On Finding the Adams Consensus Tree

This paper presents a fast algorithm for finding the Adams consensus tree of a set of conflicting phylogenetic trees with identical leaf labels, for the first time improving the time complexity of a widely used algorithm invented by Adams in 1972 [1]. Our algorithm applies\r\nthe centroid path decomposition technique [9] in a new way to traverse the input trees\u0027 centroid paths in unison, and runs in O(k n \\log n) time, where k is the number of input trees and n is the size of the leaf label set. (In comparison, the old algorithm from 1972 has a worst-case running time of O(k n^2).) For the special case of k = 2, an even faster algorithm running in O(n \\cdot \\frac{\\log n}{\\log\\log n}) time is provided, which relies on an extension of the wavelet tree-based technique by Bose et al. [6] for orthogonal range counting on a grid.\r\nOur extended wavelet tree data structure also supports truncated\r\nrange maximum queries efficiently and may be of independent interest to algorithm designers.