes all of them in 𝒪(n log log n) expected time or 𝒪(n log² log n / log log log n) worst-case time. A string of length n can have Θ(n²) non-trivial b-covers; our algorithm can report one b-cover per length (if it exists) or all shortest b-covers in 𝒪(n log n log log n) expected time or 𝒪(n log n log² log n / log log log n) worst-case time. All our algorithms use linear space. The problem in scope can be generalized to λ > 2 equal-length strings, resulting in the notion of λ-cover. Cole et al. (2005) showed that the λ-cover problem is NP-complete. Our algorithms generalize to λ-covers, with (the first component of) the algorithm’s complexity multiplied by n^{λ-2}.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Pattern matching
##### Keywords
• quasiperiodicity
• cover of a string
• 2-cover
• lambda-cover

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