New Gromov-Inspired Metrics on Phylogenetic Tree Space

Abstract

We present a new class of metrics for unrooted phylogenetic X-trees inspired by the Gromov–Hausdorff distance for (compact) metric spaces. These metrics can be efficiently computed by linear or quadratic programming. They are robust under NNI operations, too. The local behaviour of the metrics shows that they are different from any previously introduced metrics. The performance of the metrics is briefly analysed on random weighted and unweighted trees as well as random caterpillars.

A On Semimetric Extensions

Several times we met the problem whether a partial dissimilarity on X, i.e. a map $q:E\rightarrow \mathbb {R}_{\ge 0}$, $E\subseteq \left( {\begin{array}{c}X\\ 2\end{array}}\right)$, has an extension to a semimetric on X. This seems to be a well-known problem, one folklore solution I found in Guénoche et al. (2004). For our needs, the following reformulation proved more useful.

We call a cycle $p=x_0x_1\dots x_m$, $x_0=x_m$, in a graph (X, E) induced, if it is simple ($x_i$, $i=0,\dots ,m-1$, are different) and chordless ($\left\{ x_i,x_j\right\} \notin E$, $0\le i,j\le m-1$, $2\le \left|i-j\right|\le m-2$).

Theorem 6

If the graph $G=(X,E)$ is connected, then $q:E\rightarrow \mathbb {R}_{\ge 0}$ extends to a semimetric on X if and only if for all induced cycles p of G and all edges e in p

$$\begin{aligned} 2q(e)\le \mathrm {len}(p). \end{aligned}$$

(17)

Proof

By Guénoche et al. (2004), Proposition 2.1, q has a semimetric extension if and only if for all $\left\{ x,y\right\} \in E$ $q(\left\{ x,y\right\} )=d^q_G(x,y)$. $d^q_G$ was introduced in (3).

Let there be an extension of q to a semimetric. Fix an induced cycle $p=x_0x_1\dots x_{m-1} x_m$, $x_m=x_0$, and the edge $e=\left\{ x_0,x_1\right\}$ in p. We obtain

$$\begin{aligned} q(\left\{ x_0,x_1\right\} )= & {} d^q_G(x_0,x_1)\le \mathrm {len}(x_1\dots x_{m-1}x_0)= \sum _{k=1}^{m-1}q(\left\{ x_k,x_{k+1}\right\} )\\ 2q(\left\{ x_0,x_1\right\} )\le & {} q(\left\{ x_0,x_1\right\} )+ \sum _{k=1}^{m-1}q(\left\{ x_k,x_{k+1}\right\} )=\mathrm {len}(p). \end{aligned}$$

Now assume (17) is fulfilled, but there is no extension to a semimetric. Thus, we find $\left\{ x,y\right\} \in E$ such that $q(\left\{ x,y\right\} )>d^q_G(x,y)$. This means there is a path $\tilde{p}=x_0x_1\dots x_{m-1}$, $x_0=x$, $x_{m-1}=y$, such that

$$\begin{aligned} q(\left\{ x_0,x_{m-1}\right\} )>\mathrm {len}(\tilde{p})=\sum _{k=0}^{m-2}q(\left\{ x_k,x_{k+1}\right\} ). \end{aligned}$$

We may assume w.l.o.g. that m is minimal. Thus, $x_i$, $i=0,\dots ,m-1$ are different. Setting $x_m=x_0$, $e=\left\{ x,y\right\} =\left\{ x_0,x_{m-1}\right\}$, the (simple) cycle $p=x_0x_1\dots x_m$ violates (17). Suppose now that p has a chord, say $\left\{ x_i,x_j\right\}$. Since m is minimal, we know

$$\begin{aligned} q\left( \left\{ x_i,x_j\right\} \right) \le \sum _{k=i}^{j-1}q(\left\{ x_k,x_{k+1}\right\} ) \end{aligned}$$

and

$$\begin{aligned} q(\left\{ x_0,x_{m-1}\right\} )\le \sum _{k=0}^{i-1}q(\left\{ x_k,x_{k+1}\right\} )+q\left( \left\{ x_i,x_j\right\} \right) +\sum _{k=j}^{m-2}q(\left\{ x_k,x_{k+1}\right\} ). \end{aligned}$$

Substituting the first inequality into the right hand side of the second one yields

$$\begin{aligned} q(\left\{ x_0,x_{m-1}\right\} )\le \sum _{k=0}^{m-1}q(\left\{ x_k,x_{k+1}\right\} ). \end{aligned}$$

This contradiction shows that p is an induced cycle and completes the proof. $\square$

We can use this result for the

Proof of Theorem 2

We apply Theorem 6 to $X\cup X'$, $E=\left( {\begin{array}{c}X\\ 2\end{array}}\right) \cup \left( {\begin{array}{c}X'\\ 2\end{array}}\right) \cup \left\{ \left\{ x,x'\right\} :x\in X\right\}$ and $q:E\rightarrow \mathbb {R}_{\ge 0}$ given by

$$\begin{aligned} q(\left\{ u,v\right\} )=\left\{ { \begin{array}{cl} \rho (u,v)&{}\quad u,v\in X\\ \rho '(x,y)&{}\quad u=x',v=y', x,y\in X\\ \delta _x&{}\quad u=x,v=x', x\in X \end{array}}\right. . \end{aligned}$$

Induced cycles in $(X\cup X',E)$ are either triangles in X, triangles in $X'$ or quadrangles $x,y,y',x',x$. For the two former, (17) is equivalent to the triangle inequalities for $\rho ,\rho '$. For the latter, (17) is the same as (5). $\square$

The following result was used in the proof of Theorem 1.

Lemma 7

Suppose X, Y, Z are disjoint sets and there are given $d_1\in M(X\cup Y)$ and $d_2\in M(Y\cup Z)$ such that $d_1|_{\left( {\begin{array}{c}Y\\ 2\end{array}}\right) }=d_2|_{\left( {\begin{array}{c}Y\\ 2\end{array}}\right) }$. Then, there exists a $d\in M(X\cup Y\cup Z)$ such that $d|_{\left( {\begin{array}{c}X\cup Y\\ 2\end{array}}\right) }=d_1$ and $d|_{\left( {\begin{array}{c}Y\cup Z\\ 2\end{array}}\right) }=d_2$.

Proof

Now we apply the theorem to the graph $\left( X\cup Y\cup Z,\left( {\begin{array}{c}X\cup Y\\ 2\end{array}}\right) \cup \left( {\begin{array}{c}Y\cup Z\\ 2\end{array}}\right) \right)$ with

$$\begin{aligned} q(\left\{ u,v\right\} )=\left\{ {\begin{array}{ll} d_1(u,v)&{}\quad u,v\in X\cup Y\\ d_2(u,v)&{}\quad u,v\in Y\cup Z \end{array}}\right. . \end{aligned}$$

Since both $X\cup Y$ and $Y\cup Z$ are complete in this graph, the only induced cycles are triangles. The triangle inequalities for $d_1,d_2$ show (17). $\square$