A computationally efficient approximation to the nearest neighbor interchange metric

Abstract

The nearest neighbor interchange (nni) metric is a distance measure providing a quantitative measure of dissimilarity between two unrooted binary trees with labeled leaves. The metric has a transparent definition in terms of a simple transformation of binary trees, but its use in nontrivial problems is usually prevented by the absence of a computationally efficient algorithm. Since recent attempts to discover such an algorithm continue to be unsuccessful, we address the complementary problem of designing an approximation to the nni metric. Such an approximation should be well-defined, efficient to compute, comprehensible to users, relevant to applications, and a close fit to the nni metric; the challenge, of course, is to compromise these objectives in such a way that the final design is acceptable to users with practical and theoretical orientations. We describe an approximation algorithm that appears to satisfy adequately these objectives. The algorithm requires O(n) space to compute dissimilarity between binary trees withn labeled leaves; it requires O(n logn) time for rooted trees and O(n2 logn) time for unrooted trees. To help the user interpret the dissimilarity measures based on this algorithm, we describe empirical distributions of dissimilarities between pairs of randomly selected trees for both rooted and unrooted cases.