\chapter{Sequence alignment}

\iffalse
\begin{algorithm}
	\caption{Needleman-Wunsch Algorithm}
	\begin{algorithmic}[1]
		\Procedure{FillMatrix}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
		\Comment{$sub(a, b)$ is the substitution score, $del(a)$ and $ins(a)$ are the deletion and insertion penalty, in regard with the reference $S_{1}$ sequence}
		\State $M = $ Array($m+1$, $n+1$)
		\Comment{Initialize the matrix first column and first row}
		\State $P = $ Array($m$, $n$) \Comment{Store the direction of the cell we chose to build the next cell up on.}
		\State $M[0][0] = 0$
		\For {($i = 1$; $i < m+1$; $i++$)}
		\State $M[i][0] = M[i-1][0] + gap\_penalty$
		\EndFor
		\For {($j = 1$; $j < n+1$; $j++$)}
		\State $M[0][j] = M[0][j-1] + gap\_penalty$
		\EndFor
		\Comment{Fill the remaining matrix}
		\For {($i = 1$; $i < m+1$; $i++$)}
		\For {($j = 1$; $j < n+1$; $j++$)}
		\State $delete = M[i-1][j] + gap\_penalty$
		\State $insert = M[i][j-1] + gap\_penalty$
		\State $substitute = M[i-1][j-1] + sub(S_{1}[i-1], S_{2}[j-1])$
		\State $choice = \min \{delete, insert, substitute\}$
		\If {$substitute = choice$}
		\State $P[i-1][j-1] = '\nwarrow'$
		\ElsIf {$deletion = choice$}
		\State $P[i-1][j-1] = '\leftarrow'$
		\Else
		\State $P[i-1][j-1] = '\uparrow'$
		\EndIf
		\State $M[i][j] = choice$
		\EndFor
		\EndFor
		\EndProcedure
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}
	\caption{Needleman-Wunsch Algorithm (Backtrack)}
	\begin{algorithmic}[1]
		\Procedure{ShowAlignment}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
		\State $extend_{1} = ''$
		\State $extend_{2} = ''$
		\State $i = m$
		\State $j = n$
		\While{$i > 0$ and $j > 0$}
		\If {$P[i-1][j-1] = '\nwarrow'$}
		\State $extend_{1} = S_{1}[i-1] \circ extend_{1}$
		\State $extend_{2} = S_{2}[j-1] \circ extend_{2}$
		\State $i--$
		\State $j--$
		\ElsIf {$P[i-1][j-1] = '\uparrow'$}
		\State $extend_{1} = S_{1}[i-1] \circ extend_{1}$
		\State $extend_{2} = '-' \circ extend_{2}$
		\State $i--$
		\Else
		\State $extend_{1} = '-' \circ extend_{1}$
		\State $extend_{2} = S_{2}[j-1] \circ extend_{2}$
		\State $j--$
		\EndIf
		\EndWhile
		\While{$i > 0$}
		\State $extend_{1} = S_{1}[i-1] \circ extend_{1}$
		\State $extend_{2} = '-' \circ extend_{2}$
		\State $i--$
		\State \Call{Insert}{0, $alignment$,$tuple$}
		\EndWhile
		\While{$j > 0$}
		\State $extend_{1} = '-' \circ extend_{1}$
		\State $extend_{2} = S_{2}[j-1] \circ extend_{2}$
		\State $j--$
		\EndWhile
		\State \Call{print}{$extend_{1}$}
		\State \Call{print}{$extend_{2}$}
		\EndProcedure
		\State \Call{FillMatrix}{$S_{1}$, $S_{2}$}
		\State \Call{ShowAlignment}{$S_{1}$, $S_{2}$}
	\end{algorithmic}
      \end{algorithm}

      \fi

\begin{algorithm}
	\caption{Needleman-Wunsch Algorithm, Build the matrix}
	\begin{algorithmic}[1]
		\Procedure{FillMatrix}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
		\State $M = $ Array($m+1$, $n+1$)
		\State $P = $ Array($m$, $n$)
		\Comment{Store the direction of the cell we chose to build the next cell up on.}
		\State $M[0][0] = 0$
		\For {($i = 1$; $i < m+1$; $i++$)}
		\State $M[i][0] = M[i-1][0] + gap\_penalty$
		\EndFor
		\For {($j = 1$; $j < n+1$; $j++$)}
		\State $M[0][j] = M[0][j-1] + gap\_penalty$
		\EndFor
		\For {($i = 1$; $i < m+1$; $i++$)}
		\For {($j = 1$; $j < n+1$; $j++$)}
		\State $delete = M[i-1][j] + gap\_penalty$
		\State $insert = M[i][j-1] + gap\_penalty$
		\State $substitute = M[i-1][j-1] + sub(S_{1}[i-1], S_{2}[j-1])$
		\State $M[i][j] = \min \{substitute, insert, delete\}$
		\EndFor
		\EndFor
		\EndProcedure
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}
	\caption{Needleman-Wunsch Algorithm, reconstruct the alignment}
	\begin{algorithmic}[1]
		\Procedure{BacktrackAlignment}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
		\State $alignment = LinkedList$
		\State $i = m$
		\State $j = n$
		\While{$i > 0$ and $j > 0$}
		\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
		\State $tuple = \begin{pmatrix} S_{1}[i-1] \\ S_{2}[j-1] \end{pmatrix}$
		\State $i--$
		\State $j--$
		\ElsIf {$M[i-1][j-1] = M[i][j-1] - gap\_penalty$}
		\State $tuple = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix}$
		\State $i--$
		\Else
		\State $tuple = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
		\State $j--$
		\EndIf
		\State \Call{Insert}{0, $alignment$,$tuple$}
		\EndWhile
		\While{$i > 0$}
		\State $tuple = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix}$
		\State $i--$
		\State \Call{Insert}{0, $alignment$,$tuple$}
		\EndWhile
		\While{$j > 0$}
		\State $tuple = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
		\State $j--$
		\State \Call{Insert}{0, $alignment$,$tuple$}
		\EndWhile
		\EndProcedure
		\State \Call{FillMatrix}{$S_{1}$, $S_{2}$}
		\State \Call{BacktrackAlignment}{$S_{1}$, $S_{2}$}
	\end{algorithmic}
\end{algorithm}


\begin{algorithm}
	\caption{Backtrack a single alignment in a recursive way}
	\begin{algorithmic}[1]
		\State $S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$),
		\Function{BacktrackRecurse}{$i$, $j$}
		\If {$i > 0$ and $j > 0$}
		\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
		\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j-1$}
		\State $z = \begin{pmatrix} S_{1}[i-1] \\ S_{2}[j-1] \end{pmatrix} \circ z$
		\ElsIf {$M[i-1][j] + gap\_penalty = M[i][j]$}
		\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$}
		\State $z = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix} \circ z$
		\Else
		\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$}
		\State $z = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix} \circ z$
		\EndIf
		\ElsIf {$i > 0$}
		\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$}
		\State $z = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix} \circ z$
		\ElsIf {$j > 0$}
		\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$}
		\State $z = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix} \circ z$
		\Else
		\State \Return []
		\EndIf
		\Else
		\State \Return $z$
		\EndIf
		\EndFunction
		\Function{Backtrack}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$)}
		\State \Return \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $m$, $n$}
		\EndFunction
	\end{algorithmic}
\end{algorithm}

\begin{algorithm}
	\caption{Backtrack all the optimum alignments in a recursive way}
	\begin{algorithmic}[1]
		\Procedure{BacktrackRecurse}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$), $i$, $j$}
		\If {$i > 0$ and $j > 0$}
		\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
		\State $value = \begin{pmatrix} S_{1}[i-1] \\ S_{2}[j-1] \end{pmatrix}$
		\State $z' = value \circ z$
		\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j-1$, $z'$}
		\EndIf
		\If {$M[i-1][j] + gap\_penalty = M[i][j]$}
		\State $value = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix}$
		\State $z' = value \circ z$
		\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$, $z'$}
		\EndIf
		\If {$M[i][j-1] + gap\_penalty = M[i][j]$}
		\State $value = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
		\State $z' = value \circ z$
		\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$, $z'$}
		\EndIf
		\ElsIf {$i > 0$}
		\State $value = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix}$
		\State $z' = value \circ z$
		\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$, $z'$}
		\ElsIf {$j > 0$}
		\State $value = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
		\State $z' = value \circ z$
		\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$, $z'$}
		\Else
		\State \Call{print}{$z$}
		\EndIf
		\EndProcedure
		\Procedure{Backtrack}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$)}
		\State \Return \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $m$, $n$, []}
		\EndProcedure
	\end{algorithmic}
\end{algorithm}



\begin{figure}
  \centering
  \includegraphics{figures/part2/needle.pdf}
  \caption{Needleman-Wunsch global alignment matrix with an example of optimal path.}
 \end{figure}