Faster computation of the Robinson–Foulds distance between phylogenetic networks

The Robinson–Foulds distance, a widely used metric for comparing phylogenetic trees, has recently been generalized to phylogenetic networks. Given two phylogenetic networks N1, N2 with n leaf labels and at most m nodes and e edges each, the Robinson–Foulds distance measures the number of clusters of descendant leaves not shared by N1 and N2. The fastest known algorithm for computing the Robinson–Foulds distance between N1 and N2 runs in O(me) time. In this paper, we improve the time complexity to O(ne/log n) for general phylogenetic networks and O(nm/log n) for general phylogenetic networks with bounded degree (assuming the word RAM model with a word length of ⌈logn⌉ bits), and to optimal O(m) time for leaf-outerplanar networks as well as optimal O(n) time for level-1 phylogenetic networks (that is, galled-trees). We also introduce the natural concept of the minimum spread of a phylogenetic network and show how the running time of our new algorithm depends on this parameter. As an example, we prove that the minimum spread of a level-k network is at most k + 1, which implies that for one level-1 and one level-k phylogenetic network, our algorithm runs in O((k + 1)e) time.