Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

In this paper we study the graph of ranked phylogenetic trees where the adjacency relation is given by a local rearrangement of the tree structure. Our work is motivated by tree inference algorithms, such as maximum likelihood and Markov Chain Monte Carlo methods, where the geometry of the search space plays a central role for efficiency and practicality of the optimisation. We hence focus on understanding the geometry of the space (graph) of ranked trees, the so-called ranked nearest neighbour interchange (RNNI) graph. For example, we find the radius and diameter of the space exactly, improving the best previously known estimates. Since the RNNI graph is a generalisation of the classical nearest neighbour interchange (NNI) graph to ranked phylogenetic trees, we compare geometric and algorithmic properties of the two graphs. Surprisingly, we discover that both geometric and algorithmic properties of RNNI and NNI are quite different. For example, we establish convexity of certain natural subspaces in RNNI, which are not convex is NNI. Furthermore, our results suggest that the complexity of computing distances in the two graphs is different.