Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology

Abstract.

We are given a set $\cal T$ = {T ₁ ,T ₂ , . . .,T _k } of rooted binary trees, each T _i leaf-labeled by a subset ${\cal L}(T_i) \subset \{1,2, . . ., n\}$ . If T is a tree on {1,2, . . .,n }, we let $T|{\cal L}$ denote the minimal subtree of T induced by the nodes of $\cal L$ and all their ancestors. The consensus tree problem asks whether there exists a tree T ^* such that, for every i , $T^* |{\cal L}(T_i)$ is homeomorphic to T _i .

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(N n ^1/2 ), O(N+ n ² log n )}, where $N = \sum_i | T_i |$ , and uses linear space. The randomized algorithm takes time O(N log³ n) and uses linear space. The previous best for this problem was a 1981 O(Nn) 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(n ² log n + b ₀ min{n ² , m log n }), where b ₀ is the number of batches which do not result in a new component. For our particular application, $b_0 \leq1$ . If all edges are deleted, then the best previously known deterministic algorithm requires time $O(m \sqrt n)$ to solve this problem. We also present two applications of these consensus tree algorithms which solve other problems in computational evolutionary biology.