The approach is further based on the ability to decide how far is each 
genome from each others. To achieve this, we combine XXX metrics which are 
detailed in this part.

\subsection{Core SNP based Metric} 
Due to the definition of the core genome, for each element $\dot{x}$ 
in this set, there is a gene $x \in \dot{x}$ in each genome. 
Let us consider a class 
$\dot{x}= \{y  | x \sim y\}$. 

\JFC{Il faudrait être cohérent: deux génomes proches devraient partout avoir 
soit une métrique élevée soit une métrique très faible}

%1/ On SNPs of the core genome strict
All the $y$ are thus aligned 
thanks to a global alignment tool. The SNPs may thus be extracted.
For each genome, one can thus compute the vector of Boolean values 
memorizing at index $i$ whether the SNP $i$ is present in one of its gene 
(positive value) or  not (null value). 
A Hamming distance between two vectors allows to build the distance 
between two genes. 
This metric is further referred as to $m_S$.

% plus il y a de diff, plus le nombre est élevé


%2/ On SNPs of the core genome strict, each gene having the same weight
The $m_S$ method does not consider genes to have the same incidence in the 
metric value. A gene with many SNPs has a larger influence in 
the metric computation than a gene with fewer ones. 
The metric further referred as to $m_{|S|}$ gives the same weight to each gene
without considering the number of SNP it contains. 

% plus il y a de diff, plus le nombre est élevé

\subsection{Symmetric Difference based Metric}
The third metric consider the symmetric difference $\Delta$ 
between the two sets $G_1$ and $G_2$ of genes recalled hereafter
$$
G_1\Delta G2 = 
(G1\cup G_2)\setminus (G1\cap G_2) = (G1\setminus G_2)\cup(G_2\setminus G1).
$$
The cardinality of $G_1\Delta G2$, give the metric.
This metric is furthered referred as to $m_{\Delta}$.

Practically, let $k$ be the number of all the equivalence classes. Due to the definition of the pan genome, this number is equal to the cardinality of this set.
For each genome, if we only consider which gene belongs into it \textit{i.e.}, if  we abstract away all the position this gene appears, this genome may be 
memorized as a vector of $k$ Boolean values. The element at index $i, 0 \le i \le k-1$ is true if and only if the $i$-th gene of the pan genome belongs to this 
one.  
This metric is equal to the Hamming distance between the two corresponding  
vectors of Boolean values.

% plus il y a de diff, plus le nombre est élevé



% 4/ Using EPFL method
\subsection{Adjacency based metric}
Following~\cite{23424133}, a sequence 
of all the adjacencies, which is present in 
a genomes at least is computed. This sequence
is augmented with the pan genome content.
Then, each genome is compared 
with such a sequence and a boolean vector is produced with the following rule. 
%If the element $i$ in the sequence of adjacencies or content is present 
%in the genome, the 






\subsection{Shared Synteny based Metric}
Given two genomes abstracted as sequences of classes, it is classical
to computes all the maximum shared synteny chains. 

% Attention ici, moins il y a de diff, plus le nombre est élevé
There are then three issues with such a set of shared synteny chains:
\begin{itemize}
\item let $m_{Y}$ be the metric, which returns the 
length of the largest chains;
\item let $m_{\overline{Y}}$ be the metric, which returns the 
average length of synteny chains;
\item finally, let $m_{|Y|}$ be the metric, which returns the 
number of synteny chains.
\end{itemize}
