Constructing Near-Perfect Phylogenies with multiple homoplasy events

Motivation: We explore the problem of constructing near-perfect phylogenies on bi-allelic haplotypes, where the deviation from perfect phylogeny is entirely due to homoplasy events. We present polynomial-time algorithms for restricted versions of the problem. We show that these algorithms can be extended to genotype data, in which case the problem is called the near-perfect phylogeny haplotyping ( NPPH) problem. We present a near-optimal algorithm for the H1-NPPH problem, which is to determine if a given set of genotypes admit a phylogeny with a single homoplasy event. The time-complexity of our algorithm for the H1-NPPH problem is O(m(2)(n + m)), where n is the number of genotypes and m is the number of SNP sites. This is a significant improvement over the earlier O( n 4) algorithm. We also introduce generalized versions of the problem. The H(1, q)-NPPH problem is to determine if a given set of genotypes admit a phylogeny with q homoplasy events, so that all the homoplasy events occur in a single site. We present an O(m(q+1)(n + m)) algorithm for the H(1, q)-NPPH problem. Results: We present results on simulated data, which demonstrate that the accuracyof our algorithm for theH1-NPPHproblemiscomparableto that of the existing methods, while being orders of magnitude faster. Availability: The implementation of our algorithm for the H1-NPPH problem is available upon request