When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.77
URN: urn:nbn:de:0030-drops-129432
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12943/
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### Efficient Computation of 2-Covers of a String

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### 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 饾挭(n log n log log n + output) expected time or 饾挭(n log n log虏 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 饾挭(n log log n) expected time or 饾挭(n log虏 log n / log log log n) worst-case time. A string of length n can have 螛(n虏) non-trivial b-covers; our algorithm can report one b-cover per length (if it exists) or all shortest b-covers in 饾挭(n log n log log n) expected time or 饾挭(n log n log虏 log n / log log log n) worst-case time. All our algorithms use linear space.
The problem in scope can be generalized to 位 > 2 equal-length strings, resulting in the notion of 位-cover. Cole et al. (2005) showed that the 位-cover problem is NP-complete. Our algorithms generalize to 位-covers, with (the first component of) the algorithm鈥檚 complexity multiplied by n^{位-2}.

### BibTeX - Entry

```@InProceedings{radoszewski_et_al:LIPIcs:2020:12943,
author =	{Jakub Radoszewski and Juliusz Straszy艅ski},
title =	{{Efficient Computation of 2-Covers of a String}},
booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
pages =	{77:1--77:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-162-7},
ISSN =	{1868-8969},
year =	{2020},
volume =	{173},
editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},