Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology

We are given a set 7 = {Tl , Tz, . . . , Tk} of rooted binary trees, each Ti leaf-labeled by a subset L(x) c {1,2 )...) n}. IfT is a tree on {1,2, . . , n}, we let T]L denote the subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks whether there exists a tree T* such that for every I, T’ IC(Ti) is homeomorphic to Ti. We present algorithms which test if a given set of trees has a consensus tree and if so, construct one. The deterministic algorithm takes time min{O(mn’/‘), O(m + n2 logn)}, where m = Ci IZl and uses linear space. The randomized algorithm takes time O(m log3 n) and uses linear space. The previous best for this problem was an 1981 O(mn) algorithm by Aho et al. Our faster deterministic algorithm uses a new efficient algorithm for the following interesting dynamic graph problem: Given a graph G with n nodes and m edges and a sequence of b batches of one or more edge deletions, then after each batch, either find a new component that has just been created or determine that there is no such component. For this problem, we have a simple algorithm with running time O(n2 log n + be min{ n2, m log n}), where be is the number of batches which do not result in a new component. For our particular application, bc 5 1. If all edges are deleted, then the best previously known deterministic algorithm requires time O(mJ;ii) to solve this problem. computational evolutionary biology. The first application is in the problem of inferring consensus of trees when there can be disagreement[l6]. There have, been several models suggested for this problem[2, 3, 4, 8, ?, 11, 17, 181, of which one is called the Local Consensus Tree[l5]. The local consensus tree model presumes that the user provides a local consensus rule which determines the form of the output tree on (perhaps) each triple of leaves, and the objective is to determine whether a tree exists which is consistent with each of the constraints. We will show that we can construct the local consensus tree of k trees on n species in O(kn3) time, improving on the O(lcn3 + n”) running time if we use the Aho et al algorithm. The second application is a heuristic for constructing the maximum likelihood tree based upon combining solutions to small subproblems. This is a simple and yet potentially significantly interesting approach to the evolutionary tree construction problem.