Inferring a level-1 phylogenetic network from a dense set of rooted triplets

We consider the following problem: Given a set T of rooted triplets with leaf set L, determine whether there exists a phylogenetic network consistent with T, and if so, construct one. We show that 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(|T|2) time when T is dense, i.e., when T contains at least one rooted triplet for each cardinality three subset of L. We also give an O(|T|5/3)-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.