TY - JOUR
T1 - Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes
T2 - Finding All Maximal α-gapped Repeats and Palindromes in Optimal Worst Case Time on Integer Alphabets
AU - Gawrychowski, Paweł
AU - I, Tomohiro
AU - Inenaga, Shunsuke
AU - Köppl, Dominik
AU - Manea, Florin
N1 - Funding Information:
We thank the anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions. The work of Florin Manea was supported by the DFG grant 596676. Parts of this work have already been presented at 33rd International Symposium on Theoretical Aspects of Computer Science [13 ] and at the 20th International Symposium on Fundamentals of Computation Theory [10 ]. This article is part of the Topical Collection on Theoretical Aspects of Computer Science This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funding Information:
Acknowledgements We thank the anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions. The work of Florin Manea was supported by the DFG grant 596676.
Publisher Copyright:
© 2017, The Author(s).
PY - 2018/1/1
Y1 - 2018/1/1
N2 - An α-gapped repeat (α ≥ 1) in a word w is a factor uvu of w such that |uv| ≤ α|u|; the two occurrences of u are called arms of this α-gapped repeat. An α-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal α-gapped repeats occurring in words of length n is upper bounded by 18αn. In the case of α-gapped palindromes, i.e., factors uvu⊺ with |uv|≤ α|u|, we show that the number of all maximal α-gapped palindromes occurring in words of length n is upper bounded by 28αn + 7n. Both upper bounds allow us to construct algorithms finding all maximal α-gapped repeats and/or all maximal α-gapped palindromes of a word of length n on an integer alphabet of size nO ( 1 ) in O(αn) time. The presented running times are optimal since there are words that have Θ(αn) maximal α-gapped repeats/palindromes.
AB - An α-gapped repeat (α ≥ 1) in a word w is a factor uvu of w such that |uv| ≤ α|u|; the two occurrences of u are called arms of this α-gapped repeat. An α-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal α-gapped repeats occurring in words of length n is upper bounded by 18αn. In the case of α-gapped palindromes, i.e., factors uvu⊺ with |uv|≤ α|u|, we show that the number of all maximal α-gapped palindromes occurring in words of length n is upper bounded by 28αn + 7n. Both upper bounds allow us to construct algorithms finding all maximal α-gapped repeats and/or all maximal α-gapped palindromes of a word of length n on an integer alphabet of size nO ( 1 ) in O(αn) time. The presented running times are optimal since there are words that have Θ(αn) maximal α-gapped repeats/palindromes.
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U2 - 10.1007/s00224-017-9794-5
DO - 10.1007/s00224-017-9794-5
M3 - Article
AN - SCOPUS:85028529401
VL - 62
SP - 162
EP - 191
JO - Theory of Computing Systems
JF - Theory of Computing Systems
SN - 1432-4350
IS - 1
ER -