Phylogenies depicting the evolutionary history of genetically heterogeneous subpopulations of cells from the same cancer i.e., cancer phylogenies, provide useful insights about cancer development and inform treatment. Cancer phylogenies can be reconstructed using data obtained from bulk DNA sequencing of multiple tissue samples from the same cancer. We introduce Orchard, a fast algorithm that reconstructs cancer phylogenies using point mutations detected in bulk DNA sequencing data. Orchard constructs cancer phylogenies progressively, one point mutation at a time, ultimately sampling complete phylogenies from a posterior distribution implied by the bulk DNA data. Orchard reconstructs more plausible phylogenies than state-of-the-art cancer phylogeny reconstruction methods on 90 simulated cancers and 14 B-progenitor acute lymphoblastic leukemias (B-ALLs). These results demonstrate that Orchard accurately reconstructs cancer phylogenies with up to 300 mutations. We then introduce a simple graph based clustering algorithm that uses a reconstructed phylogeny to infer unique groups of mutations i.e., mutation clusters, that characterize the genetic differences between cancer cell populations, and show that this approach is competitive with state-of-the-art mutation clustering methods.

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.