Distorted Metrics on Trees and Phylogenetic Forests

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 [symbol: see text] of d such that, for all leaves u and v, it holds that absolute value(d(u,v) - [symbol: see text](u,v)) < f/2 if either d(u,v) < M + f/2 or [symbol: see text](u,v) < M + f/2, where d satisfies f < or = d(e) < or = g for all edges e. Given such distortions, we show how to reconstruct in polynomial time a forest T1, ..., Talpha such that the true tree T may be obtained from that forest by adding alpha - 1 edges and alpha - 1 < or = 2(-omega(M/g)) n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight.