Computing the blocks of a quasi-median graph

TL;DR

This paper introduces an efficient algorithm to compute the blocks of a quasi-median graph derived from sequence alignments, which can facilitate faster computation of evolutionary trees in biology.

Contribution

The authors present a novel algorithm that computes the blocks of a quasi-median graph directly from sequence data, avoiding full graph construction and improving efficiency.

Findings

Algorithm runs in O(n^2 m^2) time, independent of alphabet size.

Precomputing blocks can speed up the computation of most parsimonious trees.

The method leverages the structure of quasi-median graphs for efficient analysis.

Abstract

Quasi-median graphs are a tool commonly used by evolutionary biologists to visualise the evolution of molecular sequences. As with any graph, a quasi-median graph can contain cut vertices, that is, vertices whose removal disconnect the graph. These vertices induce a decomposition of the graph into blocks, that is, maximal subgraphs which do not contain any cut vertices. Here we show that the special structure of quasi-median graphs can be used to compute their blocks without having to compute the whole graph. In particular we present an algorithm that, for a collection of $n$ aligned sequences of length $m$, can compute the blocks of the associated quasi-median graph together with the information required to correctly connect these blocks together in run time $\mathcal O(n^2m^2)$, independent of the size of the sequence alphabet. Our primary motivation for presenting this algorithm is the fact that the quasi-median graph associated to a sequence alignment must contain all most parsimonious trees for the alignment, and therefore precomputing the blocks of the graph has the potential to help speed up any method for computing such trees.