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We present the first sub-quadratic time algorithm that with high probability correctly reconstructs phylogenetic trees for short sequences generated by a Markov model of evolution. Due to rapid expansion in sequence databases, such very fast algorithms are becoming necessary. Other fast heuristics have been developed for building trees from very large alignments (Price et al, and Brown et al), but they lack theoretical performance guarantees. Our new algorithm runs in $O(n^{1+\gamma(g)}\log^2n)$ time, where $\gamma$ is an increasing function of an upper bound on the branch lengths in the phylogeny, the upper bound $g$ must be below$1/2-\sqrt{1/8} \approx 0.15$, and $\gamma(g)<1$ for all $g$. For phylogenies with very short branches, the running time of our algorithm is close to linear. For example, if all branch lengths correspond to a mutation probability of less than 0.02, the running time of our algorithm is roughly $O(n^{1.2}\log^2n)$. Via a prototype and a sequence of large-scale experiments, we show that many large phylogenies can be reconstructed fast, without compromising reconstruction accuracy. |
