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An \(O(n \log n)\) Time Algorithm for Computing the Path-Length Distance Between Trees

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Abstract

Tree comparison metrics have proven to be an invaluable aide in the reconstruction and analysis of phylogenetic (evolutionary) trees. The path-length distance between trees is a particularly attractive measure as it reflects differences in tree shape as well as differences between branch lengths. The distance equals the sum, over all pairs of taxa, of the squared differences between the lengths of the unique path connecting them in each tree. We describe an \(O(n \log n)\) time for computing this distance, making extensive use of tree decomposition techniques introduced by Brodal et al. (Algorithmica 38(2):377–395, 2004).

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Notes

  1. A bipartition A|B with \(A\cup B=X\) is in a phylogenetic tree \(T=(V,E)\) if there exists an edge \(e\in E\) such that its removal creates two trees with taxon sets A and B.

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Acknowledgements

This research was made possible due to travel funds made available from a Marsden Grant to DB. Both authors thank David Swofford for help finding an error in an earlier version of Proposition 1.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

    David Bryant

  2. ISEM, CNRS, Université de Montpellier, IRD, EPHE, Montpellier, France

    Celine Scornavacca

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  1. David Bryant
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  2. Celine Scornavacca
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Correspondence to David Bryant.

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Bryant, D., Scornavacca, C. An \(O(n \log n)\) Time Algorithm for Computing the Path-Length Distance Between Trees. Algorithmica 81, 3692–3706 (2019). https://doi.org/10.1007/s00453-019-00594-5

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  • DOI: https://doi.org/10.1007/s00453-019-00594-5

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