Listing Conflicting Triples in Optimal Time

TL;DR

This paper presents an efficient algorithm to enumerate all conflicting triples between two phylogenetic trees in linear time relative to the number of conflicts, significantly improving over the naive cubic-time approach.

Contribution

The authors develop a novel linear-time algorithm for listing all conflict triples between two phylogenetic trees, optimizing performance when conflicts are sparse.

Findings

Enumeration time is O(n + d), where d is the number of conflicts.

The algorithm is optimal up to constant factors, matching input and output reading/writing lower bounds.

Efficient conflict detection aids in faster consensus tree computation.

Abstract

Different sources of information might tell different stories about the evolutionary history of a given set of species. This leads to (rooted) phylogenetic trees that "disagree" on triples of species, which we call "conflict triples". An important subtask of computing consensus trees which is interesting in its own regard is the enumeration of all conflicts exhibited by a pair of phylogenetic trees (on the same set of $n$ taxa). As it is possible that a significant part of the $n^3$ triples are in conflict, the trivial ${\Theta}(n^3)$-time algorithm that checks for each triple whether it constitutes a conflict, was considered optimal. It turns out, however, that we can do way better in the case that there are only few conflicts. In particular, we show that we can enumerate all d conflict triples between a pair of phylogenetic trees in $O(n + d)$ time. Since any deterministic algorithm has to spend ${\Theta}(n)$ time reading the input and ${\Theta}(d)$ time writing the output, no deterministic algorithm can solve this task faster than we do (up to constant factors).