Advertisement

Abstract

A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arises in areas such as phylogenetics, linguistics, etc. e.g., a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree. Research on perfect phylogeny is usually focused on finding a tree so that few predetermined partial colorings of its vertices are convex.

When a coloring of a tree is not convex, it is desirable to know “how far” it is from a convex one. In [MS05], a natural measure for this distance, called the recoloring distance was defined: the minimal number of color changes at the vertices needed to make the coloring convex. This can be viewed as minimizing the number of “exceptional vertices” w.r.t. to a closest convex coloring. The problem was proved to be NP-hard even for colored strings.

In this paper we continue the work of [MS05], and present a 2-approximation algorithm of convex recoloring of strings whose running time O(cn), where c is the number of colors and n is the size of the input, and an O(cn 2) 3-approximation algorithm for convex recoloring of trees.

Keywords

Covered Vertex Partial Coloring Colored Tree Perfect Phylogeny Versus Support 
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. [AFB96]
    Agrawala, R., Fernandez-Baca, D.: Simple algorithms for perfect phylogeny and triangulating colored graphs. International Journal of Foundations of Computer Science 7(1), 11–21 (1996)CrossRefGoogle Scholar
  2. [BBF99]
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. on Discrete Mathematics 12, 289–297 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BDFY01]
    Ben-Dor, A., Friedman, N., Yakhini, Z.: Class discovery in gene expression data. In: RECOMB, pp. 31–38 (2001)Google Scholar
  4. [Be00]
    Bittner, M., et al.: Molecular classification of cutaneous malignant melanoma by gene expression profiling. Nature 406(6795), 536–540 (2000)CrossRefGoogle Scholar
  5. [BFW92]
    Bodlaender, H.L., Fellows, M.R., Warnow, T.: Two strikes against perfect phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)Google Scholar
  6. [BY00]
    Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27, 131–144 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [BYE85]
    Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)MathSciNetGoogle Scholar
  8. [DS92]
    Dress, A., Steel, M.A.: Convex tree realizations of partitions. Applied Mathematics Letters 5(3), 3–6 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [FBL03]
    Fernández-Baca, D., Lagergren, J.: A polynomial-time algorithm for near-perfect phylogeny. SIAM Journal on Computing 32(5), 1115–1127 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Fit81]
    Fitch, W.M.: A non-sequential method for constructing trees and hierarchical classifications. Journal of Molecular Evolution 18(1), 30–37 (1981)CrossRefMathSciNetGoogle Scholar
  11. [GGP+96]
    Goldberg, L.A., Goldberg, P.W., Phillips, C.A.: Minimizing phylogenetic number to find good evolutionary trees. Discrete Applied Mathematics 71, 111–136 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  13. [Gus91]
    Gusfield, D.: Efficient algorithms for inferring evolutionary history. Networks 21, 19–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [Hoc97]
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problem. PWS Publishing Company (1997)Google Scholar
  15. [HTD+04]
    Hirsh, A., Tsolaki, A., DeRiemer, K., Feldman, M., Small, P.: From the cover: Stable association between strains of mycobacterium tuberculosis and their human host populations. PNAS 101, 4871–4876 (2004)CrossRefGoogle Scholar
  16. [KW94]
    Kannan, S., Warnow, T.: Inferring evolutionary history from DNA sequences. SIAM J. Computing 23(3), 713–737 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [KW97]
    Kannan, S., Warnow, T.: A fast algorithm for the computation and enumeration of perfect phylogenies when the number of character states is fixed. SIAM J. Computing 26(6), 1749–1763 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [MS03]
    Moran, S., Snir, S.: Convex recoloring of strings and trees. Technical Report CS-2003-13, Technion (November 2003)Google Scholar
  19. [MS05]
    Moran, S., Snir, S.: Convex recoloring of strings and trees: Definitions, hardness results and algorithms. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. [PM81]
    Paz, A., Moran, S.: Non deterministic polynomial optimization probems and their approximabilty. ICALP 1977 15, 251–277 (1981); Abridged version: Proc. of the 4th ICALP conference (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [San75]
    Sankoff, D.: Minimal mutation trees of sequences. SIAM Journal on Applied Mathematics 28, 35–42 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [SS03]
    Semple, C., Steel, M.A.: Phylogenetics. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  23. [Ste92]
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9(1), 91–116 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Vaz01]
    Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Shlomo Moran
    • 1
  • Sagi Snir
    • 2
  1. 1.Computer Science deptTechnionHaifaIsrael
  2. 2.Mathematics deptUniversity of CaliforniaBerkeleyUSA

Personalised recommendations