research-article

Approximating Edit Distance in Truly Subquadratic Time: Quantum and MapReduce

Authors:

Mahdi Boroujeni,

Mohammad Ghodsi,

Mohammadtaghi Hajiaghayi,

Saeed SeddighinAuthors Info & Claims

Journal of the ACM (JACM), Volume 68, Issue 3

Article No.: 19, Pages 1 - 41

https://doi.org/10.1145/3456807

Published: 13 May 2021 Publication History

Abstract

The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [37]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an

quantum algorithm that approximates the edit distance within a factor of 3. We further extend this result to an

quantum algorithm that approximates the edit distance within a larger constant factor.

Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of

, with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.

References

[1]

Andris Ambainis. 2002. Quantum lower bounds by quantum arguments. J. Comput. System Sci. 64, 4 (2002), 750–767.

Digital Library

[2]

Alexandr Andoni, Piotr Indyk, and Robert Krauthgamer. 2009. Overcoming the

non-embeddability barrier: Algorithms for product metrics. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’09). SIAM, 865–874.

Digital Library

[3]

Alexandr Andoni and Robert Krauthgamer. 2010. The computational hardness of estimating edit distance. SIAM J. Comput. 39, 6 (April 2010), 2398–2429.

[4]

Alexandr Andoni and Robert Krauthgamer. 2012. The smoothed complexity of edit distance. ACM Trans. Algorithms 8, 4 (Oct. 2012), Article 44, 25 pages.

Digital Library

[5]

Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. 2010. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS’10). IEEE, 377–386.

Digital Library

[6]

Alexandr Andoni, Aleksandar Nikolov, Krzysztof Onak, and Grigory Yaroslavtsev. 2014. Parallel algorithms for geometric graph problems. In Proceedings of the 46th Annual ACM SIGACT Symposium on Theory of Computing (STOC’14). ACM, 574–583.

Digital Library

[7]

Alexandr Andoni and Negev Shekel Nosatzki. 2020. Edit distance in near-linear time: It’s a constant factor. arxiv:cs.DS/2005.07678

[8]

Alexandr Andoni and Krzysztof Onak. 2012. Approximating edit distance in near-linear time. SIAM J. Comput. 41, 6 (2012), 1635–1648.

Digital Library

[9]

Alberto Apostolico, Mikhail J. Atallah, Lawrence L. Larmore, and Scott McFaddin. 1990. Efficient parallel algorithms for string editing and related problems. SIAM J. Comput. 19, 5 (1990), 968–988.

Digital Library

[10]

Arturs Backurs and Piotr Indyk. 2015. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proceedings of the 47th Annual ACM SIGACT Symposium on Theory of Computing (STOC’15). ACM, 51–58.

Digital Library

[11]

Ziv Bar-Yossef, T. S. Jayram, Robert Krauthgamer, and Ravi Kumar. 2004. Approximating edit distance efficiently. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04). IEEE, 550–559.

Digital Library

[12]

Tuğkan Batu, Funda Ergun, and Cenk Sahinalp. 2006. Oblivious string embeddings and edit distance approximations. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA’06). SIAM, 792–801.

Digital Library

[13]

Robert Beals. 1997. Quantum computation of Fourier transforms over symmetric groups. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC’97). ACM, 48–53.

Digital Library

[14]

Aleksandrs Belovs. 2012. Learning-graph-based quantum algorithm for k-distinctness. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE, 207–216.

Digital Library

[15]

Mahdi Boroujeni, Soheil Ehsani, Mohammad Ghodsi, MohammadTaghi HajiAghayi, and Saeed Seddighin. 2018. Approximating edit distance in truly subquadratic time: Quantum and MapReduce. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’18). SIAM, 1170–1189.

Digital Library

[16]

Mahdi Boroujeni, Masoud Seddighin, and Saeed Seddighin. 2020. Improved algorithms for edit distance and LCS: Beyond worst case. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). SIAM, 1601–1620.

Digital Library

[17]

Mahdi Boroujeni and Saeed Seddighin. 2019. Improved MPC algorithms for edit distance and Ulam distance. In The 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA’19). ACM, 31–40.

Digital Library

[18]

Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp. 1998. Tight bounds on quantum searching. Fortschritte der Physik 46, 4–5 (1998), 493–505.

[19]

Joshua Brakensiek, Moses Charikar, and Aviad Rubinstein. 2020. A simple sublinear algorithm for gap edit distance. arxiv:cs.DS/2007.14368

[20]

Joshua Brakensiek and Aviad Rubinstein. 2020. Constant-factor approximation of near-linear edit distance in near-linear time. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC’20). ACM, 685–698.

Digital Library

[21]

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. 2002. Quantum amplitude amplification and estimation. Contemp. Math. 305 (2002), 53–74.

[22]

Diptarka Chakraborty, Debarati Das, Elazar Goldenberg, Michal Koucky, and Michael Saks. 2018. Approximating edit distance within constant factor in truly sub-quadratic time. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS’18). IEEE, 979–990.

[23]

Diptarka Chakraborty, Debarati Das, Elazar Goldenberg, Michal Koucký, and Michael Saks. 2020. Approximating edit distance within constant factor in truly sub-quadratic time. J. ACM 67, 6, Article 36 (Oct. 2020), 22 pages.

Digital Library

[24]

Moses Charikar and Robert Krauthgamer. 2006. Embedding the Ulam metric into

. Theory Comput. 2, 11 (2006), 207–224.

[25]

Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. 2006. Quantum query complexity of some graph problems. SIAM J. Comput. 35, 6 (2006), 1310–1328.

Digital Library

[26]

Alireza Farhadi, MohammadTaghi HajiAghayi, Aviad Rubinstein, and Saeed Seddighin. 2020. Asymmetric streaming algorithms for edit distance and LCS. arxiv:cs.DS/2002.11342

[27]

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. 1998. A limit on the speed of quantum computation in determining parity. Phys. Rev. Lett. 81 (1998), 5442–5444.

[28]

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. 1999. Invariant quantum algorithms for insertion into an ordered list. arxiv:quant-ph/9901059

[29]

Elazar Goldenberg, Robert Krauthgamer, and Barna Saha. 2019. Sublinear algorithms for gap edit distance. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS’19). IEEE, 1101–1120.

[30]

Elazar Goldenberg, Aviad Rubinstein, and Barna Saha. 2020. Does preprocessing help in fast sequence comparisons? In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC’20). ACM, 657–670.

Digital Library

[31]

Lov K. Grover. 1996. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC’96). ACM, 212–219.

Digital Library

[32]

Bernhard Haeupler, Aviad Rubinstein, and Amirbehshad Shahrasbi. 2019. Near-linear time insertion-deletion codes and (1+

)-approximating edit distance via indexing. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC’19). ACM, 697–708.

Digital Library

[33]

MohammadTaghi HajiAghayi, Silvio Lattanzi, Saeed Seddighin, and Cliff Stein. 2019. MapReduce meets fine-grained complexity: MapReduce algorithms for APSP, matrix multiplication, 3-SUM, and beyond. arxiv:cs.DS/1905.01748

[34]

MohammadTaghi HajiAghayi, Saeed Seddighin, and Xiaorui Sun. 2019. Massively parallel approximation algorithms for edit distance and longest common subsequence. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). SIAM, 1654–1672.

Digital Library

[35]

Ramesh Hariharan and V. Vinay. 2003. String matching in

quantum time. J. Discrete Algorithms 1, 1 (2003), 103–110.

Digital Library

[36]

Sungjin Im, Benjamin Moseley, and Xiaorui Sun. 2017. Efficient massively parallel methods for dynamic programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC’17). ACM, 798–811.

Digital Library

[37]

Piotr Indyk. 2001. Algorithmic applications of low-distortion geometric embeddings. In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’01). IEEE, 10–33.

Digital Library

[38]

Stacey Jeffery, Robin Kothari, and Frédéric Magniez. 2013. Nested quantum walks with quantum data structures. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’13). SIAM, 1474–1485.

Digital Library

[39]

Shagun Jhaver, Latifur Khan, and Bhavani Thuraisingham. 2014. Calculating edit distance for large sets of string pairs using MapReduce. Paper presented at ASE International Conference on Big Data.

[40]

Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. 2010. A model of computation for MapReduce. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10). SIAM, 938–948.

Digital Library

[41]

Tomasz Kociumaka and Barna Saha. 2020. Sublinear-time algorithms for computing & embedding gap edit distance. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS’20). IEEE, Virtual Conference.

[42]

Michal Kouckỳ and Michael Saks. 2020. Constant factor approximations to edit distance on far input pairs in nearly linear time. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC’20). ACM, 699–712.

Digital Library

[43]

Robert Krauthgamer and Yuval Rabani. 2009. Improved lower bounds for embeddings into

. SIAM J. Comput. 38, 6 (2009), 2487–2498.

Digital Library

[44]

Hari Krovi and Alexander Russell. 2015. Quantum Fourier transforms and the complexity of link invariants for quantum doubles of finite groups. Comm. Math. Phys. 334, 2 (2015), 743–777.

[45]

Gad M. Landau, Eugene W. Myers, and Jeanette P. Schmidt. 1998. Incremental string comparison. SIAM J. Comput. 27, 2 (1998), 557–582.

Digital Library

[46]

Silvio Lattanzi, Benjamin Moseley, Siddharth Suri, and Sergei Vassilvitskii. 2011. Filtering: A method for solving graph problems in MapReduce. In Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA’11). ACM, 85–94.

Digital Library

[47]

François Le Gall. 2014. Improved quantum algorithm for triangle finding via combinatorial arguments. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS’14). IEEE, 216–225.

Digital Library

[48]

William J. Masek and Michael S. Paterson. 1980. A faster algorithm computing string edit distances. J. Comput. Syst. Sci. 20, 1 (1980), 18–31.

[49]

Michael Mitzenmacher and Saeed Seddighin. 2020. Dynamic algorithms for LIS and distance to monotonicity. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC’20). ACM, 671–684.

Digital Library

[50]

Ashley Montanaro, Richard Jozsa, and Graeme Mitchison. 2015. On exact quantum query complexity. Algorithmica 71, 4 (2015), 775–796.

Digital Library

[51]

Rajeev Motwani and Prabhakar Raghavan. 1995. Randomized Algorithms. Cambridge University Press, UK.

Digital Library

[52]

Aran Nayebi and Virginia Vassilevska Williams. 2014. Quantum algorithms for shortest paths problems in structured instances. arxiv:quant-ph/1410.6220

[53]

Rafail Ostrovsky and Yuval Rabani. 2007. Low distortion embeddings for edit distance. J. ACM 54, 5 (Oct. 2007), Article 23, 16 pages.

Digital Library

[54]

Aviad Rubinstein, Saeed Seddighin, Zhao Song, and Xiaorui Sun. 2019. Approximation algorithms for LCS and LIS with truly improved running times. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS’19). IEEE, 1121–1145.

[55]

Aviad Rubinstein and Virginia Vassilevska Williams. 2019. SETH vs approximation. ACM SIGACT News 50, 4 (2019), 57–76.

Digital Library

[56]

Peter W. Shor. 1997. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 5 (1997), 1484–1509.

Digital Library

[57]

Esko Ukkonen. 1985. Algorithms for approximate string matching. Inf. Control 64, 1–3 (1985), 100–118.

Digital Library

Cited By

Jin CNogler J(2024)Quantum Speed-Ups for String Synchronizing Sets, Longest Common Substring, and k-mismatch MatchingACM Transactions on Algorithms10.1145/367239520:4(1-36)Online publication date: 5-Aug-2024
https://dl.acm.org/doi/10.1145/3672395
Koo JAgrawal KPetrank E(2024)An Optimal MPC Algorithm for Subunit-Monge Matrix Multiplication, with Applications to LISProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659974(145-154)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659974
Cassis AKociumaka TWellnitz P(2023)Optimal Algorithms for Bounded Weighted Edit Distance2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00135(2177-2187)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00135
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Index Terms

Approximating Edit Distance in Truly Subquadratic Time: Quantum and MapReduce
1. Computing methodologies
  1. Parallel computing methodologies
    1. Parallel algorithms
      1. MapReduce algorithms
      2. Massively parallel algorithms
2. Theory of computation
  1. Design and analysis of algorithms
  2. Models of computation
    1. Quantum computation theory

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 68, Issue 3

June 2021

244 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3456663

Editor:
Éva Tardos
Cornell University, United States

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Copyright © 2021 Copyright held by the owner/author(s). Publication rights licensed to ACM.

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Publication History

Published: 13 May 2021

Accepted: 01 January 2021

Revised: 01 September 2020

Received: 01 April 2018

Published in JACM Volume 68, Issue 3

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Cited By

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https://dl.acm.org/doi/10.1145/3672395
Koo JAgrawal KPetrank E(2024)An Optimal MPC Algorithm for Subunit-Monge Matrix Multiplication, with Applications to LISProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659974(145-154)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659974
Cassis AKociumaka TWellnitz P(2023)Optimal Algorithms for Bounded Weighted Edit Distance2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00135(2177-2187)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00135
Kociumaka TMukherjee ASaha B(2023)Approximating Edit Distance in the Fully Dynamic Model2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00098(1628-1638)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00098
Xu YZhang SLi L(2023)Quantum algorithm for learning secret strings and its experimental demonstrationPhysica A: Statistical Mechanics and its Applications10.1016/j.physa.2022.128372609(128372)Online publication date: Jan-2023
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Akmal SJin C(2023)Near-Optimal Quantum Algorithms for String ProblemsAlgorithmica10.1007/s00453-022-01092-x85:8(2260-2317)Online publication date: 27-Jan-2023
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Srivastava PEachempati P(2022)Applications of Big Data Analytics in Investment ManagementJournal of Database Management10.4018/JDM.29955733:1(1-32)Online publication date: 23-May-2022
https://dl.acm.org/doi/10.4018/JDM.299557
Das DGilbert JHajiaghayi MKociumaka TSaha BSaleh H(2022)Õ(n+poly(k))-time Algorithm for Bounded Tree Edit Distance2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00071(686-697)Online publication date: Oct-2022
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Goldenberg EKociumaka TKrauthgamer RSaha B(2022)Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00070(674-685)Online publication date: Oct-2022
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