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Measurement Errors Make the Partial Digest Problem NP-Hard

  • Mark Cieliebak
  • Stephan Eidenbenz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

The Partial Digest problem asks for the coordinates of m points on a line such that the pairwise distances of the points form a given multiset of \(\left({m \atop 2}\right)\) distances. Partial Digest is a well-studied problem with important applications in physical mapping of DNA molecules. Its computational complexity status is open. Input data for Partial Digest from real-life experiments are always prone to error, which suggests to study variations of Partial Digest that take this fact into account. In this paper, we study the computational complexity of the variation of Partial Digest in which each distance is known only up to some error, due to experimental inaccuracies. The error can be specified either by some additive offset or by a multiplicative factor. We show that both types of error make the Partial Digest problem strongly NP-complete, by giving reductions from 3-Partition. In the case of relative errors, we show that the problem is hard to solve even for constant relative error.

Keywords

Pairwise Distance Additive Error Adjacent Point Partial Digestion Multiplicative Error 
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.
    Allison, L., Yee, C.N.: Restriction site mapping is in separation theory. Computer Applications in the Biosciences 4(1), 97–101 (1988)Google Scholar
  2. 2.
    Błażewicz, J., Formanowicz, P., Kasprzak, M., Jaroszewski, M., Markiewicz, W.T.: Construction of DNA restriction maps based on a simplified experiment. Bioinformatics 17(5), 398–404 (2001)CrossRefGoogle Scholar
  3. 3.
    Cieliebak, M., Eidenbenz, S., Penna, P.: Noisy data make the partial digest problem NP-hard. In: Benson, G., Page, R.D.M. (eds.) WABI 2003. LNCS (LNBI), vol. 2812, pp. 111–123. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Dakić, T.: On the Turnpike Problem. PhD thesis, Simon Fraser University (2000)Google Scholar
  5. 5.
    Dix, T.I., Kieronska, D.H.: Errors between sites in restriction site mapping. Computer Applications in the Biosciences 4(1), 117–123 (1988)Google Scholar
  6. 6.
    Fütterer, J.: Personal communication, ETH Zurich, Institute of Plant Sciences (2002)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Inglehart, J., Nelson, P.C.: On the limitations of automated restriction mapping. Computer Applications in the Biosciences 10(3), 249–261 (1994)Google Scholar
  9. 9.
    Lemke, P., Skiena, S.S., Smith, W.: Reconstructing sets from interpoint distances. Technical Report TR2002–37, DIMACS (2002)Google Scholar
  10. 10.
    Lemke, P., Werman, M.: On the complexity of inverting the autocorrelation function of a finite integer sequence, and the problem of locating n points on a line, given the \(\left({n \atop 2}\right)\) unlabelled distances between them. Preprint 453, Institute for Mathematics and its Application IMA (1988)Google Scholar
  11. 11.
    Pandurangan, G., Ramesh, H.: The restriction mapping problem revisited. Journal of Computer and System Sciences 65(3), 526–544 (2002); Special issue on Computational BiologyzbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pevzner, P.A.: Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  13. 13.
    Rosenblatt, J., Seymour, P.: The structure of homometric sets. SIAM Journal of Algorithms and Discrete Mathematics 3(3), 343–350 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Searls, D.B.: Formal grammars for intermolecular structure. In: Proc. of the 1st International Symposium on Intelligence in Neural and Biological Systems (INBS 1995), pp. 30–37 (1995)Google Scholar
  15. 15.
    Setubal, J., Meidanis, J.: Introduction to Computational Molecular Biology. PWS Boston (1997)Google Scholar
  16. 16.
    Skiena, S.S., Smith, W., Lemke, P.: Reconstructing sets from interpoint distances. In: Proc. of the 6th ACM Symposium on Computational Geometry (SoCG 1990), pp. 332–339 (1990)Google Scholar
  17. 17.
    Skiena, S.S., Sundaram, G.: A partial digest approach to restriction site mapping. Bulletin of Mathematical Biology 56, 275–294 (1994)zbMATHGoogle Scholar
  18. 18.
    Tuffery, P., Dessen, P., Mugnier, C., Hazout, S.: Restriction map construction using a ’complete sentence compatibility’ algorithm. Computer Applications in the Biosciences 4(1), 103–110 (1988)Google Scholar
  19. 19.
    Waterman, M.S.: Introduction to Computational Biology. Chapman & Hall, Boca Raton (1995)zbMATHGoogle Scholar
  20. 20.
    Zhang, Z.: An exponential example for a partial digest mapping algorithm. Journal of Computational Biology 1(3), 235–239 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mark Cieliebak
    • 1
  • Stephan Eidenbenz
    • 2
  1. 1.Institute of Theoretical Computer ScienceETH Zurich 
  2. 2.Los Alamos National Laboratory 

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