Algorithms for reticulate networks of multiple phylogenetic trees.

A reticulate network N of multiple phylogenetic trees may have vertices with two or more parents (called reticulation vertices). There are two ways to define the reticulation number of N. One is to define it as the number of reticulation vertices in N; in this case, a reticulate network with the smallest reticulation number is called an optimal type-I reticulate network of the trees. The other is to define it as the total number of parents of reticulation vertices in N minus the number of reticulation vertices in N; in this case, a reticulate network with the smallest reticulation number is called an optimal type-II reticulate network of the trees. In this paper, we present a fast algorithm for constructing one or all optimal type-I reticulate networks of multiple phylogenetic trees. We then use the algorithm together with other ideas to obtain an algorithm for estimating a lower bound on the reticulation number of an optimal type-II reticulate network of the input trees. To our knowledge, these are the first fast algorithms for the problems. Our experimental data shows that our algorithms can construct optimal type-I reticulate networks rapidly and can compute better lower bounds for optimal type-II reticulate networks within much shorter time than the previously best program.