Near-Optimal Quantum Algorithms for String Problems
We study quantum algorithms for several fundamental string problems, including Longest Common Substring , Lexicographically Minimal String Rotation , and Longest Square Substring . These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by...
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| Vydáno v: | Algorithmica Ročník 85; číslo 8; s. 2260 - 2317 |
|---|---|
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer US
01.08.2023
Springer Nature B.V |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | We study quantum algorithms for several fundamental string problems, including
Longest Common Substring
,
Lexicographically Minimal String Rotation
, and
Longest Square Substring
. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with
near-optimal
query complexities and time complexities. Specifically, we show that:
Longest Common Substring
can be solved by a quantum algorithm in
O
~
(
n
2
/
3
)
time, improving upon the recent
O
~
(
n
5
/
6
)
-time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022.
https://doi.org/10.4230/LIPIcs.ITCS.2022.97
). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019.
https://doi.org/10.1145/3313276.3316368
) and generalized difference covers.
Lexicographically Minimal String Rotation
can be solved by a quantum algorithm in
n
1
/
2
+
o
(
1
)
time, improving upon the recent
O
~
(
n
3
/
4
)
-time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020.
arXiv:2012.09376
). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding.
Longest Square Substring
can be solved by a quantum algorithm in
O
~
(
n
)
time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix. |
|---|---|
| AbstractList | We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in O~(n2/3) time, improving upon the recent O~(n5/6)-time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n1/2+o(1) time, improving upon the recent O~(n3/4)-time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in O~(n) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix. We study quantum algorithms for several fundamental string problems, including Longest Common Substring , Lexicographically Minimal String Rotation , and Longest Square Substring . These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in O ~ ( n 2 / 3 ) time, improving upon the recent O ~ ( n 5 / 6 ) -time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97 ). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368 ) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in n 1 / 2 + o ( 1 ) time, improving upon the recent O ~ ( n 3 / 4 ) -time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376 ). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in O ~ ( n ) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix. We study quantum algorithms for several fundamental string problems, including Longest Common Substring , Lexicographically Minimal String Rotation , and Longest Square Substring . These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in $$\tilde{O}(n^{2/3})$$ O ~ ( n 2 / 3 ) time, improving upon the recent $$\tilde{O}(n^{5/6})$$ O ~ ( n 5 / 6 ) -time algorithm by Le Gall and Seddighin (in: Proceedings of the 13th innovations in theoretical computer science conference (ITCS 2022), pp 97:1–97:23, 2022. https://doi.org/10.4230/LIPIcs.ITCS.2022.97 ). Our algorithm uses the MNRS quantum walk framework, together with a careful combination of string synchronizing sets (Kempa and Kociumaka, in: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing (STOC 2019), ACM, pp 756–767, 2019. https://doi.org/10.1145/3313276.3316368 ) and generalized difference covers. Lexicographically Minimal String Rotation can be solved by a quantum algorithm in $$n^{1/2 + o(1)}$$ n 1 / 2 + o ( 1 ) time, improving upon the recent $$\tilde{O}(n^{3/4})$$ O ~ ( n 3 / 4 ) -time algorithm by Wang and Ying (in: Quantum algorithm for lexicographically minimal string rotation. CoRR, 2020. arXiv:2012.09376 ). We design our algorithm by first giving a new classical divide-and-conquer algorithm in near-linear time based on exclusion rules, and then speeding it up quadratically using nested Grover search and quantum minimum finding. Longest Square Substring can be solved by a quantum algorithm in $$\tilde{O}(\sqrt{n})$$ O ~ ( n ) time. Our algorithm is an adaptation of the algorithm by Le Gall and Seddighin (2022) for the Longest Palindromic Substring problem, but uses additional techniques to overcome the difficulty that binary search no longer applies. Our techniques naturally extend to other related string problems, such as Longest Repeated Substring, Longest Lyndon Substring, and Minimal Suffix. |
| Author | Akmal, Shyan Jin, Ce |
| Author_xml | – sequence: 1 givenname: Shyan surname: Akmal fullname: Akmal, Shyan organization: MIT – sequence: 2 givenname: Ce surname: Jin fullname: Jin, Ce email: cejin@mit.edu organization: MIT |
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| CitedBy_id | crossref_primary_10_1145_3723884 crossref_primary_10_1145_3672395 crossref_primary_10_1137_23M1567618 crossref_primary_10_1016_j_tcs_2025_115120 |
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| Keywords | Quantum walks Longest common substring String synchronizing sets String processing |
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| Snippet | We study quantum algorithms for several fundamental string problems, including
Longest Common Substring
,
Lexicographically Minimal String Rotation
, and... We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Gall Mathematics of Computing Rotation Strings Synchronism Theory of Computation |
| Title | Near-Optimal Quantum Algorithms for String Problems |
| URI | https://link.springer.com/article/10.1007/s00453-022-01092-x https://www.proquest.com/docview/2847446783 |
| Volume | 85 |
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