### Distorted metrics on trees and phylogenetic forests [article]

Elchanan Mossel
<span title="2004-03-30">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances {d(u,v)} between leaves. It is well known that these determine the tree and the d length of all edges. Here we consider distortions of d such that for all leaves u and v it holds that |d(u,v) - (̣u,v)| < f/2 if either d(u,v) < M or (̣u,v) < M, where d satisfies f ≤ d(e) ≤ g for all edges e. Given such distortions we
more &raquo; ... how to reconstruct in polynomial time a forest T_1,...,T_α such that the true tree T may be obtained from that forest by adding α-1 edges and α-1 ≤ 2^-Ω(M/g) n. Metric distortions arise naturally in phylogeny, where d(u,v) is defined by the log-det of a covariance matrix associated with u and v. of a covariance matrix associated with u and v. When u and v are "far", the entries of the covariance matrix are small and therefore (̣u,v), which is defined by log-det of an associated empirical-correlation matrix may be a bad estimate of d(u,v) even if the correlation matrix is "close" to the covariance matrix. Our metric results are used in order to show how to reconstruct phylogenetic forests with small number of trees from sequences of length logarithmic in the size of the tree. Our method also yields an independent proof that phylogenetic trees can be reconstructed in polynomial time from sequences of polynomial length under the standard assumptions in phylogeny. Both the metric result and its applications to phylogeny are almost tight.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0403508v1">arXiv:math/0403508v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3jhhgnt6izgnfmupwpawtlax3a">fatcat:3jhhgnt6izgnfmupwpawtlax3a</a> </span>
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