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Quasi-Periodicity in Streams

Authors Paweł Gawrychowski, Jakub Radoszewski, Tatiana Starikovskaya

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • Streaming algorithms
  • quasi-periodicity
  • covers
  • seeds

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Abstract

In this work, we show two streaming algorithms for computing the length of the shortest cover of a string of length n. We start by showing a two-pass algorithm that uses O(log^2 n) space and then show a one-pass streaming algorithm that uses O(sqrt{n log n}) space. Both algorithms run in near-linear time. The algorithms are randomized and compute the answer incorrectly with probability inverse-polynomial in n. We also show that there is no sublinear-space streaming algorithm for computing the length of the shortest seed of a string.

Cite As Get BibTex

Paweł Gawrychowski, Jakub Radoszewski, and Tatiana Starikovskaya. Quasi-Periodicity in Streams. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CPM.2019.22

Author Details

Paweł Gawrychowski
  • University of Wrocław, 50-137 Wrocław, Poland
Jakub Radoszewski
  • Institute of Informatics, University of Warsaw, 02-097 Warsaw, Poland
Tatiana Starikovskaya
  • DIENS, École normale supérieure, PSL Research University, 75005 Paris, France

Funding

Jakub Radoszewski is supported by the "Algorithms for text processing with errors and uncertainties" project carried out within the HOMING programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.

References

  1. Ali Alatabbi, Abu Sayed Md. Sohidull Islam, Mohammad Sohel Rahman, Jamie Simpson, and William F. Smyth. Enhanced Covers of Regular and Indeterminate Strings Using Prefix Tables. Journal of Automata, Languages and Combinatorics, 21(3):131-147, 2016. URL: http://dx.doi.org/10.25596/jalc-2016-131.
  2. Ali Alatabbi, M. Sohel Rahman, and William F. Smyth. Computing covers using prefix tables. Discrete Applied Mathematics, 212:2-9, 2016. URL: http://dx.doi.org/10.1016/j.dam.2015.05.019.
  3. Amihood Amir, Costas S. Iliopoulos, and Jakub Radoszewski. Two strings at Hamming distance 1 cannot be both quasiperiodic. Information Processing Letters, 128:54-57, 2017. URL: http://dx.doi.org/10.1016/j.ipl.2017.08.005.
  4. Amihood Amir, Avivit Levy, Moshe Lewenstein, Ronit Lubin, and Benny Porat. Can We Recover the Cover? In Proceedings of the Annual Symposium on Combinatorial Pattern Matching, CPM 2017, pages 25:1-25:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CPM.2017.25.
  5. Amihood Amir, Avivit Levy, Ronit Lubin, and Ely Porat. Approximate Cover of Strings. In Proceedings of the Annual Symposium on Combinatorial Pattern Matching, CPM 2017, volume 78, pages 26:1-26:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CPM.2017.26.
  6. Amihood Amir, Avivit Levy, and Ely Porat. Quasi-Periodicity Under Mismatch Errors. In Proceedings of the Annual Symposium on Combinatorial Pattern Matching, CPM 2018, pages 4:1-4:15, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.CPM.2018.4.
  7. Alberto Apostolico and Andrzej Ehrenfeucht. Efficient detection of quasiperiodicities in strings. Theoretical Computer Science, 119(2):247-265, 1993. URL: http://dx.doi.org/10.1016/0304-3975(93)90159-Q.
  8. Alberto Apostolico, Martin Farach, and Costas S. Iliopoulos. Optimal superprimitivity testing for strings. Information Processing Letters, 39(1):17-20, 1991. URL: http://dx.doi.org/10.1016/0020-0190(91)90056-N.
  9. Ziv Bar-Yossef, T. S. Jayram, Robert Krauthgamer, and Ravi Kumar. The Sketching Complexity of Pattern Matching. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and of the International Workshop on Randomization and Computation, APPROX-RANDOM 2004, pages 261-272, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27821-4_24.
  10. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. Information theory methods in communication complexity. In Proceedings of the IEEE Annual Conference on Computational Complexity, CCC 2002, pages 93-102, 2002. URL: http://dx.doi.org/10.1109/CCC.2002.1004344.
  11. Dany Breslauer. An on-line string superprimitivity test. Information Processing Letters, 44(6):345-347, 1992. URL: http://dx.doi.org/10.1016/0020-0190(92)90111-8.
  12. Dany Breslauer and Zvi Galil. Real-Time Streaming String-Matching. ACM Transasctions on Algorithms, 10(4):22:1-22:12, 2014. URL: http://dx.doi.org/10.1145/2635814.
  13. Manolis Christodoulakis, Costas S. Iliopoulos, Kunsoo Park, and Jeong Seop Sim. Approximate seeds of strings. Journal of Automata, Languages and Combinatorics, 10:609-626, 2005. URL: http://dx.doi.org/10.25596/jalc-2005-609.
  14. Michalis Christou, Maxime Crochemore, Ondrej Guth, Costas S. Iliopoulos, and Solon P. Pissis. On left and right seeds of a string. Journal of Discrete Algorithms, 17:31-44, 2012. URL: http://dx.doi.org/10.1016/j.jda.2012.10.004.
  15. Michalis Christou, Maxime Crochemore, and Costas S. Iliopoulos. Quasiperiodicities in Fibonacci strings. Ars Combinatoria, 129:211-225, 2016. Google Scholar
  16. Michalis Christou, Maxime Crochemore, Costas S. Iliopoulos, Marcin Kubica, Solon P. Pissis, Jakub Radoszewski, Wojciech Rytter, Bartosz Szreder, and Tomasz Waleń. Efficient seed computation revisited. Theoretical Computer Science, 483:171-181, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2011.12.078.
  17. Richard Cole, Costas S. Iliopoulos, Manal Mohamed, William F. Smyth, and Lu Yang. The Complexity of the Minimum k-cover Problem. Journal of Automata, Languages and Combinatorics, 10(5-6):641-653, 2005. URL: http://dx.doi.org/10.25596/jalc-2005-641.
  18. Funda Ergün, Elena Grigorescu, Erfan Sadeqi Azer, and Samson Zhou. Streaming Periodicity with Mismatches. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and of the International Workshop on Randomization and Computation, APPROX-RANDOM 2017, pages 42:1-42:21, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.42.
  19. Funda Ergün, Elena Grigorescu, Erfan Sadeqi Azer, and Samson Zhou. Periodicity in Data Streams with Wildcards. In Proceedings of the International Computer Science Symposium in Russia, CSR 2018, pages 90-105, 2018. URL: http://dx.doi.org/10.1007/978-3-319-90530-3_9.
  20. Funda Ergün, Hossein Jowhari, and Mert Sağlam. Periodicity in Streams. In Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and of the International Workshop on Randomization and Computation, APPROX-RANDOM 2010, pages 545-559, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15369-3_41.
  21. Nathan J. Fine and Herbert S. Wilf. Uniqueness Theorems for Periodic Functions. Proceedings of the American Mathematical Society, 16:109-114, 1965. Google Scholar
  22. Tomás̆ Flouri, Costas S. Iliopoulos, Tomasz Kociumaka, Solon P. Pissis, Simon J. Puglisi, William F. Smyth, and Wojciech Tyczyński. Enhanced string covering. Theoretical Computer Science, 506:102-114, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2013.08.013.
  23. Qing Guo, Hui Zhang, and Costas S. Iliopoulos. Computing the λ-Seeds of a String. In Proceedings of Algorithmic Aspects in Information and Management, AAIM 2006, pages 303-313, 2006. URL: http://dx.doi.org/10.1007/11775096_28.
  24. Qing Guo, Hui Zhang, and Costas S. Iliopoulos. Computing the λ-covers of a string. Information Sciences, 177(19):3957-3967, 2007. URL: http://dx.doi.org/10.1016/j.ins.2007.02.020.
  25. Ondřej Guth. On approximate enhanced covers under Hamming distance. Discrete Applied Mathematics, 2019. URL: http://dx.doi.org/10.1016/j.dam.2019.01.015.
  26. Costas S. Iliopoulos, Manal Mohamed, and William F. Smyth. New complexity results for the k-covers problem. Information Sciences, 181(12):2571-2575, 2011. URL: http://dx.doi.org/10.1016/j.ins.2011.02.009.
  27. Costas S. Iliopoulos, Dennis W. G. Moore, and Kunsoo Park. Covering a string. Algorithmica, 16(3):288-297, September 1996. URL: http://dx.doi.org/10.1007/BF01955677.
  28. Richard M. Karp and Michael O. Rabin. Efficient randomized pattern-matching algorithms. IBM Journal of Research and Development, 31(2):249-260, 1987. Google Scholar
  29. Donald E. Knuth, James H. Morris, and Vaughan R. Pratt. Fast pattern matching in strings. SIAM Journal of Computing, 6:322-350, 1977. URL: http://dx.doi.org/10.1137/0206024.
  30. Tomasz Kociumaka, Marcin Kubica, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. A Linear Time Algorithm for Seeds Computation. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pages 1095-1112, 2012. URL: http://dl.acm.org/citation.cfm?id=2095116.2095202.
  31. Tomasz Kociumaka, Marcin Kubica, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. A Linear Time Algorithm for Seeds Computation. CoRR, abs/1107.2422v2, 2019. URL: http://arxiv.org/abs/1107.2422v2.
  32. Tomasz Kociumaka, Solon P. Pissis, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. Fast Algorithm for Partial Covers in Words. Algorithmica, 73(1):217-233, September 2015. URL: http://dx.doi.org/10.1007/s00453-014-9915-3.
  33. Tomasz Kociumaka, Solon P. Pissis, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. Efficient algorithms for shortest partial seeds in words. Theoretical Computer Science, 710:139-147, 2018. Advances in Algorithms &Combinatorics on Strings (Honoring 60th birthday for Prof. Costas S. Iliopoulos). URL: http://dx.doi.org/10.1016/j.tcs.2016.11.035.
  34. Yin Li and William F. Smyth. Computing the Cover Array in Linear Time. Algorithmica, 32(1):95-106, January 2002. URL: http://dx.doi.org/10.1007/s00453-001-0062-2.
  35. Neerja Mhaskar and William F. Smyth. Frequency Covers for Strings. Fundamenta Informaticae, 163(3):275-289, 2018. URL: http://dx.doi.org/10.3233/FI-2018-1744.
  36. Neerja Mhaskar and William F. Smyth. String covering with optimal covers. Journal of Discrete Algorithms, 51:26-38, 2018. URL: http://dx.doi.org/10.1016/j.jda.2018.09.003.
  37. Dennis Moore and William F. Smyth. A Correction to "An Optimal Algorithm to Compute All the Covers of a String". Information Processing Letters, 54(2):101-103, April 1995. URL: http://dx.doi.org/10.1016/0020-0190(94)00235-Q.
  38. Dennis W. G. Moore and William F. Smyth. An Optimal Algorithm to Compute all the Covers of a String. Information Processing Letters, 50:239-246, 1994. URL: http://dx.doi.org/10.1016/0020-0190(94)00045-X.
  39. Alexandru Popa and Andrei Tanasescu. Hardness and algorithmic results for the approximate cover problem. CoRR, abs/1806.08135, 2018. URL: http://arxiv.org/abs/1806.08135.
  40. Benny Porat and Ely Porat. Exact And Approximate Pattern Matching In The Streaming Model. In Proceedings of the Annual Symposium on Foundations of Computer Science, FOCS 2009, pages 315-323, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.11.
  41. Hui Zhang, Qing Guo, and Costas S. Iliopoulos. Algorithms for Computing the λ-regularities in Strings. Fundamenta Informaticae, 84(1):33-49, 2008. URL: http://content.iospress.com/articles/fundamenta-informaticae/fi84-1-04.
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