Phylogenetic Tree Construction from a Distance Matrix

Keywords and Synonyms

Phylogenetic tree construction from a dissimilarity matrix            

Problem Definition

Let n be a positive integer. A distance matrix of order n (also called a dissimilarity matrix of order n) is a matrix D of size ${ (n \times n) }$ which satisfies: (1) ${ D_{i,j} > 0 }$ for all ${ i,j \in \{1,2,\dots,n\} }$ with ${ i \neq j }$; (2) ${ D_{i,j} = 0 }$ for all ${ i,j \in \{1,2,\dots,n\} }$ with ${ i = j }$; and (3) ${ D_{i,j} = D_{j,i} }$ for all ${ i,j \in \{1,2,\dots,n\} }$.

Below, all trees are assumed to be unrooted and edge-weighted. For any tree ${ \mathcal{T} }$, the distance between two nodes u and v in ${ \mathcal{T} }$ is defined as the sum of the weights of all edges on the unique path in ${ \mathcal{T} }$ between u and v, and is denoted by ${ d_{u,v}^{\mathcal{T}} }$. A tree ${ \mathcal{T} }$ is said to realize a given distance matrix D of order n if and only if it holds that ${ \{1,2,\dots,n\} }$is...