fix: Needleman-Wunsch backtrack was faulty
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1953baa198
commit
80f4669d23
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@ -1,5 +1,5 @@
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build/
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**/.bak*
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**/*.bak*
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.auctex-auto
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## Core latex/pdflatex auxiliary files:
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@ -1,5 +1,6 @@
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\chapter{Sequence alignment}
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\iffalse
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\begin{algorithm}
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\caption{Needleman-Wunsch Algorithm}
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\begin{algorithmic}[1]
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@ -77,10 +78,12 @@
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\State \Call{FillMatrix}{$S_{1}$, $S_{2}$}
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\State \Call{ShowAlignment}{$S_{1}$, $S_{2}$}
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\end{algorithmic}
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\end{algorithm}
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\end{algorithm}
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\fi
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\begin{algorithm}
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\caption{Needleman-Wunsch Algorithm, using proper notation }
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\caption{Needleman-Wunsch Algorithm, Build the matrix}
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\begin{algorithmic}[1]
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\Procedure{FillMatrix}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
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\State $M = $ Array($m+1$, $n+1$)
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@ -106,7 +109,7 @@
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\end{algorithm}
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\begin{algorithm}
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\caption{Needleman-Wunsch Algorithm, using proper notation (Backtrack)}
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\caption{Needleman-Wunsch Algorithm, reconstruct the alignment}
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\begin{algorithmic}[1]
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\Procedure{BacktrackAlignment}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
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\State $alignment = LinkedList$
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@ -149,14 +152,10 @@
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\State $S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$),
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\Function{BacktrackRecurse}{$i$, $j$}
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\If {$i > 0$ and $j > 0$}
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\State $substitute = M[i-1][j-1]$
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\State $delete = M[i-1][j]$
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\State $insert = M[i][j-1]$
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\State $min = \min \{ substitute, delete, insert \}$
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\If {$substitute = min$}
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\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
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\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j-1$}
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\State $z = \begin{pmatrix} S_{1}[i-1] \\ S_{2}[j-1] \end{pmatrix} \circ z$
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\ElsIf {$delete = min$}
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\ElsIf {$M[i-1][j] + gap\_penalty = M[i][j]$}
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\State $z = $ \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$}
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\State $z = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix} \circ z$
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\Else
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@ -172,7 +171,9 @@
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\Else
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\State \Return []
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\EndIf
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\Else
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\State \Return $z$
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\EndIf
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\EndFunction
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\Function{Backtrack}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$)}
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\State \Return \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $m$, $n$}
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@ -185,21 +186,17 @@
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\begin{algorithmic}[1]
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\Procedure{BacktrackRecurse}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$), $i$, $j$}
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\If {$i > 0$ and $j > 0$}
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\State $substitute = M[i-1][j-1]$
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\State $delete = M[i-1][j]$
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\State $insert = M[i][j-1]$
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\State $min = \min \{ substitute, delete, insert \}$
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\If {$substitute = min$}
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\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
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\State $value = \begin{pmatrix} S_{1}[i-1] \\ S_{2}[j-1] \end{pmatrix}$
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\State $z' = value \circ z$
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\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j-1$, $z'$}
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\EndIf
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\If {$delete = min$}
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\If {$M[i-1][j] + gap\_penalty = M[i][j]$}
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\State $value = \begin{pmatrix} S_{1}[i-1] \\ \varepsilon \end{pmatrix}$
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\State $z' = value \circ z$
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\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i-1$, $j$, $z'$}
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\EndIf
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\If {$insert = min$}
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\If {$M[i][j-1] + gap\_penalty = M[i][j]$}
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\State $value = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
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\State $z' = value \circ z$
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\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$, $z'$}
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@ -212,11 +209,20 @@
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\State $value = \begin{pmatrix} \varepsilon \\ S_{2}[j-1] \end{pmatrix}$
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\State $z' = value \circ z$
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\State \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$, $z'$}
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\EndIf
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\Else
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\State \Call{print}{$z$}
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\EndIf
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\EndProcedure
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\Procedure{Backtrack}{$S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$)}
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\State \Return \Call{BacktrackRecurse}{$S_{1}$, $S_{2}$, $M$, $m$, $n$, []}
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\EndProcedure
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\end{algorithmic}
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\end{algorithm}
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\begin{figure}
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\centering
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\includegraphics{figures/part2/needle.pdf}
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\caption{Needleman-Wunsch global alignment matrix with an example of optimal path.}
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\end{figure}
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@ -1,22 +1,105 @@
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needle = require("./needle")
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function table.shallow_copy(t)
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local t2 = {}
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for k,v in pairs(t) do
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t2[k] = v
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end
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return t2
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end
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function multiple_path_backtrack_trace(matrix, seq1, seq2)
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local stack = {}
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local m=string.len(seq1)
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local n=string.len(seq2)
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local i=m
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local j=n
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table.insert(1, {i, j, nil})
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table.insert(stack, 1, {i, j, {}})
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local trace = {}
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while #stack ~= 0 do
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local state = table.remove(stack, 1)
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table.insert(trace, #trace+1, state)
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local i=state[1]
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local j=state[2]
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local alignment = state[3]
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if (i > 0 and j > 0) then
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local nt1 = string.sub(seq1, i-1, i-1)
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local nt2 = string.sub(seq2, j-1, j-1)
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if (matrix[i][j] == matrix[i-1][j-1] + needle.sub(nt1, nt2)) then
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local new_alignment = table.shallow_copy(alignment)
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table.insert(new_alignment, 1, {nt1, nt2})
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table.insert(stack, 1, {i - 1, j - 1, new_alignment})
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end
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if (matrix[i][j] == matrix[i-1][j] + needle.gap_penalty) then
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local new_alignment = table.shallow_copy(alignment)
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table.insert(new_alignment, 1, {nt1, '-'})
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table.insert(stack, 1, {i-1, j, new_alignment})
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end
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if (matrix[i][j] == matrix[i][j-1] + needle.gap_penalty) then
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local new_alignment = table.shallow_copy(alignment)
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table.insert(new_alignment, 1, {'-', nt2})
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table.insert(stack, 1, {i, j-1, new_alignment})
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end
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end
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if (i > 0) then
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local nt1 = string.sub(seq1, i-1, i-1)
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local new_alignment = table.shallow_copy(alignment)
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table.insert(new_alignment, 1, {nt1, '-'})
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table.insert(stack, 1, {i-1, j, new_alignment})
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end
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if (j > 0) then
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local nt2 = string.sub(seq2, j-1, j-1)
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local new_alignment = table.shallow_copy(alignment)
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table.insert(new_alignment, 1, {'-', nt2})
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table.insert(stack, 1, {i, j-1, new_alignment})
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end
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end
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return trace
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end
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function repr_alignment(alignment)
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local repr = [[\begin{pmatrix}]]
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for i, vector in ipairs(alignment) do
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repr = repr .. vector[1]
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if i < #alignment then
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repr = repr .. " & "
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end
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end
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repr = repr .. [[\\]] .. " \n"
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for i, vector in ipairs(alignment) do
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repr = repr .. vector[2]
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if i < #alignment then
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repr = repr .. " & "
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end
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end
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repr = repr .. [[\end{pmatrix}]]
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return repr
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end
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function trace_repr(trace)
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local repr = ""
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-- for stack_index, stack in ipairs(trace) do
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-- repr = repr .. "iteration " .. stack_index .. " :" .. [[\\]]
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repr = repr .. [[\begin{tabular}{|c|} \\ \hline ]]
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for call_index, call in ipairs(trace) do
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local i = call[1]
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local j = call[2]
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local aligment = call[3]
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repr = repr .. [[ $\langle ]] .. i ..", " .. j .. ", " .. repr_alignment(alignment).. [[\rangle$ ]]
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repr = repr .. [[\\ \hline]]
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end
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repr = repr .. [[\end{tabular}]]
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-- end
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return repr
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end
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function main()
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local seq1 = "ATCTGAT"
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local seq2 = "TGCATA"
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local matrix = needle.needle_matrix(seq1, seq2)
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local trace = multiple_path_backtrack_trace(matrix, seq1, seq2)
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print(#trace)
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print(trace_repr(trace))
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end
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main()
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@ -191,5 +191,7 @@ return {
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draw=draw_needle_matrix_graph,
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gap_penalty=gap_penalty,
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mismatch_penalty=mismatch_penalty,
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match_penalty=match_penalty
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match_penalty=match_penalty,
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needle_matrix=needle_matrix,
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sub=sub
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}
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