Fast Algorithms for Computing the Tripartition-Based Distance Between Phylogenetic Networks

Consider two phylogenetic networks N and N ′ of size n. The tripartition-based distance finds the proportion of tripartitions which are not shared by N and N . This distance is proposed by Moret et al (2004) and is a generalization of Robinson-Foulds distance, which is orginally used to compare two phylogenetic trees. This paper gives an O(min{kn log n, n logn+ hn})-time algorithm to compute this distance, where h is the number of hybrid nodes in N and N ′ while k is the maximum number of hybrid nodes among all biconnected components in N and N . Note that k << h << n in a phylogenetic network. In addition, we propose algorithms for comparing galled-trees, which are an important, biological meaningful special case of phylogenetic network. We give an O(n)-time algorithm for comparing two galled-trees. We also give an O(n+ kh)-time algorithm for comparing a galled-tree with another general network, where h and k are the number of hybrid nodes in the latter network and its biggest biconnected component respectively.