Advertisement

Approximate Period Detection and Correction

  • Amihood Amir
  • Avivit Levy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7608)

Abstract

Periodicity has been historically well studied and has numerous applications. In nature, however, few cyclic phenomena have an exact period.

This paper surveys some recent results in approximate periodicity: concept definition, discovery or recovery, techniques and efficient algorithms. We will also show some interesting connections between error correction codes and periodicity.

We will try to pinpoint the issues involved, the context in the literature, and possible future research directions.

Keywords

Edit Distance Input String Substitution Error Cyclic Phenomenon Approximate Period 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amir, A., Aumann, Y., Benson, G., Levy, A., Lipsky, O., Porat, E., Skiena, S., Vishne, U.: Pattern matching with address errors: rearrangement distances. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1221–1229 (2006)Google Scholar
  2. 2.
    Amir, A., Aumann, Y., Landau, G., Lewenstein, M., Lewenstein, N.: Pattern matching with swaps. Journal of Algorithms 37, 247–266 (2000); Preliminary version appeared at FOCS 1997MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Amir, A., Benson, G.: Two-dimensional periodicity and its application. SIAM J. Comp. 27(1), 90–106 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Amir, A., Eisenberg, E., Levy, A.: Approximate Periodicity. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 25–36. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Amir, A., Eisenberg, E., Levy, A., Lewenstein, N.: Closest Periodic Vectors in L p Spaces. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 714–723. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Amir, A., Eisenberg, E., Levy, A., Porat, E., Shapira, N.: Cycle Detection and Correction. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 43–54. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Amir, A., Hartman, T., Kapah, O., Levy, A., Porat, E.: On the Cost of Interchange Rearrangement in Strings. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 99–110. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Apostolico, A., Giancarlo, R.: Periodicity and repetitions in parameterized strings. Discrete Appl. Math. 156(9), 1389–1398 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Apostolico, A., Preparata, F.P.: Data structures and algorithms for the string statistics problem. Algorithmica 15(5), 481–494 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bafna, V., Pevzner, P.A.: Sorting by transpositions. SIAM J. on Discrete Mathematics 11, 221–240 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Benson, G.: Sequence alignment with tandem duplication. J. Computational Biology 4(3), 351–368 (1997)CrossRefGoogle Scholar
  12. 12.
    Berman, P., Hannenhalli, S.: Fast Sorting by Reversal. In: Hirschberg, D.S., Meyers, G. (eds.) CPM 1996. LNCS, vol. 1075, pp. 168–185. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  13. 13.
    Cayley, A.: Note on the theory of permutations. Philosophical Magazine (34), 527–529 (1849)Google Scholar
  14. 14.
    Christie, D.A.: Sorting by block-interchanges. Information Processing Letters 60, 165–169 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Crochemore, M.: An optimal algorithm for computing the repetitions in a word. Information Processing Letters 12(5), 244–250 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fischer, M.J., Paterson, M.S.: String matching and other products. In: Karp, R.M. (ed.) Complexity of Computation. SIAM-AMS Proceedings, vol. 7, pp. 113–125 (1974)Google Scholar
  17. 17.
    Galil, Z., Giancarlo, R.: Improved string matching with k mismatches. SIGACT News 17(4), 52–54 (1986)CrossRefGoogle Scholar
  18. 18.
    Galil, Z., Park, K.: Alphabet-independent two-dimensional witness computation. SIAM J. Comp. 25(5), 907–935 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gfeller, B.: Finding longest approximate periodic patterns. In: WADS (2011)Google Scholar
  20. 20.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestor. Computer and System Science 13, 338–355 (1984)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kärkkäinen, J., Sanders, P.: Simple Linear Work Suffix Array Construction. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 943–955. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Kasai, T., Lee, G.H., Arimura, H., Arikawa, S., Park, K.: Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its Applications. In: Amir, A., Landau, G.M. (eds.) CPM 2001. LNCS, vol. 2089, pp. 181–192. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  23. 23.
    Kolpakov, R.M., Bana, G., Kucherov, G.: mreps: efficient and flexible detection of tandem repeats in DNA. Nucleic Acids Research 31(13), 3672–3678 (2003)CrossRefGoogle Scholar
  24. 24.
    Kolpakov, R.M., Kucherov, G.: Finding approximate repetitions under hamming distance. Theoretical Computer Science 1(303), 135–156 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Landau, G.M., Vishkin, U.: Efficient string matching with k mismatches. Theoretical Computer Science 43, 239–249 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lothaire, M.: Combinatorics on words. Addison-Wesley, Reading (1983)Google Scholar
  27. 27.
    Régnier, M., Rostami, L.: A Unifying Look at d-dimensional Periodicities and Space Coverings. In: Apostolico, A., Crochemore, M., Galil, Z., Manber, U. (eds.) CPM 1993. LNCS, vol. 684, pp. 215–227. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  28. 28.
    Sim, J.S., Park, K., Kim, S., Lee, J.: Finding approximate covers of strings. J. Korea Information Science Society 29(1), 16–21 (2001)Google Scholar
  29. 29.
    Sokol, D., Benson, G., Tojeira, J.: Tandem repeats over the edit distance. Bioinformatics 23(2), 30–35 (2007)CrossRefGoogle Scholar
  30. 30.
    Weiner, P.: Linear pattern matching algorithm. In: Proc. 14 IEEE Symposium on Switching and Automata Theory, pp. 1–11 (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Amihood Amir
    • 1
    • 2
  • Avivit Levy
    • 3
    • 4
  1. 1.Department of Computer ScienceBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of Computer ScienceJohns Hopkins UniversityBaltimoreUSA
  3. 3.Department of Software EngineeringShenkar CollegeRamat-GanIsrael
  4. 4.CRIHaifa UniversityMount CarmelIsrael

Personalised recommendations