A faster algorithm for the tree containment problem for binary nearly stable phylogenetic networks

Phylogenetic networks and phylogenetic trees are leaf-labelled graphs used in biology to describe evolutionary histories of species whose leaves correspond to a set of taxa in the study. Given a phylogenetic network N and a phylogenetic tree T over the same set of taxa, if one can obtain T from N by edge deletions and contractions, we say that N contains T. A fundamental problem, called the tree containment problem, is to determine if N contains T. In general networks, this problem is NP-complete, but can be solved in polynomial time when N is a normal network, a binary tree-child network, or a level-k network. Recently, Gambette, Gunawan, Labarre, Vialette and Zhang showed that it is possible to solve the problem for a more general class of networks called binary nearly stable networks. Not only that binary nearly stable networks include normal and tree-child networks, they claim that important evolution histories also match this generalization. Their algorithm is also more efficient than previous algorithms as it runs in time O(n²) where n is the number of taxa. This paper presents a faster O(n log n) algorithm. We obtain this improvement from a simple observation that the iterative algorithm of Gambette et al. only performs very local modifications of the networks. Our algorithm employs elementary data structures to dynamically maintain certain internal data structures used in their algorithm instead of recomputing at every iteration.