\chapter{Matrices}

Let $S_{1}$ and $S_{2}$ be two sequences.

$S_{1} = $ ACGUUCC
$S_{2} = $ GUU

\begin{table}
    \centering
    \begin{tabular}{c|ccccccc}
          & A & C & G & U & U & C & C \\
        \hline
        G & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
        U & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
        U & 0 & 0 & 0 & 1 & 1 & 0 & 0
    \end{tabular}
    \caption{Comparison matrix}
\end{table}


Let $n = |S_{1}|$, $m = |S_{2}|$
The complexity of this algorithm is $\mathcal{O}(n \cdot m)$ to build the matrix, and it requires also to find the diagonals and thus it is a bit less efficient than the \autoref{alg:naive-motif-matching}.


To find repetitions, we can use a comparison matrix with a single sequence against itself. A repetition would appear as a diagonal of ones, not on the main diagonal.

Let $S = $ ACGUUACGUU. Let's write the comparison matrix.


\begin{table}
    \includegraphics{./figures/part1/comparison_matrix_repetitions.pdf}
    \caption{Comparison matrix for $seq = $``ACGUUACGUUGUU"}
\end{table}



\begin{algorithm}
    \caption{Construct a comparison matrix}
    \begin{algorithmic}[1]
        \Function{ComparisonMatrix}{$S$: Array($n$)}
        \State $M \gets $ Array($n$, $n$)
        \For{($i = 0$; $i < n$; $i++$)}
        \For{$j = 0$; $j < n$; $j++$}
        \If {$S[i] = S[j]$}
        \State $M[i][j] = 1$
        \Else
        \State $M[i][j] = 0$
        \EndIf
        \EndFor
        \EndFor
        \State \Return $M$
        \EndFunction
    \end{algorithmic}
\end{algorithm}

\begin{algorithm}
    \caption{Construct the top half of a comparison matrix}
    \begin{algorithmic}[1]
        \Function{ComparisonMatrix}{$S$: Array($n$)}
        \State $M \gets$ Array($n$,$n$)
        \For{($i = 0$; $i < n$; $i++$)}
        \For{j=i; j < n; j++}
        \If {S[i] = S[j]}
        \State M[i][j] = 1
        \Else
        \State M[i][j] = 0
        \EndIf
        \EndFor
        \EndFor
        \State \Return M
        \EndFunction
    \end{algorithmic}
\end{algorithm}



\begin{algorithm}
    \caption{Find repetitions (with a set of visited segments)}
    \begin{algorithmic}[1]
        \Function{FindRepetions}{$S$: Array($n$)}
        \Returns{A list of start and end positions for repeated sequences}
        \State $M = $ \Call{ComparisonMatrix}{S}
        \State $pos = \{\}$
        \State $visited = \{\}$
        \For {($i_{start} = 0$; $i_{start} < n$; $i_{start}++$)}
        \For {($j_{start} = i_{start}+1$; $j_{start} < n$; $j_{start}++$)}
        \If{$M[i_{start}][j_{start}] = 1$ and $(i_{start}, j_{start}) \notin visited$}
        \State $i = i_{start}$
        \State $j = j_{start}$
        \While {$M[i][j] = 1$}
        \State $i++$
        \State $j++$
        \State $visited = visited \cup \{(i, j)\}$
        \EndWhile
        \State $pos = pos \cup \{(i_{start}, i), (j_{start},j)\}$
        \EndIf
        \EndFor
        \EndFor
        \EndFunction
    \end{algorithmic}
\end{algorithm}


\begin{algorithm}
    \caption{Find repetitions with an exploration of diagonals}
    \begin{algorithmic}[1]
        \Function{FindRepetions}{$S$: Array($n$)}
        \Returns{A list of start and end positions for repeted sequences}
        \State $M$ = \Call{ComparisonMatrix}{S}
        \State $pos = \{\}$
        \For {($diag = 1$; $diag < n$; $diag++$)}
        \State $j = diag$
        \State $i = 0$
        \While {$i < n$ and $j < n$}
        \If {$M[i][j] = 1$}
        \State $i_{start} = i$
        \State $j_{start} = j$
        \While {$i < n$ and $j < n$ and $M[i][j] = 1$}
        \State i++
        \State j++
        \EndWhile
        \State $pos = pos \cup \{((i_{start},i-1),(j_{start},j-1))\}$
        \EndIf
        \State $i++$
        \State $j++$
        \State
        \EndWhile
        \EndFor
        \EndFunction
    \end{algorithmic}
\end{algorithm}

\begin{algorithm}
    \caption{Find repetitions with an exploration of diagonals, without nested while}
    \begin{algorithmic}[1]
        \Function{FindRepetions}{$S$: Array($n$)}
        \Returns{A list of start positions for repeted sequences and match length}
        \State $M$ = \Call{ComparisonMatrix}{S}
        \State $pos = \{\}$
        \For {($diag = 1$; $diag < n$; $diag++$)}
        \State $j = diag$
        \State $i = 0$
        \State $l = 0$
        \While {$i < n$ and $j < n$}
        \If {$M[i][j] = 1$}
        \State $l++$
        \Else
        \If {$l > 0$}
        \State $pos = pos \cup \{(i-l,j-l,l)\}$
        \State $l = 0$
        \EndIf
        \EndIf
        \State $i++$
        \State $j++$
        \EndWhile
        \If {$l > 0$}
        \State $pos = pos \cup \{((i-l,j-l,l))\}$
        \EndIf
        \EndFor
        \State \Return $pos$
        \EndFunction
    \end{algorithmic}
\end{algorithm}


\begin{algorithm}
    \caption{Find repetitions}
    \begin{algorithmic}[1]
        \Function{FindRepetions}{$S$: Array($n$)}
        \Returns{A list of start and end positions for repeted sequences}
        \State $M$ = \Call{ComparisonMatrix}{S}
        \State $pos = \{\}$
        \For {$i_{start} = 0$; $i_{start} < n$; $i_{start}++$}
        \For {$j_{start} = i_{start}+1$; $j_{start} < n$; $j_{start}++$}
        \If{$M[i_{start}][j_{start}] = 1$}
        \State $i = i_{start}$
        \State $j = j_{start}$
        \While {$M[i][j] = 1$}
        \State $M[i][j] = 0$ \Comment{Ensure that the segment is not explored again}
        \State $i++$
        \State $j++$
        \EndWhile
        \State $pos = pos \cup \{((i_{start}, i-1), (j_{start},j-1))\}$
        \EndIf
        \EndFor
        \EndFor
        \EndFunction
    \end{algorithmic}
\end{algorithm}


\section{Automata}


An automaton is a tuple $\langle S, s_{0}, T, \Sigma,f\rangle$
\begin{itemize}
    \item $S$ the set of states
    \item $s_{0}$ the initial state
    \item $T$ the set of terminal states
    \item $\Sigma$ the alphabet
    \item $f$ the transition function $f: (s_{1}, c) \to s_{2}$
\end{itemize}

\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 $f(p, q) = q$
\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 $f(p, q) = q$.
\end{definition}

\begin{algorithm}
    \caption{Check wether a word belong to a language for which we have an automaton}
    \begin{algorithmic}[1]
        \Function{WordInLanguage}{$W$: Array($n$), $A$: $\langle S, s_{0}, T, \Sigma,f \rangle$}
        \Returns{A Boolean valued to \True{} if the word is recognized by the language automaton}
        \State $s \gets s_{0}$
        \State $i \gets 0$
        \While {$i < n$}
        \State $a \gets W[i]$
        \If {$\exists f(s, a)$}
        \State $s \gets f(s, a)$
        \Else
        \State \Return \False
        \EndIf
        \State i++
        \EndWhile
        \If {$s \in T$}
        \State \Return \True
        \Else
        \State \Return \False
        \EndIf
        \EndFunction
    \end{algorithmic}
\end{algorithm}

\section{Suffix Automaton}

Let $S = $ AACTACT

A suffix automata recognize all suffix of a given sequence.


The suffix language of $S$ is $\{S, ACTACT, CTACT, TACT, ACT, CT, T\}$.


\begin{figure}
    \centering
    \includegraphics{./figures/part1/minimal_suffix_automaton_exercise.pdf}
    \caption{Suffix automaton for $S = $ AACTACT}
\end{figure}

\begin{figure}
    \centering
    \includegraphics{./figures/part1/minimal_suffix_automaton_exercise_bis.pdf}
    \caption{Suffix automaton for $S = $ TCATCATT}
\end{figure}

\begin{algorithm}
    \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}
        \State $s \gets s_{0}$
        \State $i \gets 0$
        \While {$i < m$ and $\exists f(s, W[i])$}
        \State $s \gets f(s, W[i])$
        \State $i++$
        \EndWhile
        \If {$i=n$}
        \State \Return \True
        \Else
        \State \Return \False
        \EndIf
        \EndFunction
    \end{algorithmic}
\end{algorithm}
The complexity of the pattern matching algorithm is $\mathcal{O}(n + m)$, because building the automaton is $\mathcal{O}(m)$