Edit distance
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Antti H S Laaksonen
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luku07.tex
146
luku07.tex
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@ -710,85 +710,88 @@ The efficiency of solution 1 depends on the weights
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of the objects, while the efficiency of solution 2
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depends on the values of the objects.
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\section{Editointietäisyys}
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\section{Edit distance}
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\index{editointietxisyys@editointietäisyys}
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\index{Levenšteinin etäisyys}
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\index{edit distance}
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\index{Levenshtein distance}
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\key{Editointietäisyys} eli
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\key{Levenšteinin etäisyys}
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kuvaa, kuinka kaukana kaksi merkkijonoa ovat toisistaan.
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Se on pienin määrä editointioperaatioita,
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joilla ensimmäisen merkkijonon saa muutettua toiseksi.
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Sallitut operaatiot ovat:
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The \key{edit distance},
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also known as the \key{Levenshtein distance},
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indicates how similar two strings are.
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It is the minimum number of editing operations
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needed for transforming the first string
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into the second string.
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The allowed editing operations are as follows:
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\begin{itemize}
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\item merkin lisäys (esim. \texttt{ABC} $\rightarrow$ \texttt{ABCA})
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\item merkin poisto (esim. \texttt{ABC} $\rightarrow$ \texttt{AC})
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\item merkin muutos (esim. \texttt{ABC} $\rightarrow$ \texttt{ADC})
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\item insert a character (e.g. \texttt{ABC} $\rightarrow$ \texttt{ABCA})
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\item remove a character (e.g. \texttt{ABC} $\rightarrow$ \texttt{AC})
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\item change a character (e.g. \texttt{ABC} $\rightarrow$ \texttt{ADC})
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\end{itemize}
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Esimerkiksi merkkijonojen \texttt{TALO} ja \texttt{PALLO}
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editointietäisyys on 2, koska voimme tehdä ensin
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operaation \texttt{TALO} $\rightarrow$ \texttt{TALLO}
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(merkin lisäys) ja sen jälkeen operaation
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\texttt{TALLO} $\rightarrow$ \texttt{PALLO}
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(merkin muutos).
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Tämä on pienin mahdollinen määrä operaatioita, koska
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selvästikään yksi operaatio ei riitä.
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For example, the edit distance between
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\texttt{LOVE} and \texttt{MOVIE} is 2
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because we can first perform operation
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\texttt{LOVE} $\rightarrow$ \texttt{MOVE}
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(change) and then operation
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\texttt{MOVE} $\rightarrow$ \texttt{MOVIE}
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(insertion).
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This is the smallest possible number of operations
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because it is clear that one operation is not enough.
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Oletetaan, että annettuna on merkkijonot
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\texttt{x} (pituus $n$ merkkiä) ja
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\texttt{y} (pituus $m$ merkkiä),
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ja haluamme laskea niiden editointietäisyyden.
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Tämä onnistuu tehokkaasti dynaamisella
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ohjelmoinnilla ajassa $O(nm)$.
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Merkitään funktiolla $f(a,b)$
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editointietäisyyttä \texttt{x}:n $a$
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ensimmäisen merkin sekä
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\texttt{y}:n $b$:n ensimmäisen merkin välillä.
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Tätä funktiota käyttäen
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merkkijonojen
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\texttt{x} ja \texttt{y} editointietäisyys
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on $f(n,m)$, ja funktio kertoo myös tarvittavat
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editointioperaatiot.
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Suppose we are given strings
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\texttt{x} of $n$ characters and
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\texttt{y} of $m$ characters,
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and we want to calculate the edit distance
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between them.
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This can be efficiently done using
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dynamic programming in $O(nm)$ time.
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Let $f(a,b)$ denote the edit distance
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between the first $a$ characters of \texttt{x}
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and the first $b$ characters of \texttt{y}.
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Using this function, the edit distance between
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\texttt{x} and \texttt{y} is $f(n,m)$,
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and the function also determines
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the editing operations needed.
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Funktion pohjatapaukset ovat
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The base cases for the function are
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\[
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\begin{array}{lcl}
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f(0,b) & = & b \\
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f(a,0) & = & a \\
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\end{array}
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\]
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ja yleisessä tapauksessa pätee kaava
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and in the general case the formula is
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\[ f(a,b) = \min(f(a,b-1)+1,f(a-1,b)+1,f(a-1,b-1)+c),\]
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missä $c=0$, jos \texttt{x}:n merkki $a$
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ja \texttt{y}:n merkki $b$ ovat samat,
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ja muussa tapauksessa $c=1$.
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Kaava käy läpi mahdollisuudet lyhentää merkkijonoja:
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where $c=0$ if the $a$th character of \texttt{x}
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equals the $b$th character of \texttt{y},
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and otherwise $c=1$.
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The formula covers all ways to shorten the strings:
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\begin{itemize}
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\item $f(a,b-1)$ tarkoittaa, että $x$:ään lisätään merkki
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\item $f(a-1,b)$ tarkoittaa, että $x$:stä poistetaan merkki
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\item $f(a-1,b-1)$ tarkoittaa, että $x$:ssä ja $y$:ssä on
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sama merkki ($c=0$) tai $x$:n merkki muutetaan $y$:n merkiksi ($c=1$)
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\item $f(a,b-1)$ means that a character is inserted to \texttt{x}
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\item $f(a-1,b)$ means that a chacater is removed from \texttt{x}
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\item $f(a-1,b-1)$ means that \texttt{x} and \texttt{y} contain
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the same character ($c=0$),
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or a character in \texttt{x} is transformed into
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a character in \texttt{y} ($c=1$)
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\end{itemize}
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Seuraava taulukko sisältää funktion $f$ arvot
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esimerkin tapauksessa:
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The following table shows the values of $f$
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in the example case:
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\begin{center}
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\begin{tikzpicture}[scale=.65]
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\begin{scope}
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%\fill [color=lightgray] (5, -3) rectangle (6, -4);
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\draw (1, -1) grid (7, -6);
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\node at (0.5,-2.5) {\texttt{T}};
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\node at (0.5,-3.5) {\texttt{A}};
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\node at (0.5,-4.5) {\texttt{L}};
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\node at (0.5,-5.5) {\texttt{O}};
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\node at (0.5,-2.5) {\texttt{L}};
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\node at (0.5,-3.5) {\texttt{O}};
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\node at (0.5,-4.5) {\texttt{V}};
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\node at (0.5,-5.5) {\texttt{E}};
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\node at (2.5,-0.5) {\texttt{P}};
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\node at (3.5,-0.5) {\texttt{A}};
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\node at (4.5,-0.5) {\texttt{L}};
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\node at (5.5,-0.5) {\texttt{L}};
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\node at (6.5,-0.5) {\texttt{O}};
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\node at (2.5,-0.5) {\texttt{M}};
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\node at (3.5,-0.5) {\texttt{O}};
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\node at (4.5,-0.5) {\texttt{V}};
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\node at (5.5,-0.5) {\texttt{I}};
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\node at (6.5,-0.5) {\texttt{E}};
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\node at (1.5,-1.5) {$0$};
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\node at (1.5,-2.5) {$1$};
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@ -824,29 +827,28 @@ esimerkin tapauksessa:
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\end{tikzpicture}
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\end{center}
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Taulukon oikean alanurkan ruutu
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kertoo, että merkkijonojen \texttt{TALO}
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ja \texttt{PALLO} editointietäisyys on 2.
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Taulukosta pystyy myös
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lukemaan, miten pienimmän editointietäisyyden
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voi saavuttaa.
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Tässä tapauksessa polku on seuraava:
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The lower-right corner of the table
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indicates that the edit distance between
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\texttt{LOVE} and \texttt{MOVIE} is 2.
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The table also shows how to construct
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the shortest sequence of editing operations.
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In this case the path is as follows:
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\begin{center}
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\begin{tikzpicture}[scale=.65]
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\begin{scope}
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\draw (1, -1) grid (7, -6);
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\node at (0.5,-2.5) {\texttt{T}};
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\node at (0.5,-3.5) {\texttt{A}};
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\node at (0.5,-4.5) {\texttt{L}};
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\node at (0.5,-5.5) {\texttt{O}};
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\node at (0.5,-2.5) {\texttt{L}};
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\node at (0.5,-3.5) {\texttt{O}};
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\node at (0.5,-4.5) {\texttt{V}};
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\node at (0.5,-5.5) {\texttt{E}};
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\node at (2.5,-0.5) {\texttt{P}};
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\node at (3.5,-0.5) {\texttt{A}};
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\node at (4.5,-0.5) {\texttt{L}};
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\node at (5.5,-0.5) {\texttt{L}};
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\node at (6.5,-0.5) {\texttt{O}};
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\node at (2.5,-0.5) {\texttt{M}};
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\node at (3.5,-0.5) {\texttt{O}};
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\node at (4.5,-0.5) {\texttt{V}};
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\node at (5.5,-0.5) {\texttt{I}};
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\node at (6.5,-0.5) {\texttt{E}};
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\node at (1.5,-1.5) {$0$};
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\node at (1.5,-2.5) {$1$};
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