Inferring a Level-1 Phylogenetic Network from a Dense Set of Rooted Triplets

Abstract

Given a set ${\cal T}$ of rooted triplets with leaf set L, we consider the problem of determining whether there exists a phylogenetic network consistent with ${\mathcal{T}}$, and if so, constructing one. If no restrictions are placed on the hybrid nodes in the solution, the problem is trivially solved in polynomial time by a simple sorting network-based construction. For the more interesting (and biologically more motivated) case where the solution is required to be a level-1 phylogenetic network, we present an algorithm solving the problem in O(n ⁶) time when ${\mathcal{T}}$ is dense (i.e., contains at least one rooted triplet for each cardinality three subset of L), where n = |L|. Note that the size of the input is Θ(n ³) if ${\mathcal{T}}$ is dense. We also give an O(n ⁵)-time algorithm for finding the set of all phylogenetic networks having a single hybrid node attached to exactly one leaf (and having no other hybrid nodes) that are consistent with a given dense set of rooted triplets.