The Complexity of Constructing Evolutionary Trees Using Experiments

We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model. We describe an algorithm which constructs an evolutionary tree of n species in time O(nd log_d n) using at most n⌈d/2⌉(log_{2ċd/2ċ 1} n+O(1)) experiments for d > 2, and at most n(log n+O(1)) experiments for d = 2, where d is the degree of the tree. This improves the previous best upper bound by a factor Θ(log d). For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments. By an explicit adversary argument, we show an Ω(nd log_d n) lower bound, matching our upper bounds and improving the previous best lower bound by a factor Θ(log_d n). Central to our algorithm is the construction and maintenance of separator trees of small height, which may be of independent interest.