  We introduce a new distance-based phylogeny reconstruction technique which
provably achieves, at sufficiently short branch lengths, a logarithmic
sequence-length requirement---improving significantly over previous polynomial
bounds for distance-based methods and matching existing results for general
methods. The technique is based on an averaging procedure that implicitly
reconstructs ancestral sequences.
  In the same token, we extend previous results on phase transitions in
phylogeny reconstruction to general time-reversible models. More precisely, we
show that in the so-called Kesten-Stigum zone (roughly, a region of the
parameter space where ancestral sequences are well approximated by "linear
combinations" of the observed sequences) sequences of length $O(\log n)$
suffice for reconstruction when branch lengths are discretized. Here $n$ is the
number of extant species.
  Our results challenge, to some extent, the conventional wisdom that estimates
of evolutionary distances alone carry significantly less information about
phylogenies than full sequence datasets.
