Ranking Top-$k$ Trees in Tree-Based Phylogenetic Networks

Tree-based phylogenetic networks provide a powerful model for representing complex data or non-tree-like evolution. Such networks consist of an underlying evolutionary tree called a “support tree” (also known as a “subdivision tree”) together with extra arcs added between the edges of that tree. However, a tree-based network can have exponentially many support trees, and this leads to a variety of computational problems. Recently, Hayamizu established a theory called the structure theorem for rooted binary phylogenetic networks and provided linear-time and linear-delay algorithms for different problems, such as counting, optimization, and enumeration of support trees. However, in practice, it is often more useful to search for both optimal and near-optimal solutions than to calculate only an optimal solution. In the present paper, we thus consider the following problem: Given a tree-based phylogenetic network $N$ where each arc is weighted by its probability, compute the ranking of top-$k$ support trees of $N$ according to their likelihood values. We provide a linear-delay (and hence optimal) algorithm for this problem.