Some notes on the nearest neighbour interchange distance
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Abstract
We present some new results on a well known distance measure between evolutionary trees. The trees we consider are free 3-trees having n leaves labeled 0,..., n − 1 (representing species), and n − 2 internal nodes of degree 3. The distance between two trees is the minimum number of nearest neighbour interchange (NNI) operations required to transform one into the other. First, we improve an upper bound on the nni-distance between two arbitrary n-node trees from 4n log n [2] to n log n. Second, we present a counterexample disproving several theorems in [13]. Roughly speaking, finding an equal partition between two trees doesn't imply decomposability of the distance finding problem. Third, we present a polynomial-time approximation algorithm that, given two trees, finds a transformation between them of length O(log n) times their distance. We also present some results of computations we performed on small size trees.
Keywords
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References
- 1.R. P. Boland, E. K. Brown and W. H. E. Day, Approximating minimum-length-sequence metrics: a cautionary note, Mathematical Social Sciences 4, 261–270, 1983.Google Scholar
- 2.Karel Culik II and Derick Wood, A note on some tree similarity measures, Information Processing Letters 15, 39–42, 1982.Google Scholar
- 3.W. H. E. Day, Properties of the Nearest Neighbour Interchange Metric for Tress of Small Size, Journal of Theoretical Biology 101, 275–288, 1983.Google Scholar
- 4.M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, 1979.Google Scholar
- 5.J. P. Jarvis, J. K. Luedeman and D. R. Shier, Counterexamples in measuring the distance between binary trees, Mathematical Social Sciences 4, 271–274, 1983.Google Scholar
- 6.J. P. Jarvis, J. K. Luedeman and D. R. Shier, Counterexamples in measuring the distance between binary trees, Journal of Theoretical Biology 100, 427–433, 1983.Google Scholar
- 7.M. Křivánek, Computing the Nearest Neighbour Interchange Metric for Unlabeled Binary Trees is NP-Complete, Journal of Classification 3, 55–60, 1986.Google Scholar
- 8.V. King and T. Warnow, On Measuring the nni Distance Between Two Evolutionary Trees, DIMACS mini workshop on combinatorial structures in molecular biology, Rutgers University, Nov 4, 1994.Google Scholar
- 9.G. W. Moore, M. Goodman and J. Barnabas, An iterative approach from the standpoint of the additive hypothesis to the dendrogram problem posed by molecular data sets, Journal of Theoretical Biology 38, 423–457, 1973.PubMedGoogle Scholar
- 10.C.H. Papadimitriou and M. Yannakakis, Optimization, Approximation, and complexity classes, Journal of Computer and System Sciences 43, 425–440, 1991.CrossRefGoogle Scholar
- 11.D. F. Robinson, Comparison of Labeled Trees with Valency Three, Journal of Combinatorial Theory 11, 105–119, 1971.Google Scholar
- 12.D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM Journal on Discrete Mathematics 5, 428–450, 1992.Google Scholar
- 13.M, S. Waterman and T. F. Smith, On the Similarity of Dendrograms, Journal of Theoretical Biology 73, 789–800, 1978.PubMedGoogle Scholar