Computing Geodesic Distances in Tree Space

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann (2001). We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of finding the shortest path through a certain region of Euclidean space. In particular, we show there is a linear time algorithm for finding the shortest path between a point in the all positive orthant and a point in the all negative orthant of R contained in the subspace of R consisting of all orthants with the first i coordinates non-positive and the remaining coordinates non-negative for 0 ≤ i ≤ k. The resulting algorithms for computing the geodesic distance appear to be the best available to date.