114 lines
3.1 KiB
TeX
114 lines
3.1 KiB
TeX
\chapter{Longest common subsequence}
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Let $S_{1} = \text{ATCTGAT}$ and $S_{2} = \text{TGCATA}$.
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In this case the longest common subsequence of $S_{1}$ and $S_{2}$ is $TCTA$.
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\begin{algorithm}
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\caption{Construct a longest common subsequence matrix}
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\begin{algorithmic}[1]
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\Function{LCSQ\_Matrix}{$S_{1}$: Array($n$), $S_{2}$: Array($m$)}
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\State $M \gets $ Array($m+1$, $n+1$)
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\For{($i = 0$; $i < n+1$; $i++$)}
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\For{$j = 0$; $j < m+1$; $j++$}
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\If {$i = 0$ or $j = 0$}
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\State $M[i][j] = 0$
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\Else
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\If {$S_{1}[i] = S_{2}[j]$}
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\State $match = M[i-1][j-1] + 1$
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\Else
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\State $match = M[i-1][j-1]$
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\EndIf
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\State $gap_{1} = M[i-1][j]$
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\State $gap_{2} = M[i][j-1]$
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\State $M[i][j] = \max \{ match, gap_{1}, gap_{2}\}$
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\EndIf
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\EndFor
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\EndFor
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\State \Return $M$
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\EndFunction
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\end{algorithmic}
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\end{algorithm}
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\begin{algorithm}
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\caption{Construct a longest common subsequence matrix keeping the path in memory}
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\begin{algorithmic}[1]
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\Function{LCSQ\_Matrix\_Path}{$S_{1}$: Array($n$), $S_{2}$: Array($m$)}
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\State $M \gets $ Array($m+1$, $n+1$)
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\State $P \gets $ Array($m+1$, $n+1$)
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\For {($i = 0$; $i < n+1$, $i++$)}
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\State $M[i][0] \gets 0$
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\EndFor
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\For {($j = 0$; $j < m+1$; $j+$)}
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\State $M[0][j] \gets 0$
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\EndFor
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\For{($i = 1$; $i < n+1$; $i++$)}
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\For{($j = 1$; $j < m+1$; $j++$)}
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\If {$i = 1$ or $j = 0$}
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\State $M[i][j] = 0$
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\Else
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\If {$S_{1}[i-1] = S_{2}[j-1]$}
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\State $M[i][j] \gets M[i-1][j-1] + 1$
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\State $P[i][j] \gets '\nwarrow'$
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\ElsIf {$M[i][j-1] \geq M[i-1][j]$}
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\State $M[i][j] \gets M[i][j-1]$
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\State $P[i][j] \gets '\leftarrow'$
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\Else
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\State $M[i][j] \gets M[i-1][j]$
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\State $P[i][j] \gets '\downarrow'$
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\EndIf
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\EndFor
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\EndFor
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\State \Return $M, P$
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\EndFunction
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\end{algorithmic}
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\end{algorithm}
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\begin{algorithm}
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\caption{Backtrack the longest common subsequence}
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\begin{algorithmic}[1]
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\Function{LCSQ}{$S_{1}$: Array($n$), $S_{2}$: Array($m$)}
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\State $M, P \gets $ \Call{LCSQ\_Matrix}{$S_{1}$, $S_{2}$}
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\State $L \gets Array(M[n][m])$
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\State $k \gets 0$
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\State $i \gets n$
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\State $j \gets m$
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\While{$i > 0$ and $j > 0$}
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\If {$P[i][j] = '\nwarrow' $}
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\State $L[k] \gets S_{1}[i]$
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\State $i--$
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\State $j--$
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\State $k++$
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\ElsIf {$P[i][j] = '\leftarrow'$}
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\State $j--$
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\Else
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\State $i--$
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\EndIf
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\EndWhile
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\State \Return $L$
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\EndFunction
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\end{algorithmic}
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\end{algorithm}
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\begin{algorithm}
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\caption{Recursive reconstruction of the longest common subsequence}
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\begin{algorithmic}[1]
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\Procedure{LCSQ}{$S_{1}$: Array($n$), $S_{2}$: Array($m$)}
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\State $M, P \gets $ \Call{LCSQ\_Matrix}{$S_{1}$, $S_{2}$}
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\State $i \gets n$
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\State $j \gets m$
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\State \Call{Aux}{$P$, $S_{1}$, $i$, $j$}
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\EndProcedure
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\Procedure{Aux}{$P$: Array($n+1$, $m+1$), $S_{1}$: Array($n$), $i$, $j$}
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\If {$P[i][j] = '\nwarrow' $}
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\State $l \gets S_{1}[i]$
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\State \Call{Aux}{$P$, $S_{1}$, $i-1$, $j-1$}
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\State \texttt{print}($l$)
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\ElsIf {$P[i][j] = '\leftarrow'$}
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\State \Call{Aux}{$P$, $S_{1}$, $i$, $j-1$}
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\Else
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\State \Call{Aux}{$P$, $S_{1}$, $i-1$, $j$}
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\EndIf
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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