# Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes

## Finding All Maximal α-gapped Repeats and Palindromes in Optimal Worst Case Time on Integer Alphabets

Paweł Gawrychowski, I. Tomohiro, Shunsuke Inenaga, Dominik Köppl, Florin Manea

7 引用 (Scopus)

### 抄録

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.

元の言語 英語 162-191 30 Theory of Computing Systems 62 1 https://doi.org/10.1007/s00224-017-9794-5 出版済み - 1 1 2018

### Fingerprint

Palindrome
Optimal Algorithm
Integer
Upper bound

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computational Theory and Mathematics

### これを引用

Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes : Finding All Maximal α-gapped Repeats and Palindromes in Optimal Worst Case Time on Integer Alphabets. / Gawrychowski, Paweł; Tomohiro, I.; Inenaga, Shunsuke; Köppl, Dominik; Manea, Florin.

：: Theory of Computing Systems, 巻 62, 番号 1, 01.01.2018, p. 162-191.

@article{4c3a0c5e286840fe8505a246e5591451,
title = "Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes: Finding All Maximal α-gapped Repeats and Palindromes in Optimal Worst Case Time on Integer Alphabets",
abstract = "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.",
author = "Paweł Gawrychowski and I. Tomohiro and Shunsuke Inenaga and Dominik K{\"o}ppl and Florin Manea",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/s00224-017-9794-5",
language = "English",
volume = "62",
pages = "162--191",
journal = "Theory of Computing Systems",
issn = "1432-4350",
publisher = "Springer New York",
number = "1",

}

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 - Tomohiro, I.

AU - Inenaga, Shunsuke

AU - Köppl, Dominik

AU - Manea, Florin

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.

UR - http://www.scopus.com/inward/record.url?scp=85028529401&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028529401&partnerID=8YFLogxK

U2 - 10.1007/s00224-017-9794-5

DO - 10.1007/s00224-017-9794-5

M3 - Article

VL - 62

SP - 162

EP - 191

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 1

ER -