The asymmetric median tree — A new model for building consensus trees

  • Cynthia Phillips
  • Tandy J. Warnow
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1075)

Abstract

Inferring the consensus of a set of different evolutionary trees for a given species set is a well-studied problem, for which several different models have been proposed. In this paper, we propose a new optimization problem for consensus tree construction, which we call the asymmetric median tree, or AMT. Our main theoretical result is the equivalence between the asymmetric median tree problem on k trees and the maximum independent set (MIS) problem on k-colored graphs. Although the problem is NP-hard for three or more trees, we have polynomial time algorithms to construct the AMT for two trees and an approximation algorithm for three or more trees. We define a measure of phylogenetic resolution and show that our algorithms (both exact and approximate) produce consensus trees that on every input are at least as resolved as the standard models (strict consensus and majority tree) in use. Finally, we show that the AMT combines desirable features of many of the standard consensus tree models in use.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amir, A., and Keselman, D. Maximum agreement subtree in a set of evolutionary trees — metrics and efficient algorithms, Proceedings, FOCS '94.Google Scholar
  2. 2.
    Barthelemy, J.P., and McMorris, F.R. (1986), The median procedure for n-trees, Journal of Classification (3), pp. 329–334.Google Scholar
  3. 3.
    M. Bellare and M. Sudan, “Improved non-approximability results”, Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, (Montreal), ACM, pp. 184–193.Google Scholar
  4. 4.
    David Bryant, “Hunting for binary trees in binary character sets: efficient algorithms for extraction, enumeration, and optimization”, Research Report #124, Department of Mathematics and Statistics, Canterbury University, Christchurch, New Zealand, April 1995.Google Scholar
  5. 5.
    Day, W.H.E. (1985), Optimal algorithms for comparing trees with labelled leaves, Journal of Classification 2, pp. 7–28.MathSciNetGoogle Scholar
  6. 6.
    Day, W.H.E., and Sankoff, D. (1986), Computational complexity of inferring phylogenies by compatibility, Systematic Zoology, 35, pp. 224–229.Google Scholar
  7. 7.
    G.F. Estabrook, C.S. Johnson, Jr., and F.R. McMorris (1976), A mathematical foundation for the analysis of cladistic character compatibility, Math. Biosci. 29, pp. 181–187.CrossRefGoogle Scholar
  8. 8.
    Estabrook, G.F., and McMorris, F.R. (1980), When is one estimate of evolutionary relationships a refinement of another?, J. Math. Biosci. 10, pp. 327–373.Google Scholar
  9. 9.
    M. Farach, T. Przytycka, and M. Thorup, On the agreement of many trees, Information Processing Letters, to appear.Google Scholar
  10. 10.
    M. Farach and M. Thorup, Sparse Dynamic Programming for Evolutionary Tree Comparison, SIAM J. on Computing.Google Scholar
  11. 11.
    Farach, M. and Thorup, M. (1994), Optimal evolutionary tree comparisons by sparse dynamic programming, Proceedings, FOCS '94.Google Scholar
  12. 12.
    Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of N P-Completeness, W.H. Freeman and Company, NY, 1979.Google Scholar
  13. 13.
    Gusfield, D. (1991), Efficient algorithms for inferring evolutionary trees, Networks 21, pp. 19–28.Google Scholar
  14. 14.
    Hopcroft, J., and Karp, R.M. (1975), An O(n2.5) algorithm for maximum matching in bipartite graphs, SIAM J. on Computing, 1975, pp. 225–231.Google Scholar
  15. 15.
    D. S. Johnson, “A catalog of complexity classes”, in Algorithms and Complexity, volume A of Handbook of Theoretical Computer Science, Elsevier science publishing company, Amsterdam, 1990, pp. 67–161.Google Scholar
  16. 16.
    Kao, Ming. Tree contractions and evolutionary trees, submitted for publication (1995).Google Scholar
  17. 17.
    McMorris, F.R. and Steel, M. (1994), The complexity of the median procedure for binary trees. Proceedings of the 4th Conference of the International Federation of Classification Societies, Paris 1993, to be published in the series “Studies in Classification, Data Analysis, and Knowledge Organization” by Springer Verlag.Google Scholar
  18. 18.
    Page, R. D. M. (1993), Genes, Organisms, and Areas: The Problem of Multiple Lineages, Systematic Biology, 42(1), 77–84.Google Scholar
  19. 19.
    Page, R. D. M. (1993), Reconciled Trees and Cladistic Analysis of Historical Associations Between Genes, Organisms, and Areas, manuscript.Google Scholar
  20. 20.
    N. Saitou and M. Nei, The neighbor-joining method: a new method for reconstructing evolutionary trees, Mol. Biol. Evol. 4: 406–25, 1987.PubMedGoogle Scholar
  21. 21.
    Steel, M.A., and Warnow, T.J. (1993), Kaikoura tree theorems: Computing the maximum agreement subtree, Information Processing Letters 48, pp. 72–82.Google Scholar
  22. 22.
    Wareham, H. T. (1985) ”An Efficient Algorithm for Computing Ml Consensus Trees”, Honors Dissertation, Department of Computer Science, Memorial University of Newfoundland, St. John's, Newfoundland.Google Scholar
  23. 23.
    Warnow, T.J. (1994), Tree compatibility and inferring evolutionary history, Journal of Algorithms, 16, pp. 388–407.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Cynthia Phillips
    • 1
    • 2
  • Tandy J. Warnow
    • 1
    • 2
  1. 1.Sandia National LabsAlbuquerqueUSA
  2. 2.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations