Rapid Evaluation of Least-Squares and Minimum-Evolution Criteria on Phylogenetic Trees

We present fast new algorithms for evaluating trees with respectto least squares and minimum evolution (ME), the most commonlyused criteria for inferring phylogenetic trees from distancedata. The new algorithms include an optimal O(N2) time algorithmfor calculating the edge (branch or internode) lengths on atree according to ordinary or unweighted least squares (OLS);an O(N3) time algorithm for edge lengths under weighted leastsquares (WLS) including the Fitch-Margoliash method; and anoptimal O(N4) time algorithm for generalized least-squares (GLS)edge lengths (where N is the number of taxa in the tree). TheME criterion is based on the sum of edge lengths. Consequently,the edge lengths algorithms presented here lead directly toO(N2), O(N3), and O(N4) time algorithms for MEunder OLS, WLS,and GLS, respectively. All of these algorithms are as fast asor faster than any of those previously published, and the algorithmsfor OLS and GLS are the fastest possible (with respect to orderof computational complexity). A major advantage of our new methodsis that they are as well adapted to multifurcating trees asthey are to binary trees. An optimal algorithm for determiningpath lengths from a tree with given edge lengths is also developed.Thisleads to an optimal O(N2) algorithm for OLS sums of squaresevaluation and corresponding O(N3) and O(N4) time algorithmsfor WLS and GLS sums of squares, respectively. The GLS algorithmis time-optimal if the covariance matrix is already inverted.The speed of each algorithm is assessed analytically—thespeed increases we calculate are confirmed by the dramatic speedincreases resulting from their implementation in PAUP* 4.0.The new algorithms enable far more extensive tree searches andstatistical evaluations (e.g., bootstrap, parametric bootstrap,or jackknife) in the same amount of time. Hopefully, the fastalgorithms for WLS and GLS will encourage the use of these criteriafor evaluating trees and their edge lengths (e.g., for approximatedivergence time estimates), since they should be more statisticallyefficient than OLS.