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Approximate Matching in Weighted Sequences

  • Amihood Amir
  • Costas Iliopoulos
  • Oren Kapah
  • Ely Porat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4009)

Abstract

Weighted sequences have been recently introduced as a tool to handle a set of sequences that are not identical but have many local similarities. The weighted sequence is a “statistical image” of this set, where the probability of every symbol’s occurrence at every text location is given.

We address the problem of approximately matching a pattern in such a weighted sequence. The pattern is a given string and we seek all locations in the set where the pattern occurs with a high enough probability. We define the notion of Hamming distance and edit distance in weighted sequences and give efficient algorithms for computing them. We compute two versions of the Hamming distance in time \(O(n \sqrt{m\log m})\), where n is the length of the weighted text and m is the pattern length. The edit distance is computed in time O(nm) and O(nm 2), depending on the edit distance definition used. Unfortunately, due to space considerations, the edit distance details are left to the journal version.

We also define the notion of weighted matching in infinite alphabets and show that exact weighted matching can be computed in time O(slog2 s), where s is the number of text symbols having non-zero probability. The weighted Hamming distance over infinite alphabets can be computed in time \(\min(O(kn\sqrt{s}+s^{3/2}\log^2s), O(s^{4/3}m^{1/3}\log s))\).

Keywords

Edit Distance Text Element Zero Probability Text Location Weighted Match 
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.

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References

  1. 1.
    Amir, A., Farach, M.: Efficient 2-dimensional approximate matching of half-rectangular figures. Information and Computation 118(1), 1–11 (1995)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Amir, A., Lewenstein, N., Lewenstein, M.: Pattern matching in hypertext. J. of Algorithms 35, 82–99 (2000)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Christodoulakis, M., Iliopoulos, C.S., Mouchard, L., Tsichlas, K.: Pattern matchnig on weighted sequences. In: Proceedings of the Algorithms and Computational Methods for Biochemical and Evolutionary Networks (CompBioNets). KCL Publications (2004)Google Scholar
  4. 4.
    Cole, R., Hariharan, R.: Tree pattern matching and subset matching in randomized o(n log3 m) time. In: Proc. 29th ACM STOC, pp. 66–75 (1997)Google Scholar
  5. 5.
    Cole, R., Hariharan, R.: Verifying candidate matches in sparse and wildcard matching. In: Proc. 34st Annual Symposium on the Theory of Computing (STOC), pp. 592–601 (2002)Google Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press and McGraw-Hill (1992)Google Scholar
  7. 7.
    Dubiner, M., Galil, Z., Magen, E.: Faster tree pattern matching. J. ACM 41(2), 205–213 (1994)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Iliopoulos, C.S., Mouchard, L., Perdikuri, K., Tsakalidis, A.: Computing the repetitions in a weighted sequence. In: Proceeding of the Prague Stringology Conference, pp. 91–98 (2003)Google Scholar
  10. 10.
    Iliopoulos, C.S., Perdikuri, K., Theodoridis, E., Tsakalidis, A., Tsichlas, K.: Motif extraction from weighted sequences. In: Apostolico, A., Melucci, M. (eds.) SPIRE 2004. LNCS, vol. 3246, pp. 286–297. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Iliopoulos, C.S., Makris, C., Panagis, I., Perdikuri, K., Theodoridis, E., Tsakalidis, A.: Computing the repetiotions in a weighted sequence using weighted suffix trees. In: European Conference on Computational Biology (ECCB), pp. 539–540 (2003)Google Scholar
  12. 12.
    Kosaraju, S.R.: Efficient tree pattern matching. In: Proc. 30th IEEE FOCS, pp. 178–183 (1989)Google Scholar
  13. 13.
    Levenshtein, V.I.: Binary codes capable of correcting, deletions, insertions and reversals. Soviet Phys. Dokl. 10, 707–710 (1966)MathSciNetGoogle Scholar
  14. 14.
    Thompson, J.D., Higgins, D.G., Gibson, T.J.: Clustal w: Improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic Acids Research 22, 4673–4680 (1994)CrossRefGoogle Scholar
  15. 15.
    Venter, J.C.: Celera Genomics Corporation. The sequence of the human genome. Science 291, 1304–1351 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Amihood Amir
    • 1
  • Costas Iliopoulos
    • 2
  • Oren Kapah
    • 3
  • Ely Porat
    • 3
  1. 1.Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel and College of Computing, Georgia TechAtlantaUSA
  2. 2.Department of Computer ScienceKing’s College LondonStrand, LondonUnited Kingdom
  3. 3.Department of Computer ScienceBar-Ilan UniversityRamat-GanIsrael

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