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.
Supported in part by the NSERC Operating Grant OGP0046506, ITRC, a CGAT grant and DIMACS.
Supported by an NSERC International Fellowship.
Supported by a CGAT grant.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
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.
Karel Culik II and Derick Wood, A note on some tree similarity measures, Information Processing Letters 15, 39–42, 1982.
W. H. E. Day, Properties of the Nearest Neighbour Interchange Metric for Tress of Small Size, Journal of Theoretical Biology 101, 275–288, 1983.
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, 1979.
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.
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.
M. Křivánek, Computing the Nearest Neighbour Interchange Metric for Unlabeled Binary Trees is NP-Complete, Journal of Classification 3, 55–60, 1986.
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.
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.
C.H. Papadimitriou and M. Yannakakis, Optimization, Approximation, and complexity classes, Journal of Computer and System Sciences 43, 425–440, 1991.
D. F. Robinson, Comparison of Labeled Trees with Valency Three, Journal of Combinatorial Theory 11, 105–119, 1971.
D. Sleator, R. Tarjan, W. Thurston, Short encodings of evolving structures, SIAM Journal on Discrete Mathematics 5, 428–450, 1992.
M, S. Waterman and T. F. Smith, On the Similarity of Dendrograms, Journal of Theoretical Biology 73, 789–800, 1978.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, M., Tromp, J., Zhang, L. (1996). Some notes on the nearest neighbour interchange distance. In: Cai, JY., Wong, C.K. (eds) Computing and Combinatorics. COCOON 1996. Lecture Notes in Computer Science, vol 1090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61332-3_168
Download citation
DOI: https://doi.org/10.1007/3-540-61332-3_168
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61332-9
Online ISBN: 978-3-540-68461-9
eBook Packages: Springer Book Archive