Fast Algorithms for Computing Path-Difference Distances

Abstract

Tree comparison metrics are an important tool for the study of phylogenetic trees. Path-difference distances measure the dissimilarity between two phylogenetic trees (on the same set of taxa) by comparing their path-length vectors. Various norms can be applied to this distance. Three important examples are the -, , and -norms. The previous best algorithms for computing path-difference distances all have running time. In this paper, we show how to compute the l_1-norm path-difference distance in time and how to compute the - and -norm path-difference distances in time. By extending the presented algorithms, we also show that the -norm path-difference distance can be computed in time for any positive integer . In addition, when the integer is even, we show that the distance can be computed in time as well.

A Faster Algorithm for Computing the Kernel of Maximum Agreement Subtrees

“…One approach is to calculate a numerical index of agreement or a distance between rival trees. Many tree comparison metrics have been proposed for this purpose, such as the cophenetic correlation coefficient [41], the path-difference distance [43], [47], the nearest-neighbor interchange (NNI) distance [14], [48], the Robinson-Foulds (RF) distance [15], [39], the quartet distance [8], and the matching distance [33].…”

Phylogenetic comparison measurements and their application towards the accurate inference of evolutionary histories

“…In this section, we introduce a novel divide-and-conquer framework for computation of the cophenetic metrics (Górecki et al, 2018). The framework avoids the explicit computation of the cophenetic vectors and it was partially inspired by the divide-and-conquer strategy in Wang and Li (2018), where they proposed first sub-quadratic algorithms for the related PD metrics.…”

“…Despite the recent algorithmic breakthroughs (partially presented in the previous chapter) on the computation of PD (Wang and Li, 2018) and cophenetic (Górecki et al, 2018) metrics in subquadratic time, these novel algorithms remain slower than simple quadratic solutions on a scale of a 1000 taxa or less (Górecki et al, 2018). Since local search is generally applied on the scale of less than a thousand taxa, naïvely, a single local search iteration can be performed in O(kn 4 ) time under both PD and cophenetic metrics, where k is the number of input trees.…”

“…1: The summary of algorithmic results enabling practical local search methods under the PD and cophenetic metrics. ) that can compute the Manhattan and Euclidean PD distances in O(n log 2 n) and O(n log n) time respectively(Wang and Li, 2018), as well as the Manhattan and Euclidean cophenetic distances in…”