Efficient merging and construction of evolutionary trees

In this paper we study the algorithmic problem of constructing rooted evolutionary trees in the so-called experiment model. This model was first presented by Kannan, Lawler, and Warnow (J. Algorithms 21 (1996), 26-50). We present a new technique of efficiently merging partial evolutionary trees in this model. We show that two partial evolutionary trees for disjoint sets of species can be merged using experiments in time O(dn), where n is the total number of species in the resulting evolutionary tree and d is its maximum degree. We prove our upper time bound on merging evolutionary trees to be asymptotically optimal. By applying our algorithm for merging evolutionary trees we obtain an O(dn log n)-time bound on the problem of constructing an evolutionary tree for n species and maximum degree d from experiments. The classic O(n log n)-time bound on sorting in the comparison-based model can be seen as a very special case of this upper bound. (C) 2001 Academic Press.