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

  • Yohei UekiEmail author
  • Diptarama
  • Masatoshi Kurihara
  • Yoshiaki Matsuoka
  • Kazuyuki Narisawa
  • Ryo Yoshinaka
  • Hideo Bannai
  • Shunsuke Inenaga
  • Ayumi Shinohara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Longest common subsequence Dynamic programming Order-isomorphism Order-preserving matching 

Notes

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.

References

  1. 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 CrossRefGoogle Scholar
  2. 2.
    Benson, G., Levy, A., Maimoni, S., Noifeld, D., Shalom, B.: LCSk: a refined similarity measure. Theor. Comput. Sci. 638, 11–26 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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 CrossRefGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cole, R., Hariharan, R.: Dynamic LCA queries on trees. SIAM J. Comput. 34(4), 894–923 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deorowicz, S., Grabowski, S.: Efficient algorithms for the longest common subsequence in \(k\)-length substrings. Inf. Process. Lett. 114(11), 634–638 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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 CrossRefGoogle Scholar
  9. 9.
    Fischer, J., Heun, V.: Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput. 40(2), 465–492 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gusfield, D.: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hasan, M.M., Islam, A., Rahman, M.S., Rahman, M.: Order preserving pattern matching revisited. Pattern Recogn. Lett. 55, 15–21 (2015)CrossRefGoogle Scholar
  12. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pavetić, F., Žužić, G., Šikić, M.: \(LCSk\)++: practical similarity metric for long strings (2014). CoRR 1407.2407
  16. 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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yohei Ueki
    • 1
    Email author
  • Diptarama
    • 1
  • Masatoshi Kurihara
    • 1
  • Yoshiaki Matsuoka
    • 2
  • Kazuyuki Narisawa
    • 1
  • Ryo Yoshinaka
    • 1
  • Hideo Bannai
    • 2
  • Shunsuke Inenaga
    • 2
  • Ayumi Shinohara
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Department of InformaticsKyushu UniversityFukuokaJapan

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