Approximating Approximate Pattern Matching

Authors Jan Studený, Przemysław Uznański



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Author Details

Jan Studený
  • Department of Computer Science, ETH Zürich, Switzerland
Przemysław Uznański
  • Institute of Computer Science, University of Wrocław, Poland

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Jan Studený and Przemysław Uznański. Approximating Approximate Pattern Matching. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CPM.2019.15

Abstract

Given a text T of length n and a pattern P of length m, the approximate pattern matching problem asks for computation of a particular distance function between P and every m-substring of T. We consider a (1 +/- epsilon) multiplicative approximation variant of this problem, for l_p distance function. In this paper, we describe two (1+epsilon)-approximate algorithms with a runtime of O~(n/epsilon) for all (constant) non-negative values of p. For constant p >= 1 we show a deterministic (1+epsilon)-approximation algorithm. Previously, such run time was known only for the case of l_1 distance, by Gawrychowski and Uznański [ICALP 2018] and only with a randomized algorithm. For constant 0 <= p <= 1 we show a randomized algorithm for the l_p, thereby providing a smooth tradeoff between algorithms of Kopelowitz and Porat [FOCS 2015, SOSA 2018] for Hamming distance (case of p=0) and of Gawrychowski and Uznański for l_1 distance.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Approximate Pattern Matching
  • l_p Distance
  • l_1 Distance
  • Hamming Distance
  • Approximation Algorithms

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References

  1. Karl R. Abrahamson. Generalized String Matching. SIAM J. Comput., 16(6):1039-1051, 1987. Google Scholar
  2. Amihood Amir, Moshe Lewenstein, and Ely Porat. Faster algorithms for string matching with k mismatches. J. Algorithms, 50(2):257-275, 2004. URL: http://dx.doi.org/10.1016/S0196-6774(03)00097-X.
  3. Amihood Amir, Ohad Lipsky, Ely Porat, and Julia Umanski. Approximate Matching in the L₁ Metric. In CPM, pages 91-103, 2005. URL: http://dx.doi.org/10.1007/11496656_9.
  4. Amit Chakrabarti and Oded Regev. An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance. SIAM J. Comput., 41(5):1299-1317, 2012. URL: http://dx.doi.org/10.1137/120861072.
  5. Peter Clifford, Raphaël Clifford, and Costas S. Iliopoulos. Faster Algorithms for δ,γ-Matching and Related Problems. In CPM, pages 68-78, 2005. URL: http://dx.doi.org/10.1007/11496656_7.
  6. Raphaël Clifford. Matrix multiplication and pattern matching under Hamming norm. http://www.cs.bris.ac.uk/Research/Algorithms/events/BAD09/BAD09/Talks/BAD09-Hammingnotes.pdf, 2009. Retrieved March 2017.
  7. Raphaël Clifford, Allyx Fontaine, Ely Porat, Benjamin Sach, and Tatiana A. Starikovskaya. The k-mismatch problem revisited. In SODA, pages 2039-2052. SIAM, 2016. Google Scholar
  8. M. J. Fischer and M. S. Paterson. STRING-MATCHING AND OTHER PRODUCTS. Technical report, Massachusetts Institute of Technology, 1974. Google Scholar
  9. Sumit Ganguly. Taylor Polynomial Estimator for Estimating Frequency Moments. In ICALP, pages 542-553, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_44.
  10. Paweł Gawrychowski and Przemysław Uznański. Towards Unified Approximate Pattern Matching for Hamming and L₁ Distance. In ICALP, pages 62:1-62:13, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.62.
  11. Daniel Graf, Karim Labib, and Przemysław Uznański. Brief Announcement: Hamming Distance Completeness and Sparse Matrix Multiplication. In ICALP, pages 109:1-109:4, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.109.
  12. T. S. Jayram, Ravi Kumar, and D. Sivakumar. The One-Way Communication Complexity of Hamming Distance. Theory of Computing, 4(1):129-135, 2008. URL: http://dx.doi.org/10.4086/toc.2008.v004a006.
  13. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. On the Exact Space Complexity of Sketching and Streaming Small Norms. In SODA, pages 1161-1178, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.93.
  14. Howard J. Karloff. Fast Algorithms for Approximately Counting Mismatches. Inf. Process. Lett., 48(2):53-60, 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90177-B.
  15. Tsvi Kopelowitz and Ely Porat. Breaking the Variance: Approximating the Hamming Distance in 1/ε Time Per Alignment. In FOCS, pages 601-613, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.43.
  16. Tsvi Kopelowitz and Ely Porat. A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance. In SOSA, pages 10:1-10:5, 2018. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.10.
  17. Yi Li and David P. Woodruff. A Tight Lower Bound for High Frequency Moment Estimation with Small Error. In APPROX-RANDOM, pages 623-638, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40328-6_43.
  18. Ohad Lipsky. Eficient Distance Computations. Master’s thesis, Bar-Ilan University. Department of Mathematics and Computer Science., 2003. Google Scholar
  19. Ohad Lipsky and Ely Porat. L₁ pattern matching lower bound. Inf. Process. Lett., 105(4):141-143, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2007.08.011.
  20. Ohad Lipsky and Ely Porat. Approximate Pattern Matching with the L₁, L₂ and L_∞ Metrics. Algorithmica, 60(2):335-348, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9345-9.
  21. Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. A Subquadratic Approximation Scheme for Partition. In SODA, pages 70-88, 2019. URL: http://dx.doi.org/10.1137/1.9781611975482.5.
  22. John Nolan. Stable distributions: models for heavy-tailed data. Birkhauser New York, 2003. Google Scholar
  23. Ely Porat and Klim Efremenko. Approximating general metric distances between a pattern and a text. In SODA, pages 419-427, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347128.
  24. Jan Studený and Przemysław Uznański. Approximating Approximate Pattern Matching. CoRR, abs/1810.01676, 2018. URL: http://arxiv.org/abs/1810.01676.
  25. David P. Woodruff. Optimal space lower bounds for all frequency moments. In SODA, pages 167-175, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982817.
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