fix: Definition on automata

Co-authored-by: Naïa Périnelle <naia.perinelle@laposte.net>

.gitignore

@@ -1,5 +1,5 @@
 build/
-**/.bak*
+**/*.bak*
 .auctex-auto
 
 ## Core latex/pdflatex auxiliary files:

content/chapters/part1/1.tex

@@ -279,11 +279,11 @@ An automaton is a tuple $\langle S, s_{0}, T, \Sigma,f\rangle$
 \paragraph{Example} Given the language $L$ on the alphabet $\Sigma = \{A, C, T\}$, $L = \{A^{*}, CTT, CA^{*}\}$
 
 \begin{definition}[Deterministic automaton]
-    An automaton is deterministic, if for each couple $(p, a) \in S \times \Sigma$ it exists at most a state $q$ such as $f(p, q) = q$
+    An automaton is deterministic, if for each couple $(p, a) \in S \times \Sigma$ it exists at most a state $q$ such as $f(p, a) = q$
 \end{definition}
 
 \begin{definition}[Complete automaton]
-    An automaton is complete, if for each couple $(p, a) \in S \times \Sigma$ it exists at least a state $q$ such as $f(p, q) = q$.
+    An automaton is complete, if for each couple $(p, a) \in S \times \Sigma$ it exists at least a state $q$ such as $f(p, a) = q$.
 \end{definition}
 
 \begin{algorithm}

content/chapters/part2/1.tex

@@ -1,5 +1,6 @@
 \chapter{Sequence alignment}
 
+\iffalse
 \begin{algorithm}
 	\caption{Needleman-Wunsch Algorithm}
 	\begin{algorithmic}[1]
@@ -79,8 +80,10 @@
 	\end{algorithmic}
       \end{algorithm}
 
+      \fi
+
 \begin{algorithm}
-	\caption{Needleman-Wunsch Algorithm, using proper notation }
+	\caption{Needleman-Wunsch Algorithm, Build the matrix}
 	\begin{algorithmic}[1]
 		\Procedure{FillMatrix}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
 		\State $M = $ Array($m+1$, $n+1$)
@@ -106,7 +109,7 @@
 \end{algorithm}
 
 \begin{algorithm}
-	\caption{Needleman-Wunsch Algorithm, using proper notation (Backtrack)}
+	\caption{Needleman-Wunsch Algorithm, reconstruct the alignment}
 	\begin{algorithmic}[1]
 		\Procedure{BacktrackAlignment}{$S_{1}$: Array($m$), $S_{2}$: Array($n$)}
 		\State $alignment = LinkedList$
@@ -149,14 +152,10 @@
 		\State $S_{1}$: Array($m$), $S_{2}$: Array($n$), $M$: Array($m+1$, $n+1$),
 		\Function{BacktrackRecurse}{$i$, $j$}
 		\If {$i > 0$ and $j > 0$}
-		\State $substitute = M[i-1][j-1]$
+		\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
-		\State $delete = M[i-1][j]$
-		\State $insert = M[i][j-1]$
-		\State $min = \min \{ substitute, delete, insert \}$
-		\If {$substitute = min$}
 		\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 {$delete = min$}
+		\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
@@ -172,7 +171,9 @@
 		\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}{$S_{1}$, $S_{2}$, $M$, $m$, $n$}
@@ -185,21 +186,17 @@
 	\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 $j > 0$}
-		\State $substitute = M[i-1][j-1]$
+		\If {$M[i-1][j-1] = M[i][j] - sub(S_{1}[i-1], S_{2}[j-1])$}
-		\State $delete = M[i-1][j]$
-		\State $insert = M[i][j-1]$
-		\State $min = \min \{ substitute, delete, insert \}$
-		\If {$substitute = min$}
 		\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 {$delete = min$}
+		\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 {$insert = min$}
+		\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}{$S_{1}$, $S_{2}$, $M$, $i$, $j-1$, $z'$}
@@ -212,11 +209,20 @@
 		\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
+		\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}

figures/part2/needle-backtrack-stack.lua

@@ -0,0 +1,105 @@
+needle = require("./needle")
+
+function table.shallow_copy(t)
+  local t2 = {}
+  for k,v in pairs(t) do
+    t2[k] = v
+  end
+  return t2
+end
+
+function multiple_path_backtrack_trace(matrix, seq1, seq2)
+    local stack = {}
+    local m=string.len(seq1)
+    local n=string.len(seq2)
+    local i=m
+    local j=n
+    table.insert(stack, 1, {i, j, {}})
+    local trace = {}
+    while #stack ~= 0 do
+        local state = table.remove(stack, 1)
+        table.insert(trace, #trace+1, state)
+        local i=state[1]
+        local j=state[2]
+        local alignment = state[3]
+        if (i > 0 and j > 0) then
+            local nt1 = string.sub(seq1, i-1, i-1)
+            local nt2 = string.sub(seq2, j-1, j-1)
+            if (matrix[i][j] == matrix[i-1][j-1] + needle.sub(nt1, nt2)) then
+                local new_alignment = table.shallow_copy(alignment)
+                table.insert(new_alignment, 1, {nt1, nt2})
+                table.insert(stack, 1, {i - 1, j - 1, new_alignment})
+            end
+            if (matrix[i][j] == matrix[i-1][j] + needle.gap_penalty) then
+                local new_alignment = table.shallow_copy(alignment)
+                table.insert(new_alignment, 1, {nt1, '-'})
+                table.insert(stack, 1, {i-1, j, new_alignment})
+            end
+            if (matrix[i][j] == matrix[i][j-1] + needle.gap_penalty) then
+                local new_alignment = table.shallow_copy(alignment)
+                table.insert(new_alignment, 1, {'-', nt2})
+                table.insert(stack, 1, {i, j-1, new_alignment})
+            end
+        end
+        if (i > 0) then
+            local nt1 = string.sub(seq1, i-1, i-1)
+            local new_alignment = table.shallow_copy(alignment)
+            table.insert(new_alignment, 1, {nt1, '-'})
+            table.insert(stack, 1, {i-1, j, new_alignment})
+        end
+        if (j > 0) then
+            local nt2 = string.sub(seq2, j-1, j-1)
+            local new_alignment = table.shallow_copy(alignment)
+            table.insert(new_alignment, 1, {'-', nt2})
+            table.insert(stack, 1, {i, j-1, new_alignment})
+        end
+    end
+    return trace
+end
+
+function repr_alignment(alignment)
+    local repr = [[\begin{pmatrix}]]
+    for i, vector in ipairs(alignment) do
+        repr = repr .. vector[1]
+        if i < #alignment then
+            repr = repr .. " & "
+        end
+    end
+    repr = repr .. [[\\]] .. " \n"
+    for i, vector in ipairs(alignment) do
+        repr = repr .. vector[2]
+        if i < #alignment then
+            repr = repr .. " & "
+        end
+    end
+    repr = repr .. [[\end{pmatrix}]]
+    return repr
+end
+
+function trace_repr(trace)
+    local repr = ""
+--     for stack_index, stack in ipairs(trace) do
+--         repr = repr .. "iteration " .. stack_index .. " :" .. [[\\]]
+        repr = repr .. [[\begin{tabular}{|c|} \\ \hline ]]
+        for call_index, call in ipairs(trace) do
+            local i = call[1]
+            local j = call[2]
+            local aligment = call[3]
+            repr = repr .. [[ $\langle ]] .. i ..", " .. j .. ", " .. repr_alignment(alignment).. [[\rangle$ ]]
+            repr = repr .. [[\\ \hline]]
+        end
+        repr = repr .. [[\end{tabular}]]
+--     end
+    return repr
+end
+
+function main()
+    local seq1 = "ATCTGAT"
+    local seq2 = "TGCATA"
+    local matrix = needle.needle_matrix(seq1, seq2)
+    local trace = multiple_path_backtrack_trace(matrix, seq1, seq2)
+    print(#trace)
+    print(trace_repr(trace))
+end
+
+main()

figures/part2/needle.lua

@@ -0,0 +1,197 @@
+gap_penalty = 1
+mismatch_penalty = 1
+match_penalty = 0
+
+function needle_matrix(seq1, seq2)
+
+    local n1 = string.len(seq1)
+    local n2 = string.len(seq2)
+    -- Create a n1 x n2 matrix
+    local matrix = {}
+    for i=0,n1 do
+        matrix[i] = {}
+        for j=0,n2 do
+            matrix[i][j] = 0
+        end
+    end
+    -- Fill first row and first column
+    for i=1,n1 do
+        matrix[i][0] = i * gap_penalty
+    end
+    for i=1,n2 do
+        matrix[0][i] = i * gap_penalty
+    end
+    -- Fill the rest of the matrix
+    local match, delete, insert
+    for i=1,n1 do
+        for j=1,n2 do
+            if string.sub(seq1, i, i) == string.sub(seq2, j, j) then
+                match = matrix[i-1][j-1] + match_penalty
+            else
+                match = matrix[i-1][j-1] + mismatch_penalty
+            end
+            delete = matrix[i-1][j] + gap_penalty
+            insert = matrix[i][j-1] + gap_penalty
+            matrix[i][j] = math.min(match, delete, insert)
+        end
+    end
+    return matrix
+end
+
+function draw_needle_matrix(seq1, seq2)
+    -- tex.print(string.format(" Path: %s -> %s", seq1, seq2))
+    matrix = needle_matrix(seq1, seq2)
+    n1 = string.len(seq1)
+    n2 = string.len(seq2)
+    -- Draw the matrix as tikz nodes
+    for i=0,n1-1 do
+        for j=0,n2-1 do
+            tex.print(string.format("\\node[draw, minimum width=1cm, minimum height=1cm] at (%d, -%d) {};", i, j, matrix[i][j]))
+        end
+    end
+    -- Draw the sequence labels
+    for i=1,n1 do
+        tex.print(string.format("\\node at (%d, -%d) {%s};", i-1, -1, string.sub(seq1, i, i)))
+    end
+    for i=1,n2 do
+        tex.print(string.format("\\node at (%d, -%d) {%s};", -1, i-1, string.sub(seq2, i, i)))
+    end
+    -- Add a path from the bottom right corner to the top left corner, following the minimum of the three possible moves at each step
+    local i, j, value, previous_value
+    i = n1-1
+    j = n2-1
+    tex.print(string.format("\\draw[-,line width=2, gray] (%d, -%d) --", i, j))
+    while i > 0 and j > 0 do
+        value = math.min(matrix[i-1][j-1], table[i-1][j], table[i][j-1])
+        if value == matrix[i-1][j-1] then
+            i = i - 1
+            j = j - 1
+        elseif value == matrix[i-1][j] then
+            i = i - 1
+        else
+            j = j - 1
+        end
+        tex.print(string.format(" (%d, -%d) -- ", i, j))
+    end
+    tex.print(string.format("(0, 0) -- (-1, 1);", i, j))
+end
+
+local function has_value (tab, val)
+    for index, value in ipairs(tab) do
+        if value == val then
+            return true
+        end
+    end
+
+    return false
+end
+
+function sub(a, b)
+    if (a==b) then
+        return match_penalty
+    else
+        return mismatch_penalty
+    end
+end
+
+-- Returns true if it could have passed from k, l, to i, j
+-- during dynamic programming matrix building
+function check_path(matrix, i, j, k, l, seq1, seq2)
+    -- diagonal
+    if ((i == k + 1) and (j == l + 1)) then
+        if (matrix[i][j] == matrix[k][l] + sub(string.sub(seq1, i, i), string.sub(seq2, j, j))) then
+            return true
+        end
+    elseif (matrix[i][j] == matrix[k][l] + 1) then
+        return true
+    end
+    return false
+end
+
+function draw_needle_matrix_graph(seq1, seq2)
+    local matrix = needle_matrix(seq1, seq2)
+    local tikz_code = ""
+    function coordinate(i, j)
+        return i .. "_" .. j
+    end
+    local steps = {
+        {-1, -1},
+        {-1, 0},
+        {0, -1}
+    }
+
+    local n1 = string.len(seq1)
+    local n2 = string.len(seq2)
+    local path = {}
+    local i = n1
+    local j = n2
+    while i >= 0 and j >= 0 do
+        path[#path+1] = coordinate(i, j)
+        local min = matrix[i][j]
+        local min_step = steps[1]
+        for index, step in ipairs(steps) do
+            local k = i + step[1]
+            local l = j + step[2]
+            if k >= 0 and l >= 0 and check_path(matrix, i, j, k, l, seq1, seq2) then
+                min_step = step
+                min = matrix[k][l]
+                break
+            end
+        end
+        i = i + min_step[1]
+        j = j + min_step[2]
+--         print(i, j)
+    end
+    -- Draw the matrix as tikz node with matrix value
+    for i=0,n1 do
+        for j=0,n2 do
+            local options = ""
+            if has_value(path, coordinate(i, j)) then
+                options = "[fill=gray, draw, minimum size=1]"
+            end
+            tikz_code = tikz_code .. "\\node" .. options .. " (" .. coordinate(i, j) .. ") at (" .. i .. ", " .. -j .. ")" .. " {" .. matrix[i][j] .. "};"
+        end
+    end
+    -- Add nucleotide labels
+    for i=1,n1 do
+        local nt = string.sub(seq1, i, i)
+        tikz_code = tikz_code .. "\\node at (".. i .. "," .. 1 .. ")" .. "{$" .. nt .."$};"
+    end
+    for i=1,n2 do
+        local nt = string.sub(seq2, i, i)
+        tikz_code = tikz_code .. "\\node at (" .. -1 .. ", " .. -i .. ")" .. "{$ ".. nt .."$};"
+    end
+    -- For seq2
+       for i=0,n1 do
+        for j=0,n2 do
+            local min = math.huge
+            for index, step in ipairs(steps) do
+                local k = i + step[1]
+                local l = j + step[2]
+                if k >= 0 and l >= 0 and matrix[k][l] < min then
+                    min = matrix[k][l]
+                end
+            end
+--             local highlighted = false
+            for index, step in ipairs(steps) do
+                local k = i + step[1]
+                local l = j + step[2]
+                if k >= 0 and l >= 0 and check_path(matrix, i, j, k, l, seq1, seq2) then
+                    tikz_code = tikz_code .. "\\draw[->] (" .. coordinate(i, j) .. ")" .. " -- " .. "(" .. coordinate (k, l) .. ");"
+                end
+            end
+        end
+    end
+    return tikz_code
+end
+
+-- print(draw_needle_matrix_graph("ATGC", "TAGCGA"))
+
+return {
+  draw=draw_needle_matrix_graph,
+  gap_penalty=gap_penalty,
+  mismatch_penalty=mismatch_penalty,
+  match_penalty=match_penalty,
+  needle_matrix=needle_matrix,
+  sub=sub
+}

figures/part2/needle.tex

@@ -0,0 +1,20 @@
+\documentclass[tikz]{standalone}
+
+\usepackage{luacode}
+\usepackage{tikz}
+
+\begin{document}
+
+\begin{tikzpicture}
+  \newcommand{\seqone}{TGCATA}
+  \newcommand{\seqtwo}{ATCTGAT}
+  \begin{luacode}
+    local needle = require("needle")
+    seq1 = \luastring{\seqone}
+    seq2 = \luastring{\seqtwo}
+    local tikz_code = needle.draw(seq1, seq2)
+    tex.print(tikz_code)
+  \end{luacode}
+\end{tikzpicture}
+
+\end{document}