Computing Geodesic Distances in Tree Space

TL;DR

This paper introduces two algorithms for efficiently computing geodesic distances between phylogenetic trees in tree space, utilizing combinatorial representations and Euclidean shortest path conversions.

Contribution

The authors develop novel algorithms that significantly improve the computation of geodesic distances in tree space, including a linear time method for specific cases.

Findings

Algorithms for geodesic distance computation are introduced.

Shortest path problem is reduced to Euclidean space analysis.

Linear time algorithm for specific orthant cases is presented.

Abstract

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^k contained in the subspace of R^k consisting of all orthants with the first i coordinates non-positive and the remaining coordinates non-negative for 0 <= i <= k.