A fast algorithm for the computation and enumeration of perfect phylogenies when the number of character states is fixed

The Perfect Phylogeny Problem is a classical problem in computational evolutionary biology, in which a set of species/taxa is described by a set of qualitative characters. In recent years, the problem has been shown to be NP-Complete in general, while the different fixed parameter versions can each be solved in polynomial time. In particular, Agarwala and Fernandez-Baca have developed an O(2^3r (nk³ + k⁴)) algorithm for the perfect phylogeny problem for n species defined by k r-state characters. Since commonly the character data is drawn from alignments of molecular sequences, k is the length of the sequences and can thus be very large (in the hundreds or thousands). Thus, it is imperative to develop algorithms which run efficiently for large values of k. In this paper we make additional observations about the structure of the problem and produce an algorithm for the problem that runs in time O(2^2rk²n). We also show how it is possible to efficiently build a structure that implicitly represents the set of all perfect phylogenies, and to randomly sample from that set.