sequence-algorithms/content/chapters/part1/3.tex

\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}