Computing the Rooted Triplet Distance Between Phylogenetic Networks

Abstract

The rooted triplet distance measures the structural dissimilarity of two phylogenetic trees or networks by counting the number of rooted trees with exactly three leaf labels that occur as embedded subtrees in one, but not both of them. Suppose that $N_1 = (V_1, E_1)$ and $N_2 = (V_2, E_2)$ are rooted phylogenetic networks over a common leaf label set of size $\lambda$, that $N_i$ has level $k_i$ and maximum in-degree $d_i$ for $i \in \{1,2\}$, and that the networks’ out-degrees are unbounded. Denote $n = \max (|V_1|, |V_2|)$, $m = \max (|E_1|, |E_2|)$, $k = \max (k_1, k_2)$, and $d = \max (d_1, d_2)$. Previous work has shown how to compute the rooted triplet distance between $N_1$ and $N_2$ in $\mathrm {O}(\lambda \log \lambda )$ time in the special case $k \le 1$. For $k > 1$, no efficient algorithms are known; a trivial approach leads to a running time of $\mathrm {\Omega }(n^{7} \lambda ^{3})$ and the only existing non-trivial algorithm imposes restrictions on the networks’ in- and out-degrees (in particular, it does not work when non-binary nodes are allowed). In this paper, we develop two new algorithms that have no such restrictions. Their running times are $\mathrm {O}(n^{2} m + \lambda ^{3})$ and $\mathrm {O}(m + k^{3} d^{3} \lambda + \lambda ^{3})$, respectively. We also provide implementations of our algorithms and evaluate their performance in practice. This is the first publicly available software for computing the rooted triplet distance between unrestricted networks of arbitrary levels.