Phylogenetic Tree Construction from a Distance Matrix

Years and Authors of Summarized Original Work

• 1968; Boesch

• 1989; Hein

• 1989; Culberson, Rudnicki

• 2003; King, Zhang, Zhou

Problem Definition

Let n be a positive integer. A distance matrix of order n is a matrix D of size (n × n) which satisfies (1) D_{i, j} > 0 for all $i,j \in \{ 1,2,\ldots ,n\}$ with i ≠ j; (2) D_{i, j} = 0 for all $i,j \in \{ 1,2,\ldots ,n\}$ with $i \neq j;$ and (3) D_{i, j} = D_{j, i} for all $i,j \in \{ 1,2,\ldots ,n\}$. In the literature, a distance matrix of order n is also called a dissimilarity matrix of order 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 $\{1,2,\ldots ,n\}$ is a subset of the nodes of \(\mathcal{T}\)...