feat: Add more algorithm on automata

content/chapters/include.tex

@@ -11,7 +11,7 @@
 		}
 }
 
-\includechapters{part1}{2}
+\includechapters{part1}{3}
 
 % \includechapters{part2}{2}
 

content/chapters/part1/2.tex

@@ -51,8 +51,8 @@ Let $S = $ ACGUUACGUU. Let's write the comparison matrix.
         \State \Return $M$
         \EndFunction
     \end{algorithmic}
-
 \end{algorithm}
+
 \begin{algorithm}
     \caption{Construct the top half of a comparison matrix}
     \begin{algorithmic}[1]
@@ -259,7 +259,7 @@ The suffix language of $S$ is $\{S, ACTACT, CTACT, TACT, ACT, CT, T\}$.
 \end{figure}
 
     \begin{algorithm}
-        \caption{Check if a sequences matches a motif, from a suffix automaton $\mathcal{O}(m)$}
+        \caption{Check if a sequences matches a motif, from a suffix automaton $\mathcal{O}(m)$, built from the automaton}
         \begin{algorithmic}[1]
             \Function{CheckMotifInSuffixAutomaton}{$W$: Array($m$), $A$: $\langle S, s_{0}, T, \Sigma,f \rangle$}
             \Returns{Boolean valued to \True{} if the motif is in the sequence}

content/chapters/part1/3.tex

@@ -0,0 +1,84 @@
+\chapter{Automata for motif search}
+
+Let $M$ be a motif $M = $ ACAT.
+
+\begin{figure}
+    \centering
+    \includegraphics{figures/part1/motif_search_automaton.pdf}
+    \caption{Motif search automaton for $M = $ ACAT}
+\end{figure}
+
+The alphabet of motif is the same as the alphabet of the sequence.
+The search automaton is complete.
+If the there exists a letter $c$ in the sequence that is not
+in the motif alphabet, we can make a virtual transition from 
+each state to the initial state whenever we encounter an unknown letter.
+
+\begin{algorithm}
+    \caption{Search a motif in a sequence with an automaton}
+    \begin{algorithmic}[1]
+        \Function{SearchMotif}{$S$: Array($n$), $A$: $\langle S, s_{0}, T, \Sigma, f \rangle$, $P$: Array($m$)}
+            \Returns{A set of positions where the motif has been found}
+            \State $s \gets s_0$
+            \State $i \gets 0$
+            \State $pos \gets \{\}$
+            \While {$i < n$} % $\exists f(s, S[i])$  We assume $S$ and $P$ are formed on the same alphabet, so we could remove the second check, as $A$ is complete
+                \If {$s \in T$}
+                    \State $pos \gets pos \cup \{ i - m \}$
+                \EndIf
+                \State $s \gets f(s, S[i])$
+                \State $i++$
+            \EndWhile
+            \State \Return $pos$
+        \EndFunction
+    \end{algorithmic}
+\end{algorithm}
+
+\begin{algorithm}
+    \caption{Check if the a motif automaton recognizes only the prefix of size $m-1$ of a motif $P$ of size $m$ }
+    \begin{algorithmic}[1]
+        \Function{SearchMotifLastPrefix}{$S$: Array($n$), $A$: $\langle S, s_{0}, T, \Sigma, f \rangle$, $P$: Array($m$)}
+            \Returns{A set of positions where the motif has been found}
+            \State $s \gets s_0$
+            \State $i \gets 0$
+            \State $T_{new} \gets \{\}$
+            \For {$s \in S$}
+                \For {$a \in \Sigma$}
+                    \For {$t \in T$}
+                        \If {$\exists f(s, a)$ and $f(s, a) = t$}
+                            \State $T_{new} \gets T_{new} \cup s$
+                        \EndIf
+                    \EndFor
+                \EndFor
+            \EndFor
+            \While {$i < n$}
+                \If {$s \in T_{new}$}
+                    \State \Return \True
+                \EndIf
+                \State $s \gets f(s, S[i])$
+                \State $i++$
+            \EndWhile
+            \State \Return \False 
+        \EndFunction
+    \end{algorithmic}
+\end{algorithm}
+
+\begin{algorithm}
+    \caption{Check if the a motif automaton recognizes only the prefix of size $m-1$ of a motif $P$ of size $m$, knowing the sequence of the motif}
+    \begin{algorithmic}[1]
+        \Function{SearchMotifLastPrefix}{$S$: Array($n$), $A$: $\langle S, s_{0}, T, \Sigma, f \rangle$, $P$: Array($m$)}
+        \Returns{A set of positions where the motif has been found}
+        \State $s \gets s_0$
+        \State $i \gets 0$
+        \While {$i < n$ and $f(s, P[m-1]) \notin T$}
+        \State $s \gets f(s, S[i])$
+        \State $i++$
+        \EndWhile
+        \If{$f(s, P[m-1]) \in T$} 
+        \State \Return \True
+        \Else
+        \State \Return \False 
+        \EndIf
+        \EndFunction
+    \end{algorithmic}
+\end{algorithm}

tmp.tex

@@ -13,4 +13,6 @@
 \begin{document}
 
 
+\end{document}
+
 \end{document}