Computing Maximum-Scoring Segments in Almost Linear Time

  • Fredrik Bengtsson
  • jingsen Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


Given a sequence, the problem studied in this paper is to find a set of k disjoint continuous subsequences such that the total sum of all elements in the set is maximized. This problem arises naturally in the analysis of DNA sequences. The previous best known algorithm requires Θ(n log n) time in the worst case. For a given sequence of length n, we present an almost linear-time algorithm for this problem. Our algorithm uses a disjoint-set data structure and requires O((n, n)) time in the worst case, where α(n, n) is the inverse Ackermann function.


Linear Time Extra Information Positive Element Optimal Cover Working Sequence 
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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  1. 1.
    Csűrös, M.: Maximum-scoring segment sets. IEEE/ACM Transactions on Computational Biology and Bioinformatics 1(4), 139–150 (2004)CrossRefGoogle Scholar
  2. 2.
    Fariselli, P., Finelli, M., Marchignoli, D., Martelli, P., Rossi, I., Casadio, R.: Maxsubseq: An algorithm for segment-length optimization. The case study of the transmembrane spanning segments. Bioinformatics 19, 500–505 (2003)CrossRefGoogle Scholar
  3. 3.
    Huang, X.: An algorithm for identifying regions of a DNA sequence that satisfy a content requirement. Computer Applications in the Biosciences 10, 219–225 (1994)Google Scholar
  4. 4.
    Auger, I.E., Lawrence, C.E.: Algorithms for the optimal identification of segment neighbourhoods. Bulletin of Mathematical Biology 51(1), 39–54 (1989)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bement, T.R., Waterman, M.S.: Locating maximum variance segments in sequential data. Mathematical Geology 9(1), 55–61 (1977)CrossRefGoogle Scholar
  6. 6.
    Bentley, J.L.: Programming pearls: Algorithm design techniques. Communications of the ACM 27, 865–871 (1984)CrossRefGoogle Scholar
  7. 7.
    Bentley, J.L.: Programming pearls: Perspective on performance. Communications of the ACM 27, 1087–1092 (1984)CrossRefGoogle Scholar
  8. 8.
    Smith, D.: Applications of a strategy for designing divide-and-conquer algorithms. Science of Computer Programming 8, 213–229 (1987)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bae, S.E., Takaoka, T.: Algorithms for the problem of k maximum sums and a VLSI algorithm for the k maximum subarrays problem. In: Proceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks, pp. 247–253 (2004)Google Scholar
  10. 10.
    Bae, S.E., Takaoka, T.: Improved algorithms for the k-maximum subarray problem for small k. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 621–631. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Bengtsson, F., Chen, J.: Efficient algorithms for k maximum sums. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 137–148. Springer, Heidelberg (2004); Revised version to appear in Algorithmica. CrossRefGoogle Scholar
  12. 12.
    Lin, T.C., Lee, D.T.: Randomized algorithm for the sum selection problem. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 515–523. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Bergkvist, A., Damaschke, P.: Fast algorithms for finding disjoint subsequences with extremal densities. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 714–723. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Chung, K.M., Lu, H.I.: An optimal algorithm for the maximum-density segment problem. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 136–147. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Ruzzo, W.L., Tompa, M.: A linear time algorithm for finding all maximal scoring subsequences. In: Proceedings of the 7th Annual International Conference on Intelligent Systems for Molecular Biology, pp. 234–241 (1999)Google Scholar
  16. 16.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press, Cambridge (1990)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Fredrik Bengtsson
    • 1
  • jingsen Chen
    • 1
  1. 1.Department of Computer Science and Electrical EngineeringLuleå University of TechnologyLuleåSweden

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