On the Linear-cost Subtree-transfer Distance between Phylogenetic Trees (revised Version of Dimacs Technical Report 97-18) 1

Di erent phylogenetic trees for the same group of species are often produced either by procedures that use diverse optimality criteria [16] or from di erent genes [12] in the study of molecular evolution. Comparing these trees to nd their similarities and dissimilarities (i.e. distance) is thus an important issue in computational molecular biology. Several distance metrics including the nearest neighbor interchange (nni) distance and the subtree-transfer distance have been proposed and extensively studied in the literature. This article considers a natural extension of the subtreetransfer distance, called the linear-cost subtree-transfer distance, and studies the complexity and e cient approximation algorithms for this distance as well as its relationship to the nni distance. The linear-cost subtree-transfer model seems more suitable than the (unit-cost) subtree-transfer model in some applications. The following is a list of our results. 1. The linear-cost subtree-transfer distance is in fact identical to the nni distance on unweighted phylogenies. 2. There is an algorithm to compute an optimal linear-cost subtree-transfer sequence between unweighted phylogenies in O(n 2O(d)) time, where d denotes the linear-cost subtree-transfer distance. Such an algorithm is useful when d is small. 3. Computing the linear-cost subtree-transfer distance between two weighted phylogenetic trees is NP-hard, provided we allow multiple leaves of a tree to share the same label (i.e. the trees are not necessarily uniquely labeled). 4. There is an e cient approximation algorithm for computing the linear-cost subtree-transfer distance between weighted phylogenies with performance ratio 2.