Document

# Improved Approximation for Longest Common Subsequence over Small Alphabets

## File

LIPIcs.ICALP.2021.13.pdf
• Filesize: 0.7 MB
• 18 pages

## Acknowledgements

We thank Jenny Kaufmann for suggesting several helpful revisions.

## Cite As

Shyan Akmal and Virginia Vassilevska Williams. Improved Approximation for Longest Common Subsequence over Small Alphabets. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.13

## Abstract

This paper investigates the approximability of the Longest Common Subsequence (LCS) problem. The fastest algorithm for solving the LCS problem exactly runs in essentially quadratic time in the length of the input, and it is known that under the Strong Exponential Time Hypothesis the quadratic running time cannot be beaten. There are no such limitations for the approximate computation of the LCS however, except in some limited scenarios. There is also a scarcity of approximation algorithms. When the two given strings are over an alphabet of size k, returning the subsequence formed by the most frequent symbol occurring in both strings achieves a 1/k approximation for the LCS. It is an open problem whether a better than 1/k approximation can be achieved in truly subquadratic time (O(n^{2-δ}) time for constant δ > 0). A recent result [Rubinstein and Song SODA'2020] showed that a 1/2+ε approximation for the LCS over a binary alphabet is possible in truly subquadratic time, provided the input strings have the same length. In this paper we show that if a 1/2+ε approximation (for ε > 0) is achievable for binary LCS in truly subquadratic time when the input strings can be unequal, then for every constant k, there is a truly subquadratic time algorithm that achieves a 1/k+δ approximation for k-ary alphabet LCS for some δ > 0. Thus the binary case is the hardest. We also show that for every constant k, if one is given two strings of equal length over a k-ary alphabet, one can obtain a 1/k+ε approximation for some constant ε > 0 in truly subquadratic time, thus extending the Rubinstein and Song result to all alphabets of constant size.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• approximation algorithms
• longest common subsequence

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Amir Abboud and Arturs Backurs. Towards hardness of approximation for polynomial time problems. In 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 11:1-11:26, 2017.
2. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 59-78. IEEE Computer Society, 2015.
3. Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating Branching Programs with Edit Distance and Friends: Or: A Polylog Shaved is a Lower Bound Made, page 375–388. Association for Computing Machinery, New York, NY, USA, 2016. URL: https://doi.org/10.1145/2897518.2897653.
4. Amir Abboud and Aviad Rubinstein. Fast and deterministic constant factor approximation algorithms for LCS imply new circuit lower bounds. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, volume 94 of LIPIcs, pages 35:1-35:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
5. Alex Andoni. Simpler constant-factor approximation to edit distance problems, 2019.
6. Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 377-386. IEEE Computer Society, 2010.
7. Alexandr Andoni and Negev Shekel Nosatzki. Edit distance in near-linear time: it’s a constant factor. CoRR, abs/2005.07678, 2020. URL: http://arxiv.org/abs/2005.07678.
8. Alexandr Andoni and Krzysztof Onak. Approximating edit distance in near-linear time. SIAM J. Comput., 41(6):1635-1648, 2012.
9. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). SIAM J. Comput., 47(3):1087-1097, 2018.
10. Ziv Bar-Yossef, T. S. Jayram, Robert Krauthgamer, and Ravi Kumar. Approximating edit distance efficiently. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 550-559. IEEE Computer Society, 2004.
11. Tugkan Batu, Funda Ergün, and Süleyman Cenk Sahinalp. Oblivious string embeddings and edit distance approximations. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 792-801. ACM Press, 2006.
12. Mahdi Boroujeni, Soheil Ehsani, Mohammad Ghodsi, Mohammad Taghi Hajiaghayi, and Saeed Seddighin. Approximating edit distance in truly subquadratic time: Quantum and mapreduce. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1170-1189. SIAM, 2018.
13. Joshua Brakensiek and Aviad Rubinstein. Constant-factor approximation of near-linear edit distance in near-linear time. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 685–698, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384282.
14. Karl Bringmann and Marvin Künnemann. Multivariate fine-grained complexity of longest common subsequence. In SODA, 2018.
15. Diptarka Chakraborty, Debarati Das, Elazar Goldenberg, Michal Koucký, and Michael E. Saks. Approximating edit distance within constant factor in truly sub-quadratic time. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 979-990. IEEE Computer Society, 2018.
16. Lijie Chen, Shafi Goldwasser, Kaifeng Lyu, Guy N. Rothblum, and Aviad Rubinstein. Fine-grained complexity meets IP = PSPACE. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1-20. SIAM, 2019.
17. MohammadTaghi Hajiaghayi, Masoud Seddighin, Saeed Seddighin, and Xiaorui Sun. Approximating LCS in linear time: Beating the √n barrier. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1181-1200. SIAM, 2019.
18. Michal Koucký and Michael Saks. Constant factor approximations to edit distance on far input pairs in nearly linear time. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 699-712, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384307.
19. Gad M. Landau, Eugene W. Myers, and Jeanette P. Schmidt. Incremental string comparison. SIAM Journal on Computing, 27(2):557-582, 1998. URL: https://doi.org/10.1137/s0097539794264810.
20. William J. Masek and Mike Paterson. A faster algorithm computing string edit distances. J. Comput. Syst. Sci., 20(1):18-31, 1980.
21. Aviad Rubinstein and Zhao Song. Reducing approximate longest common subsequence to approximate edit distance. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’20, page 1591–1600, USA, 2020. Society for Industrial and Applied Mathematics.
X

Feedback for Dagstuhl Publishing