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Longest Common Subsequence in at Least k Length Order-Isomorphic Substrings

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SOFSEM 2017: Theory and Practice of Computer Science (SOFSEM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10139))

Abstract

We consider the longest common subsequence (LCS) problem with the restriction that the common subsequence is required to consist of at least k length substrings. First, we show an O(mn) time algorithm for the problem which gives a better worst-case running time than existing algorithms, where m and n are lengths of the input strings. Furthermore, we mainly consider the LCS in at least k length order-isomorphic substrings problem. We show that the problem can also be solved in O(mn) worst-case time by an easy-to-implement algorithm.

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Notes

  1. 1.

    Since the problem is motivated by the order-preserving matching problem, we abbreviate it to the op-LCS\(_{k^{+}}\) problem.

  2. 2.

    The formal definition given by Pavetić et al. [15] contains a minor error, i.e., they do not require that each chunk is identical, while Benson et al. [2] and we do (confirmed by F. Pavetić, personal communication, October 2016).

  3. 3.

    Hasan et al. [11] assume that characters in a string are distinct. If the assumption is false, use Lemma 4 in [4] in order to verify the order-isomorphism, that is, modify line 10 of Algorithm 4 in [11] and line 7 and 12 in Algorithm 1. Note that \( Prev \) and \( Next \) are denoted as \( LMax \) and \( LMin \) in [4], respectively, with slight differences.

References

  1. Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000). doi:10.1007/10719839_9

    Chapter  Google Scholar 

  2. Benson, G., Levy, A., Maimoni, S., Noifeld, D., Shalom, B.: LCSk: a refined similarity measure. Theor. Comput. Sci. 638, 11–26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouvel, M., Rossin, D., Vialette, S.: Longest common separable pattern among permutations. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 316–327. Springer, Heidelberg (2007). doi:10.1007/978-3-540-73437-6_32

    Chapter  Google Scholar 

  4. Cho, S., Na, J.C., Park, K., Sim, J.S.: A fast algorithm for order-preserving pattern matching. Inf. Process. Lett. 115(2), 397–402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cole, R., Hariharan, R.: Dynamic LCA queries on trees. SIAM J. Comput. 34(4), 894–923 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crochemore, M., Iliopoulos, C.S., Kociumaka, T., Kubica, M., Langiu, A., Pissis, S.P., Radoszewski, J., Rytter, W., Waleń, T.: Order-preserving indexing. Theor. Comput. Sci. 638, 122–135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deorowicz, S., Grabowski, S.: Efficient algorithms for the longest common subsequence in \(k\)-length substrings. Inf. Process. Lett. 114(11), 634–638 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fischer, J.: Inducing the LCP-array. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 374–385. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22300-6_32

    Chapter  Google Scholar 

  9. Fischer, J., Heun, V.: Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput. 40(2), 465–492 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gusfield, D.: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, New York (1997)

    Book  MATH  Google Scholar 

  11. Hasan, M.M., Islam, A., Rahman, M.S., Rahman, M.: Order preserving pattern matching revisited. Pattern Recogn. Lett. 55, 15–21 (2015)

    Article  Google Scholar 

  12. Khan, R., Ahmad, M., Zakarya, M.: Longest common subsequence based algorithm for measuring similarity between time series: a new approach. World Appl. Sci. J. 24(9), 1192–1198 (2013)

    Google Scholar 

  13. Kim, J., Eades, P., Fleischer, R., Hong, S.H., Iliopoulos, C.S., Park, K., Puglisi, S.J., Tokuyama, T.: Order-preserving matching. Theor. Comput. Sci. 525(13), 68–79 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kubica, M., Kulczynski, T., Radoszewski, J., Rytter, W., Walen, T.: A linear time algorithm for consecutive permutation pattern matching. Inf. Process. Lett. 113(12), 430–433 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pavetić, F., Žužić, G., Šikić, M.: \(LCSk\)++: practical similarity metric for long strings (2014). CoRR 1407.2407

  16. Sović, I., Šikić, M., Wilm, A., Fenlon, S.N., Chen, S., Nagarajan, N.: Fast and sensitive mapping of nanopore sequencing reads with GraphMap. Nat. Commun. 7, Article No. 11307 (2016). doi:10.1038/ncomms11307

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Acknowledgements

This work was funded by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan), Tohoku University Division for Interdisciplinary Advance Research and Education, and JSPS KAKENHI Grant Numbers JP24106010, JP16H02783, JP26280003.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Yohei Ueki,  Diptarama, Masatoshi Kurihara, Kazuyuki Narisawa, Ryo Yoshinaka & Ayumi Shinohara

  2. Department of Informatics, Kyushu University, Fukuoka, Japan

    Yoshiaki Matsuoka, Hideo Bannai & Shunsuke Inenaga

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  1. Yohei Ueki
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  2. Diptarama
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Correspondence to Yohei Ueki .

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Editors and Affiliations

  1. TU Dortmund , Dortmund, Germany

    Bernhard Steffen

  2. TU Dresden , Dresden, Germany

    Christel Baier

  3. Eindhoven University of Technology , Eindhoven, The Netherlands

    Mark van den Brand

  4. Alpen Adria University Klagenfurt , Klagenfurt, Austria

    Johann Eder

  5. Lero - Irish Software Research Center , Limerick, Ireland

    Mike Hinchey

  6. Lero - Irish Software Research Center , Limerick, Ireland

    Tiziana Margaria

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Ueki, Y. et al. (2017). Longest Common Subsequence in at Least k Length Order-Isomorphic Substrings. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_28

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  • DOI: https://doi.org/10.1007/978-3-319-51963-0_28

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  • Print ISBN: 978-3-319-51962-3

  • Online ISBN: 978-3-319-51963-0

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