Abstract
The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map. From this point of view, NJ is "optimal" when the algorithm outputs the tree which minimizes the balanced minimum evolution criterion. We use the fact that the NJ tree topology and the BME tree topology are determined by polyhedral subdivisions of the spaces of dissimilarity maps to study the optimality of the neighbor-joining algorithm. In particular, we investigate and compare the polyhedral subdivisions for n ≤ 8. This requires the measurement of volumes of spherical polytopes in high dimension, which we obtain using a combination of Monte Carlo methods and polyhedral algorithms. Our results include a demonstration that highly unrelated trees can be co-optimal in BME reconstruction, and that NJ regions are not convex. We obtain the l2 radius for neighbor-joining for n = 5 and we conjecture that the ability of the neighbor-joining algorithm to recover the BME tree depends on the diameter of the BME tree.
Highlights
The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map
The popular neighbor-joining algorithm used for phylogenetic tree reconstruction [1] has recently been "revealed" to be a greedy algorithm for finding the balanced minimum evolution tree associated to a dissimilarity map [2]
We found that the edge graph of the BME polytope is the complete graph for n = 4, 5, 6 which means that for every pair of trees T1 and T2 with the same number (≤ 6) of leaves, there is a dissimilarity map for which T1 and T2 are co-optimal BME trees
Summary
The popular neighbor-joining algorithm used for phylogenetic tree reconstruction [1] has recently been "revealed" to be a greedy algorithm for finding the balanced minimum evolution tree associated to a dissimilarity map [2]. The NJ algorithm will pick pairs of leaves to merge in a particular order and output a particular tree T if and only if the pairwise distances satisfy a system of linear inequalities, whose solution set forms a polyhedral cone in We call such a cone a neighbor-joining cone. Our main result is a comparison of the neighbor-joining cones with the normal fan of the balanced minimum evolution polytope This means that we characterize those dissimilarity maps for which neighbor-joining, despite being a greedy algorithm, is able to identify the balanced minimum evolution tree.
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