Advertisement

Sublinear Time Width-Bounded Separators and Their Application to the Protein Side-Chain Packing Problem

  • Bin Fu
  • Zhixiang Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)

Abstract

Given d > 2 and a set of n grid points Q in \(\Re^d\), we design a randomized algorithm that finds a w-wide separator, which is determined by a hyper-plane, in \(O(n^{2\over d}\log n)\) sublinear time such that Q has at most \(({d\over d+1}+o(1))n\) points one either side of the hyper-plane, and at most \(c_dwn^{d-1\over d}\) points within \(\frac{w}{2}\) distance to the hyper-plane, where c d is a constant for fixed d. In particular, c 3 = 1.209. To our best knowledge, this is the first sublinear time algorithm for finding geometric separators. Our 3D separator is applied to derive an algorithm for the protein side-chain packing problem, which improves and simplifies the previous algorithm of Xu [26].

Keywords

Planar Graph Signed Distance Balance Partition Sublinear Time Geometric Separator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akutsu, T.: NP-hardness results for protein side-chain packing. In: Miyano, S., Takagi, T. (eds.) Genome Informatics, vol. 8, pp. 180–186 (1997)Google Scholar
  2. 2.
    Alon, N., Seymour, P., Thomas, R.: Planar Separator. SIAM J. Discr. Math. 7(2), 184–193 (1990)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Seymour, P., Thomas, R.: A separator theorem for graphs with an excluded minor and its applications. In: STOC 1990, pp. 293–299 (1990)Google Scholar
  4. 4.
    Canutescu, A.A., Shelenkov, A.A., Dunbrack Jr., R.L.: A graph-theory algorithm for rapid protein side-chain prediction. Protein science 12, 2001–2014 (2003)CrossRefGoogle Scholar
  5. 5.
    Chazelle, B., Kingsford, C., Singh, M.: A semidefinite programming approach to side-chain positioning with new rounding strategies. INFORMS Journal on Computing, 86–94 (2004)Google Scholar
  6. 6.
    Chen, Z., Fu, B., Tang, Y., Zhu, B.: A PTAS for a DISC covering problem using width-bounded separator. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 490–503. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Djidjev, H.N.: On the problem of partitioning planar graphs. SIAM journal on discrete mathematics 3(2), 229–240 (1982)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Djidjev, H.N., Venkatesan, S.M.: Reduced constants for simple cycle graph separation. Acta informatica 34, 231–234 (1997)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Eppstein, D., Miller, G.L., Teng, S.: A Deterministic Linear Time Algorithm for Geometric Separators and its Applications. In: SOCG 1993, pp. 99–108 (1993)Google Scholar
  10. 10.
    Fu, B., Wang, W.: A \(2^{O(n^{1-1/d}\log n)}\)-time algorithm for d-dimensional protein folding in the HP-model. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 630–644. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Fu, B.: Theory and application of width bounded geometric separator. In: ECCC 2005, TR05-13 (2005) (Full draft); In: STACS 2006, pp. 277–288 (Extended abstract)Google Scholar
  12. 12.
    Gazit, H.: An improved algorithm for separating a planar graph, USC (manuscript, 1986)Google Scholar
  13. 13.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separation theorem for graphs of bounded genus. Journal of algorithm (5), 391–407 (1984)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jadhar, S., Mukhopadhyay, A.: Computing a center of a finite planar set of points in linear time. In: SOCG 1993, pp. 83–90 (1993)Google Scholar
  15. 15.
    Lichtenstein, D.: Planar formula and their uses. SIAM journal on computing 11(2), 329–343 (1982)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lipton, R.J., Tarjan, R.: A separator theorem for planar graph. SIAM Journal on Applied Mathematics 36, 177–189 (1979)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Miller, G.L., Teng, S.-H., Vavasis, S.A.: An unified geometric approach to graph separators. In: FOCS 1991, pp. 538–547 (1991)Google Scholar
  18. 18.
    Miller, G.L., Thurston, W.: Separators in two and three dimensions. In: STOC 1990, pp. 300–309 (1990)Google Scholar
  19. 19.
    Miller, G.L., Vavasis, S.A.: Density graphs and separators. In: SODA 1991, pp. 331–336 (1991)Google Scholar
  20. 20.
    Motwani, R., Raghavan, P.: Randomized algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  21. 21.
    Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Publication, Chichester (1995)MATHGoogle Scholar
  22. 22.
    Plotkin, S., Rao, S., Smith, W.D.: Shallow excluded minors and improved graph decomposition. In: SODA 1990, pp. 462–470 (1990)Google Scholar
  23. 23.
    Ponter, J.W., Richards, F.M.: Tertiary templates for proteins: use of packing criteria and the enumeration of allowed sequences for different structural classes. J. Molecular Biology 193, 775–791 (1987)CrossRefGoogle Scholar
  24. 24.
    Smith, W.D., Wormald, N.C.: Application of geometric separator theorems. In: FOCS 1998, pp. 232–243 (1998)Google Scholar
  25. 25.
    Spielman, D.A., Teng, S.H.: Disk packings and planar separators. In: SOCG 1996, pp. 349–358 (1996)Google Scholar
  26. 26.
    Xu, J.: Rapid protein side-chain packing via tree decomposition. In: Miyano, S., Mesirov, J., Kasif, S., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2005. LNCS (LNBI), vol. 3500, pp. 408–422. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bin Fu
    • 1
    • 2
  • Zhixiang Chen
    • 3
  1. 1.Dept. of Computer ScienceUniversity of New OrleansUSA
  2. 2.Research Institute for ChildrenNew OrleansUSA
  3. 3.Dept. of Computer ScienceUniversity of Texas – Pan AmericanUSA

Personalised recommendations