This Article 
 Bibliographic References 
 Add to: 
A Fast Algorithm for Computing Geodesic Distances in Tree Space
January-February 2011 (vol. 8 no. 1)
pp. 2-13
Megan Owen, North Carolina State University, Raleigh
J. Scott Provan, University of North Carolina, Chapel Hill
Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.

[1] R.K. Ahuja, T.L. Magnanti, and J.B. Orlin, Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
[2] B.L. Allen and M. Steel, "Subtree Transfer Operations and Their Induced Metrics on Evolutionary Trees," Annals of Combinatorics, vol. 5, pp. 1-15, 2001.
[3] N. Amenta, M. Godwin, N. Postarnakevich, and K. St. John, "Approximating Geodesic Tree Distance," Information Processing Letters, vol. 103, pp. 61-65, 2007.
[4] L. Billera, S. Holmes, and K. Vogtmann, "Geometry of the Space of Phylogenetic Trees," Advances in Applied Math., vol. 27, pp. 733-767, 2001.
[5] M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature. Springer-Verlag, 1999.
[6] M.A. Charleston, "Toward a Characterization of Landscapes of Combinatorial Optimization Problems, with Special Attention to the Phylogeny Problem," J. Computational Biology, vol. 2, pp. 439-450, 1995.
[7] J. Hein, "Reconstructing Evolution of Sequences Subject to Recombination Using Parsimony," Math. Biosciences, vol. 98, pp. 185-200, 1990.
[8] D.M. Hillis, T.A. Heath, and K. St. John, "Analysis and Visualization of Tree Space," Systematic Biology, vol. 54, pp. 471-482, 2005.
[9] S. Holmes, "Statistics for Phylogenetic Trees," Theoretical Population Biology, vol. 63, pp. 17-32, 2003.
[10] S. Holmes, "Statistical Approach to Tests Involving Phylogenetics," Proc. Math. of Evolution and Phylogeny. Oxford Univ. Press, 2005.
[11] M.K. Kuhner and J. Felsenstein, "A Simulation Comparison of Phylogeny Algorithms under Equal and Unequal Evolutionary Rates," Molecular Biology and Evolution, vol. 11, pp. 459-468, 1994.
[12] A. Kupczok, A. von Haeseler, and S. Klaere, "An Exact Algorithm for the Geodesic Distance between Phylogenetic Trees," J. Computational Biology, vol. 15, pp. 577-591, 2008.
[13] D.R. Maddison, "The Discovery and Importance of Multiple Islands of Most-Parsimonious Trees," Systematic Zoology, vol. 40, pp. 315-328, 1991.
[14] T.M.W. Nye, "Trees of Trees: An Approach to Comparing Multiple Alternative Phylogenies," Systematic Biology, vol. 57, pp. 785-794, 2008.
[15] M. Owen, "Computing Geodesic Distances in Tree Space," arXiv:0903.0696, 2009.
[16] A. Robinson and S. Whitehouse, "The Tree Representation of $\sigma_{n+1}$ ," J. Pure and Applied Algebra, vol. 111, pp. 245-253, 1996.
[17] D.F. Robinson, "Comparison of Labeled Trees with Valency Three," J. Combinatorial Theory, vol. 11, pp. 105-119, 1971.
[18] D.F. Robinson and L.R. Foulds, "Comparison of Phylogenetic Trees," Math. Biosciences, vol. 53, pp. 131-147, 1981.
[19] A. Rokas, B.L. Williams, N. King, and S.B. Carroll, "Genome-Scale Approaches to Resolving Incongruence in Molecular Phylogenies," Nature, vol. 425, pp. 798-804, 2003.
[20] C. Semple and M. Steel, Phylogenetics. Oxford Univ. Press, 2003.
[21] H. Trappmann and G.M. Ziegler, "Shellability of Complexes of Trees," J. Combinatorial Theory Series A, vol. 82, pp. 168-178, 1998.
[22] L. Wang and J.S. Marron, "Object Oriented Data Analysis: Sets of Trees," The Annals of Statistics, vol. 35, pp. 1849-1873, 2007.

Index Terms:
Geometrical problems and computations, trees, graph theory, biology and genetics, phylogenetics, distance.
Megan Owen, J. Scott Provan, "A Fast Algorithm for Computing Geodesic Distances in Tree Space," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 8, no. 1, pp. 2-13, Jan.-Feb. 2011, doi:10.1109/TCBB.2010.3
Usage of this product signifies your acceptance of the Terms of Use.