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On the String Consensus Problem and the Manhattan Sequence Consensus Problem

  • Tomasz Kociumaka
  • Jakub W. Pachocki
  • Jakub Radoszewski
  • Wojciech Rytter
  • Tomasz Waleń
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8799)

Abstract

In the Manhattan Sequence Consensus problem (MSC problem) we are given k integer sequences, each of length ℓ, and we are to find an integer sequence x of length ℓ (called a consensus sequence), such that the maximum Manhattan distance of x from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient \(\mathcal{O}(\ell)\)-time algorithm solving MSC for k ≤ 5 sequences. Practicality of our algorithms has been verified experimentally. It improves upon the quadratic algorithm by Amir et al. (SPIRE 2012) for string consensus problem for k = 5 binary strings. Similarly as in Amir’s algorithm we use a column-based framework. We replace the implied general integer linear programming by its easy special cases, due to combinatorial properties of the MSC for k ≤ 5. We also show that for a general parameter k any instance can be reduced in linear time to a kernel of size k!, so the problem is fixed-parameter tractable. Nevertheless, for k ≥ 4 this is still too much for any naive solution to be feasible in practice.

Keywords

Integer Linear Programming Manhattan Distance Consensus Problem Integer Sequence Basic Interval 
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., Paryenty, H., Roditty, L.: Configurations and minority in the string consensus problem. In: Calderón-Benavides, L., González-Caro, C., Chávez, E., Ziviani, N. (eds.) SPIRE 2012. LNCS, vol. 7608, pp. 42–53. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Amir, A., Paryenty, H., Roditty, L.: On the hardness of the consensus string problem. Inf. Process. Lett. 113(10-11), 371–374 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andoni, A., Indyk, P., Patrascu, M.: On the optimality of the dimensionality reduction method. In: FOCS, pp. 449–458. IEEE Computer Society (2006)Google Scholar
  4. 4.
    Badoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Reif, J.H. (ed.) STOC, pp. 250–257. ACM (2002)Google Scholar
  5. 5.
    Boucher, C., Brown, D.G., Durocher, S.: On the structure of small motif recognition instances. In: Amir, A., Turpin, A., Moffat, A. (eds.) SPIRE 2008. LNCS, vol. 5280, pp. 269–281. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Cohen, G.D., Honkala, I.S., Litsyn, S., Solé, P.: Long packing and covering codes. IEEE Transactions on Information Theory 43(5), 1617–1619 (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fischer, K., Gärtner, B., Kutz, M.: Fast smallest-enclosing-ball computation in high dimensions. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 630–641. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Frances, M., Litman, A.: On covering problems of codes. Theory Comput. Syst. 30(2), 113–119 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gärtner, B., Schönherr, S.: An efficient, exact, and generic quadratic programming solver for geometric optimization. In: Symposium on Computational Geometry, pp. 110–118 (2000)Google Scholar
  11. 11.
    Graham, R.L., Sloane, N.J.A.: On the covering radius of codes. IEEE Transactions on Information Theory 31(3), 385–401 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for closest string and related problems. Algorithmica 37(1), 25–42 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kannan, R.: Minkowski’s convex body theorem and integer programming. Mathematics of Operations Reasearch 12, 415–440 (1987)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kociumaka, T., Pachocki, J.W., Radoszewski, J., Rytter, W., Waleń, T.: On the string consensus problem and the Manhattan sequence consensus problem (full version). CoRR, abs/1407.6144 (2014)Google Scholar
  15. 15.
    Kumar, P., Mitchell, J.S.B., Yildirim, E.A.: Computing core-sets and approximate smallest enclosing hyperspheres in high dimensions. In: 5th Workshop on Algorithm Engineering and Experiments (2003)Google Scholar
  16. 16.
    Lanctôt, J.K., Li, M., Ma, B., Wang, S., Zhang, L.: Distinguishing string selection problems. In: Tarjan, R.E., Warnow, T. (eds.) SODA, pp. 633–642. ACM/SIAM (1999)Google Scholar
  17. 17.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8, 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lokshtanov, D.: New Methods in Parameterized Algorithms and Complexity. PhD thesis, University of Bergen (2009)Google Scholar
  19. 19.
    Ma, B., Sun, X.: More efficient algorithms for closest string and substring problems. SIAM J. Comput. 39(4), 1432–1443 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mazumdar, A., Polyanskiy, Y., Saha, B.: On Chebyshev radius of a set in Hamming space and the closest string problem. In: ISIT, pp. 1401–1405. IEEE (2013)Google Scholar
  21. 21.
    Ritter, J.: An efficient bounding sphere. In: Glassner, A.S. (ed.) Gems. Academic Press, Boston (1990)Google Scholar
  22. 22.
    Sylvester, J.J.: A question in the geometry of situation. Quarterly Journal of Pure and Applied Mathematics 1, 79 (1857)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Tomasz Kociumaka
    • 1
  • Jakub W. Pachocki
    • 2
  • Jakub Radoszewski
    • 1
  • Wojciech Rytter
    • 1
    • 3
  • Tomasz Waleń
    • 1
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Carnegie Mellon UniversityUSA
  3. 3.Faculty of Mathematics and Computer ScienceCopernicus UniversityToruńPoland

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