  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.
