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100 lines
4.1 KiB
TeX
100 lines
4.1 KiB
TeX
\subsection{Hamiltonkreisproblem}
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\begin{frame}{Hamiltonkreisproblem}{Hamiltonian path}
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\begin{block}{Erklärung}
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Ein Hamiltonkreis ist ein Kreis in einem Graphen, in dem jeder Knoten genau einmal benutzt wird.\\
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Das Hamiltonkreisproblem ist NP-vollständig.
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\end{block}
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\end{frame}
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\begin{frame}{Hamiltonkreisproblem}{Bedingungen und Kriterien (Auszug)}
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\begin{block}{Kriterien (notwendig)}
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\begin{itemize}
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\item G hat keine Schnittknoten
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\item G hat keine Brückenkanten
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\item G hat Minimalgrad $\geq 2$
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\end{itemize}
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\end{block}
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\begin{block}{Bedingungen}
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Bei Graphen G mit $n \geq 3$ Knoten
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\begin{itemize}
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\item G hat Minimalgrad $\frac{n}{2} \Rightarrow$ $\exists $ Hamiltonkreis
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\item G ist vollständig $\Rightarrow \exists$ Hamiltonkreis
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\item G ist Kantengraph eines eulerschen oder hamiltonschen Graphen
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}{Hamilton- und Eulerkreisproblem}{Anwendungsbeispiel}
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\begin{block}{Gegeben}
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Eine Menge von Wörtern
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\end{block}
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\begin{block}{Gesucht}
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Aneinanderreihung von Wörtern, sodass jeweils Anfangs- und Endbuchstaben gleich sind
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(auch im Ringschluss).
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\end{block}
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\end{frame}
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\begin{frame}{Hamilton- und Eulerkreisproblem}{Anwendungsbeispiel}
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Wortmenge: $\{$as, man, meet, nets, set, sum, tea, team$\}$
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Menge der Anfangs- und Endbuchstaben: $\{$a, m, n, s, t$\}$
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\begin{figure}
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\begin{tikzpicture}[->,scale=1.3, auto,swap]
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% First we draw the vertices
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\foreach \pos/\name in {{(1,0)/nets}, {(3,0)/set}, {(5,0)/tea},
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{(0,2)/man}, {(6,2)/team}, {(1,4)/as},
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{(3,4)/sum}, {(5,4)/meet}}
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\node[vertex] (\name) at \pos {$\name$};
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% Connect vertices with edges and draw weights
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\foreach \source/ \dest /\pos in {nets/set/, nets/sum/, set/tea/, set/team/,
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tea/as/, man/nets/, team/man/, team/meet/, as/set/,
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as/sum/, sum/meet/, sum/man/, meet/team/bend left, meet/tea/}
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\path (\source) edge [\pos] node {} (\dest);
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% Start animating the edge selection.
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% For convenience we use a background layer to highlight edges
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% This way we don't have to worry about the highlighting covering
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% weight labels.
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\begin{pgfonlayer}{background}
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\foreach \source / \dest / \fr / \pos in { as/as/1/, as/sum/2/, sum/man/3/, man/nets/4/,
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nets/set/5/, set/team/6/, team/meet/7/, meet/tea/8/, tea/as/9/}
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\path<\fr->[selected edge] (\source.center) edge [\pos] node {} (\dest.center);
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\end{pgfonlayer}
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\end{tikzpicture}
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\end{figure}
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% Animierter Graph mit einem Hamiltonkreis. (ohne speziellen Algorithmus?) Wie ausführlich sollen Hamiltonkreise noch genau behandelt werden?
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\end{frame}
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\begin{frame}{Hamilton- und Eulerkreisproblem}{Anwendungsbeispiel}
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Wortmenge: $\{$as, man, meet, nets, set, sum, tea, team$\}$
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Menge der Anfangs- und Endbuchstaben: $\{$a, m, n, s, t$\}$
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\begin{figure}
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\begin{tikzpicture}[->,scale=1.8, auto,swap]
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% Draw a 7,11 network
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% First we draw the vertices
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\foreach \pos/\name in {{(0,0)/a}, {(0,2)/s},
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{(4,0)/m}, {(2,2)/t}, {(3,2)/n}}
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\node[vertex] (\name) at \pos {$\name$};
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% Connect vertices with edges and draw weights
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\tikzstyle{LabelStyle}=[fill=white, fill opacity=0.0, text opacity=1,sloped]
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\foreach \source/ \dest /\foo /\pos in {a/s/as/,s/m/sum/bend right,
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s/t/set/, m/t/meet/bend left,m/n/man/bend right,
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t/a/tea/bend left, t/m/team/bend left, n/s/nets/bend right}
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%\path (\source) edge [\pos] node {\foo} (\dest);
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\Edge[label=\foo,style={\pos}](\source)(\dest);
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% Start animating the edge selection.
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% For convenience we use a background layer to highlight edges
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% This way we don't have to worry about the highlighting covering
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% weight labels.
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\begin{pgfonlayer}{background}
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\foreach \source / \dest / \fr / \pos in {s/s/1/, s/t/2/, t/m/3/bend left, m/n/4/bend right, n/s/5/bend right,
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s/m/6/bend right, m/t/7/bend left, t/a/8/bend left, a/s/9/}
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\path<\fr->[selected edge] (\source.center) edge [\pos] node {} (\dest.center);
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\end{pgfonlayer}
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\end{tikzpicture}
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\end{figure}
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\end{frame}
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