Incomplete Directed Perfect Phylogeny

Perfect phylogeny is one of the fundamental models for studying evolution. We investigate the following variant of the problem: The input is an n ×m species-characters matrix. The characters are binary and directed, i.e., a species can only gain characters. The difference from standard perfect phylogeny is that for some species the state of some characters is unknown. The question is whether one can complete the missing states in a way admitting a perfect phylogeny. We call this problem Incomplete Directed Perfect phylogeny (IDP). The problem arises in classical phylogenetic studies, when some states are missing or undetermined. Swofford’s PAUP software package [4] provides an exponential solution to the problem by exhaustive search. Quite recently, a novel kind of genomic data has given rise to the same problem: Nikaido et al. [3] use inserted repetitive genomic elements, particularly SINEs, as a source of evolutionary information. The specific insertion events are identifiable by the flanking sequences on both sides of the insertion site. These insertions are assumed to be unique, irreversible events in evolution. However, the site and its flanking sequences may be lost when a large region of the genome which includes them is deleted. In that case we do not know whether an insertion had occurred in the missing site. One can model such data by assigning each locus a character, whose state is ‘1’ if the SINE occurred in that locus, ‘0’ if the locus is present but does not contain the SINE, and ‘?’ if the locus is missing. An example of the problem input and solution is given in Figure 1.