  Several algorithms build on the perfect phylogeny model to infer evolutionary
trees. This problem is particularly hard when evolutionary trees are inferred
from the fraction of genomes that have mutations in different positions, across
different samples. Existing algorithms might do extensive searches over the
space of possible trees. At the center of these algorithms is a projection
problem that assigns a fitness cost to phylogenetic trees. In order to perform
a wide search over the space of the trees, it is critical to solve this
projection problem fast. In this paper, we use Moreau's decomposition for
proximal operators, and a tree reduction scheme, to develop a new algorithm to
compute this projection. Our algorithm terminates with an exact solution in a
finite number of steps, and is extremely fast. In particular, it can search
over all evolutionary trees with fewer than 11 nodes, a size relevant for
several biological problems (more than 2 billion trees) in about 2 hours.
