Computing nearest neighbour interchange distances between ranked phylogenetic trees

Abstract Many popular algorithms for searching the space of leaf-labelled (phylogenetic) trees are based on tree rearrangement operations. Under any such operation, problem is reduced to a graph where vertices and (undirected) edges given by pairs connected one operation (sometimes called move). Most classical nearest neighbour interchange, subtree prune regraft, bisection reconnection moves. The computing distances, however, $${\mathbf {N}}{\mathbf {P}}$$ N P -hard in each these graphs, making inference comparison challenging design practice. Although ranked phylogenetic central objects interest applications as cancer research, immunology, epidemiology, computational complexity shortest path remained unsolved decades. In this paper, we settle interchange establishing that depends weight difference between two types rearrangements (rank moves edge moves), varies from quadratic, which lowest possible problem, -hard, highest. particular, our result provides first example paths, hence distance, can be computed efficiently. Specifically, algorithm scales with tens thousands leaves (and likely hundreds if implemented efficiently).