License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.77
URN: urn:nbn:de:0030-drops-129432
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12943/
Radoszewski, Jakub ;
Straszyński, Juliusz
Efficient Computation of 2-Covers of a String
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 ?(n log n log log n + output) expected time or ?(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 ?(n log log n) expected time or ?(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 ?(n log n log log n) expected time or ?(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}.
BibTeX - Entry
@InProceedings{radoszewski_et_al:LIPIcs:2020:12943,
author = {Jakub Radoszewski and Juliusz Straszyński},
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},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12943},
URN = {urn:nbn:de:0030-drops-129432},
doi = {10.4230/LIPIcs.ESA.2020.77},
annote = {Keywords: quasiperiodicity, cover of a string, 2-cover, lambda-cover}
}
|
Keywords: |
|
quasiperiodicity, cover of a string, 2-cover, lambda-cover |
|
Collection: |
|
28th Annual European Symposium on Algorithms (ESA 2020) |
|
Issue Date: |
|
2020 |
|
Date of publication: |
|
26.08.2020 |
