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On the Chain Pair Simplification Problem

  • Chenglin Fan
  • Omrit Filtser
  • Matthew J. Katz
  • Tim Wylie
  • Binhai ZhuEmail author
Conference paper
  • 1.2k Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

The problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones in 3D has led Bereg et al. [4] to pose the Chain Pair Simplification problem (CPS). In this problem, given two polygonal chains A and B of lengths m and n, respectively, one needs to simplify them simultaneously, such that each of the resulting simplified chains, \(A'\) and \(B'\), is of length at most k and the discrete Fréchet distance between \(A'\) and \(B'\) is at most \(\delta \), where k and \(\delta \) are given parameters. In this paper we study the complexity of CPS under the discrete Fréchet distance (CPS-3F), i.e., where the quality of the simplifications is also measured by the discrete Fréchet distance. Since CPS-3F was posed in 2008, its complexity has remained open. In this paper, we prove that CPS-3F is actually polynomially solvable, by presenting an \(O(m^2n^2\min \{m,n\})\) time algorithm for the corresponding minimization problem. On the other hand, we prove that if the vertices of the chains have integral weights then the problem is weakly NP-complete.

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References

  1. 1.
    Agarwal, P.K., Avraham, R.B., Kaplan, H., Sharir, M.: Computing the discrete Fréchet distance in subquadratic time. SIAM J. Comput. 43(2), 429–449 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Internat. J. Comput. Geometry Appl. 5, 75–91 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avraham, R.B., Filtser, O., Kaplan, H., Katz, M.J., Sharir, M.: The discrete Fréchet distance with shortcuts via approximate distance counting and selection. In: Proc. 30th Annual ACM Sympos. on Computational Geometry, SOCG 2014, p. 377 (2014)Google Scholar
  4. 4.
    Bereg, S., Jiang, M., Wang, W., Yang, B., Zhu, B.: Simplifying 3d polygonal chains under the discrete Fréchet distance. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 630–641. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  5. 5.
    Buchin, K., Buchin, M., Meulemans, W., Mulzer, W.: Four soviets walk the dog – with an application to alt’s conjecture. In: Proc. 25th Annual ACM-SIAM Sympos. on Discrete Algorithms, SODA 2014, pp. 1399–1413 (2014)Google Scholar
  6. 6.
    Driemel, A., Har-Peled, S.: Jaywalking your dog: Computing the Fréchet distance with shortcuts. SIAM J. Comput. 42(5), 1830–1866 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eiter, T., Mannila, H.: Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Information Systems Dept., Technical University of Vienna (1994)Google Scholar
  8. 8.
    Fréchet, M.: Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Matematico di Palermo 22(1), 1–72 (1906)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jiang, M., Xu, Y., Zhu, B.: Protein structure-structure alignment with discrete Fréchet distance. J. Bioinformatics and Computational Biology 6(1), 51–64 (2008)CrossRefGoogle Scholar
  10. 10.
    Wylie, T., Luo, J., Zhu, B.: A Practical solution for aligning and simplifying pairs of protein backbones under the discrete Fréchet distance. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part III. LNCS, vol. 6784, pp. 74–83. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  11. 11.
    Wylie, T., Zhu, B.: Protein chain pair simplification under the discrete Fréchet distance. IEEE/ACM Trans. Comput. Biology Bioinform. 10(6), 1372–1383 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Chenglin Fan
    • 1
  • Omrit Filtser
    • 2
  • Matthew J. Katz
    • 2
  • Tim Wylie
    • 3
  • Binhai Zhu
    • 1
    Email author
  1. 1.Montana State UniversityBozemanUSA
  2. 2.Ben-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.The University of Texas-Pan AmericanEdinburgUSA

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