On Two Measures of Distance between Fully-Labelled Trees

TL;DR

This paper investigates two measures of distance between fully-labelled trees, providing complexity results for permutation distance and introducing a linear-time constant-factor approximation for the combined rearrangement distance.

Contribution

It establishes the complexity of permutation distance computation and presents a new efficient approximation algorithm for the rearrangement distance.

Findings

Permutation distance computation is equivalent to maximum bipartite matching.

An $O(n^{4/3+o(1)})$ time algorithm for permutation distance.

A linear-time constant-factor approximation algorithm for rearrangement distance.

Abstract

The last decade brought a significant increase in the amount of data and a variety of new inference methods for reconstructing the detailed evolutionary history of various cancers. This brings the need of designing efficient procedures for comparing rooted trees representing the evolution of mutations in tumor phylogenies. Bernardini et al. [CPM 2019] recently introduced a notion of the rearrangement distance for fully-labelled trees motivated by this necessity. This notion originates from two operations: one that permutes the labels of the nodes, the other that affects the topology of the tree. Each operation alone defines a distance that can be computed in polynomial time, while the actual rearrangement distance, that combines the two, was proven to be NP-hard. We answer two open question left unanswered by the previous work. First, what is the complexity of computing the permutation distance? Second, is there a constant-factor approximation algorithm for estimating the rearrangement distance between two arbitrary trees? We answer the first one by showing, via a two-way reduction, that calculating the permutation distance between two trees on $n$ nodes is equivalent, up to polylogarithmic factors, to finding the largest cardinality matching in a sparse bipartite graph. In particular, by plugging in the algorithm of Liu and Sidford [ArXiv 2020], we obtain an $O(n^{4/3+o(1)})$ time algorithm for computing the permutation distance between two trees on $n$ nodes. Then we answer the second question positively, and design a linear-time constant-factor approximation algorithm that does not need any assumption on the trees.