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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.77
URN: urn:nbn:de:0030-drops-146581
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14658/
Radoszewski, Jakub ;
Rytter, Wojciech ;
Straszyński, Juliusz ;
Waleń, Tomasz ;
Zuba, Wiktor
Hardness of Detecting Abelian and Additive Square Factors in Strings
Abstract
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality of the state-of-the-art algorithms for finding such factors.
Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of an odd half-length, (b) computing centers of all Abelian square factors, (c) detecting an additive square factor in a length-n string of integers of magnitude n^{?(1)}, and (d) a problem of computing a double 3-term arithmetic progression (i.e., finding indices i ≠ j such that (x_i+x_j)/2 = x_{(i+j)/2}) in a sequence of integers x₁,… ,x_n of magnitude n^{?(1)}.
Problem (d) is essentially a convolution version of the AVERAGE problem that was proposed in a manuscript of Erickson. We obtain a conditional lower bound for it with the aid of techniques recently developed by Dudek et al. [STOC 2020]. Problem (d) immediately reduces to problem (c) and is a step in reductions to problems (a) and (b). In conditional lower bounds for problems (a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it using several string gadgets that include arbitrarily long Abelian-square-free strings.
Our reductions also imply conditional lower bounds for detecting Abelian squares in strings over a constant-sized alphabet. We also show a subquadratic upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].
BibTeX - Entry
@InProceedings{radoszewski_et_al:LIPIcs.ESA.2021.77,
author = {Radoszewski, Jakub and Rytter, Wojciech and Straszy\'{n}ski, Juliusz and Wale\'{n}, Tomasz and Zuba, Wiktor},
title = {{Hardness of Detecting Abelian and Additive Square Factors in Strings}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {77:1--77:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14658},
URN = {urn:nbn:de:0030-drops-146581},
doi = {10.4230/LIPIcs.ESA.2021.77},
annote = {Keywords: Abelian square, additive square, 3SUM problem}
}
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Keywords: |
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Abelian square, additive square, 3SUM problem |
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Collection: |
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29th Annual European Symposium on Algorithms (ESA 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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31.08.2021 |
