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