Approximating the Nearest Neighbor Interchange Distance for Non-Uniform-Degree Evolutionary Trees

Abstract

The nearest neighbor interchange (nni) distance is a classical metric for measuring the distance (dissimilarity) between evolutionary trees. It has been known that computing the nni distance is NP-complete. Existing approximation algorithms can attain an approximation ratio log n for unweighted trees and 4 log n for weighted trees; yet these algorithms are limited to degree-3 trees. This paper extends the study of nni distance to trees with non-uniform degrees. We formulate the necessary and sufficient conditions for nni transformation and devise more topology-sensitive approximation algorithms to handle trees with non-uniform degrees. The approximation ratios are respectively and for unweighted and weighted trees, where d ≥ 4 is the maximum degree of the input trees.