aufgabe 2
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übung_6/agt_übung_6.pdf
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\documentclass[parskip=half]{ngexrcs}
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\documentclass[parskip=half]{ngexrcs}
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\usepackage{hyperref}
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\usepackage{hyperref}
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\usepackage{subcaption}
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\setkeys{Gin}{pagebox=artbox, width=0.2\textwidth}
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\setkeys{Gin}{pagebox=artbox, width=0.2\textwidth}
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\subject{Algorithmische Graphentheorie}
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\subject{Algorithmische Graphentheorie}
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\newcommand{\OPT}{\mathrm{OPT}}
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\section{LP-Runden}
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\section{Paarungen (Matchings) in Bäumen}
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\begin{tasks}
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\begin{tasks}
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\item
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\item
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\label{1a}
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\item
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Ein Baum ist ein kreisfreier, zusammenhängender Graph. Ein Baum besitzt ein
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\item
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perfektes Matching genau dann, wenn die Anzahl der Knoten gerade ist und
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jedes Blatt keine Geschwisterknoten hat.
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Die Anzahl der Knoten muss gerade sein, da in einem perfekten Match\-ing jeder
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Knoten mit einem anderen gematcht wird.
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Jedes Blatt muss Einzelblatt sein, da der Elternknoten nur mit einem Blatt
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gematcht werden kann.
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Also ist ein perfektes Matching im Baum eindeutig, da jedes Blatt mit seinem
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Elternknoten gematcht werden muss. Diese Kante $uv$ ist sicher im Matching enthalten und alle zu $u$ oder $v$ inzidenten Kanten können aus dem Baum gelöscht werden. So entsteht ein neues Blatt, für das
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die selbe Regel gilt.
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\points{3}
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\item
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Der \autoref{alg:treematching} berechnet ein größtes Matching.
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Der Algorithmus ist korrekt, da der Baum $T$ entweder ein perfektes Matching
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enthält und somit ein perfektes Matching berechnet (Siehe \autoref{1a}),
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oder I dont fucking know.
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Die Laufzeit für \algt{Initialize} liegt in $\Oh(V)$, da sich der Baum in linearer Zeit augmentieren lässt und es $\abs{V} - 1$ viele Kanten gibt.
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Die Laufzeit für \algt{MatchaTeeRec} liegt in $\Oh(V)$. Der Algorithmus wird genau
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einmal für jeden Knoten aufgerufen. Wenn eine Kante zum
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Match\-ing hinzugefügt wird, werden die Knoten und Kanten aus dem Baum gelöscht
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die nicht mehr zum Matching hinzugefügt werden können und somit
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nicht mehr bearbeitet werden.
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Also liegt auch \algt{MatchaTee} in $\Oh(V)$.
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\begin{algorithm}
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\centering
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\caption{Größtes Matching in Bäumen}
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\label{alg:treematching}
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\begin{algorithmic}
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\alg{MatchaTee}{$T = \tup{V, E}$} \+ \\
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\com{Den ungerichten Baum zu einem gerichteten Baum mit $parent$ und} \\
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\com{$children$ pro Knoten und einer Wurzel $root$ augmentieren.} \\
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\alg{Initialize}{$T$} \\
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return \alg{MatchaTeeRec}{$T$, $T.root$} \- \\
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\\
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\alg{MatchaTeeRec}{$T$, $r$} \+ \\
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\com{Rekursiv für die Kinder aufrufen} \\
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$m \gets \emptyset$ \\
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foreach $v \in r.children$ do \+ \\
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$m \gets m \cup \text{\alg{MatchaTeeRec}{T, v}}$ \- \\
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\com{Wenn Knoten Blatt ist, dann die Kante zum Elternknoten hinzufügen.} \\
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if $r \neq nil$ and $deg(r) = 1$ then \+ \\
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\com{Die Knoten und inzidente Kanten aus dem Baum entfernen.} \\
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$T \gets T \setminus \set{r.parent, r.parent.children}$ \\
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$m \gets m \cup \set{\set{r, r.parent}}$ \- \\
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return $m$
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\end{algorithmic}
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\end{algorithm}
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\points{5}
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\item
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Wenn eine Kante $rv$ zu einem Matching hinzugefügt wird, kann keine zu $v$
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inzidente Kante hinzugefügt werden. Das müssen wir auch bei unserem dynamischen
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Programm beachten.
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\[
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\OPT(r) = \max_{v \in r.children}
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\begin{cases}
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rv + \sum_{i \in v.children} \OPT(i) + \sum_{j \in r.children \setminus v} \OPT(j) \\
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\sum_{k \in r.children} \OPT(k)
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\end{cases}
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\]
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Der Algorithmus hält sich an die Eigenschaften eines Matchings, also ist
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er korrekt.
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Der Algorithmus probiert alle Kindknoten von $r$ aus. Geht man mit $r$ im Baum von
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unten nach oben sind das also höchstens so viele Iterationen wie, Knoten im Baum
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existieren (ohne die Blätter). Also läuft das Dynamische Programm in $\Oh(V)$.
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\points{5}
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\end{tasks}
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\end{tasks}
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\section{Hamiltonkreise}
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\section{Christofides’ Algorithmus}
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\begin{tasks}
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\begin{tasks}
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\item
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\item
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Ein Kreis hat im allgemeinen mindestens die Länge 3. Da der Turniergraph mindestens
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Siehe \autoref{fig:christofides}.
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3 Knoten enthält und stark zusammenhängend ist, liegt jeder Knoten auf einem Kreis
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\points{4}
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mit Länge $\geq 3$.
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\points{1}
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\item
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Gegeben ist ein Kreis $K$ mit Länge $\geq 3$.
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Um $K$ zu verkleinern wählen wir uns drei aufeinanderfolgende Knoten $a, b, c$ aus.
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Nach Definition gibt es für jedes Knotenpaar eine gerichtete Kante. Wir ersetzen
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jetzt einen Weg der Länge 2 durch eine Kante die nach Annahme existieren muss. Das
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machen wir solange, bis der Kreis die Länge 3 hat.
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\points{2}
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\item
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\points{1}
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\item
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\points{1}
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\item
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\points{1}
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\item
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\item
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Siehe \autoref{fig:checkmate}. Diese Tour kann nicht von \algt{Christofides}
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berechnet werden, denn egal mit welcher Kante in welche Richtung gestartet
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wird, nimmt \algt{Cristofides} immer eine Kante in die Tour, die nicht in
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\autoref{fig:checkmate} ist.
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\begin{figure}
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\centering
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\includegraphics[page=10, width=0.5\textwidth]{figures.pdf}
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\caption{Eine kleinere TSP-Tour, die von \algt{Cristofides} nicht berechnet werden kann.}
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\label{fig:checkmate}
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\end{figure}
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\points{1}
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\points{1}
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\end{tasks}
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\end{tasks}
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\newpage
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\KOMAoptions{mpinclude=false}
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\recalctypearea
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\begin{figure}
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\centering
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=1, width=0.9\textwidth]{figures.pdf}
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\caption{Der gewichtete, vollständige Graph $G$.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=2, width=0.9\textwidth]{figures.pdf}
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\caption{Minimalen Spannbaum finden.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=3, width=0.9\textwidth]{figures.pdf}
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\caption{Knoten $U$ mit ungeraden Graden finden.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=4, width=0.9\textwidth]{figures.pdf}
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\caption{Minimales, perfektes Matching auf $G[U]$ finden.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=5, width=0.9\textwidth]{figures.pdf}
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\caption{Eulertour konstruieren.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=6, width=0.9\textwidth]{figures.pdf}
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\caption{Eulertour konstruieren.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=7, width=0.9\textwidth]{figures.pdf}
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\caption{Eulertour konstruieren.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=8, width=0.9\textwidth]{figures.pdf}
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\caption{Eulertour konstruieren.}
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\end{subfigure}
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\begin{subfigure}{.3\linewidth}
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\centering
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\includegraphics[page=9, width=0.9\textwidth]{figures.pdf}
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\caption{Schon besuchte Knoten überspringen. TSP vollständig.}
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\end{subfigure}
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\caption{Cristofides' Algorithmus}
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\label{fig:christofides}
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\end{figure}
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\KOMAoptions{mpinclude=true}
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\recalctypearea
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% vim: ft=tex
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\section{Matchings in allgemeinen Graphen}
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\section{Perfekte Matchings in bipartiten Graphen}
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\begin{tasks}
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\begin{quote}
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Ein bipartiter Graph $G = \tup{A \cupdot B, E}$ mit $\abs{A} = \abs{B} = k$
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und jeder Knoten hat mindestens Grad $\frac{k}{2}$ $\iff$ $G$ enthält
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perfektes Matching.
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\item
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\end{quote}
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\item
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\begin{itemize}
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\end{tasks}
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\item[$\seilpmi$] Wenn $G$ ein perfektes Matching enthält, dann muss
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$\abs{A} = \abs{B}$, sonst würden Knoten übrig bleiben.
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Außerdem muss jeder Knoten mindestens Grad $1$ haben, da sonst Knoten
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nicht gematcht werden könnten.
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\item[$\implies$] Man kann jeden Knoten in $A$ einem
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Knoten in $B$ zuordnen, da $\abs{A} = \abs{B}$.
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Alle Knoten sind im Matching enthalten. Wenn wir eine Kante zum Matching
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hinzufügen, verringert sich die Anzahl der offenen Knoten um $2$.
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Insbesondere verringert sich auch der Grad von $\frac{k}{2} - 1$
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Knoten um $1$ in $A$ und $B$. Da jeder Knoten mindestens Grad $\frac{k}{2}$ hat und jedes mal
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ein offenes Knotenpaar entfernt wird.
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$k$ mal eine Kante zum Matching hinzufügen. Also wird jeder Knoten gematcht.
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\end{itemize}
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\points{5}
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-0.57 0.43 l
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h
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</symbol>
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h
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h
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<symbol name="arrow/fpointed(spx)">
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h
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-1 -0.333 l
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|
h
|
||||||
|
-1 0 m
|
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|
-2 0.333 l
|
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-2 -0.333 l
|
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h
|
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-1 -0.333 l
|
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h
|
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|
-1 0 m
|
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|
-2 0.333 l
|
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-2 -0.333 l
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-0.5 -0.333 l
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h
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0.5 0 m
|
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|
-0.5 0.333 l
|
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|
-0.5 -0.333 l
|
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h
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<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
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0.5 0 m
|
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-0.5 0.333 l
|
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-0.3 0 l
|
||||||
|
-0.5 -0.333 l
|
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|
h
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|
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<symbol name="arrow/mid-fpointed(spx)">
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|
<path stroke="sym-stroke" fill="white" pen="sym-pen">
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0.5 0 m
|
||||||
|
-0.5 0.333 l
|
||||||
|
-0.3 0 l
|
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-0.5 -0.333 l
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|
h
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1 0 m
|
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0 0.333 l
|
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0 -0.333 l
|
||||||
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h
|
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0 0 m
|
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-1 0.333 l
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||||||
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-1 -0.333 l
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h
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</symbol>
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<symbol name="arrow/mid-fdouble(spx)">
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||||||
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<path stroke="sym-stroke" fill="white" pen="sym-pen">
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1 0 m
|
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0 0.333 l
|
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0 -0.333 l
|
||||||
|
h
|
||||||
|
0 0 m
|
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-1 0.333 l
|
||||||
|
-1 -0.333 l
|
||||||
|
h
|
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|
</path>
|
||||||
|
</symbol>
|
||||||
|
<anglesize name="22.5 deg" value="22.5"/>
|
||||||
|
<anglesize name="36 deg" value="36"/>
|
||||||
|
<anglesize name="45 deg" value="45"/>
|
||||||
|
<anglesize name="60 deg" value="60"/>
|
||||||
|
<anglesize name="90 deg" value="90"/>
|
||||||
|
<arrowsize name="large" value="10"/>
|
||||||
|
<arrowsize name="small" value="5"/>
|
||||||
|
<arrowsize name="tiny" value="3"/>
|
||||||
|
<color name="blue" value="0 0 1"/>
|
||||||
|
<color name="brown" value="0.647 0.165 0.165"/>
|
||||||
|
<color name="darkblue" value="0 0 0.545"/>
|
||||||
|
<color name="darkcyan" value="0 0.545 0.545"/>
|
||||||
|
<color name="darkgray" value="0.663"/>
|
||||||
|
<color name="darkgreen" value="0 0.392 0"/>
|
||||||
|
<color name="darkmagenta" value="0.545 0 0.545"/>
|
||||||
|
<color name="darkorange" value="1 0.549 0"/>
|
||||||
|
<color name="darkred" value="0.545 0 0"/>
|
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|
<color name="gold" value="1 0.843 0"/>
|
||||||
|
<color name="gray" value="0.745"/>
|
||||||
|
<color name="green" value="0 1 0"/>
|
||||||
|
<color name="lightblue" value="0.678 0.847 0.902"/>
|
||||||
|
<color name="lightcyan" value="0.878 1 1"/>
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|
<color name="lightgray" value="0.827"/>
|
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|
<color name="lightgreen" value="0.565 0.933 0.565"/>
|
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|
<color name="lightyellow" value="1 1 0.878"/>
|
||||||
|
<color name="navy" value="0 0 0.502"/>
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<color name="orange" value="1 0.647 0"/>
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<color name="pink" value="1 0.753 0.796"/>
|
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<color name="purple" value="0.627 0.125 0.941"/>
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<color name="seagreen" value="0.18 0.545 0.341"/>
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<color name="turquoise" value="0.251 0.878 0.816"/>
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<color name="violet" value="0.933 0.51 0.933"/>
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<color name="yellow" value="1 1 0"/>
|
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<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
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|
<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
|
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|
<dashstyle name="dashed" value="[4] 0"/>
|
||||||
|
<dashstyle name="dotted" value="[1 3] 0"/>
|
||||||
|
<gridsize name="10 pts (~3.5 mm)" value="10"/>
|
||||||
|
<gridsize name="14 pts (~5 mm)" value="14"/>
|
||||||
|
<gridsize name="16 pts (~6 mm)" value="16"/>
|
||||||
|
<gridsize name="20 pts (~7 mm)" value="20"/>
|
||||||
|
<gridsize name="28 pts (~10 mm)" value="28"/>
|
||||||
|
<gridsize name="32 pts (~12 mm)" value="32"/>
|
||||||
|
<gridsize name="4 pts" value="4"/>
|
||||||
|
<gridsize name="56 pts (~20 mm)" value="56"/>
|
||||||
|
<gridsize name="8 pts (~3 mm)" value="8"/>
|
||||||
|
<opacity name="10%" value="0.1"/>
|
||||||
|
<opacity name="30%" value="0.3"/>
|
||||||
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<opacity name="50%" value="0.5"/>
|
||||||
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<opacity name="75%" value="0.75"/>
|
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<pen name="fat" value="1.2"/>
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<pen name="heavier" value="0.8"/>
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||||||
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<pen name="ultrafat" value="2"/>
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||||||
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<symbolsize name="large" value="5"/>
|
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|
<symbolsize name="small" value="2"/>
|
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|
<symbolsize name="tiny" value="1.1"/>
|
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|
<textsize name="Huge" value="\Huge"/>
|
||||||
|
<textsize name="LARGE" value="\LARGE"/>
|
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|
<textsize name="Large" value="\Large"/>
|
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|
<textsize name="footnote" value="\footnotesize"/>
|
||||||
|
<textsize name="huge" value="\huge"/>
|
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|
<textsize name="large" value="\large"/>
|
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<textsize name="script" value="\scriptsize"/>
|
||||||
|
<textsize name="small" value="\small"/>
|
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|
<textsize name="tiny" value="\tiny"/>
|
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|
<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
|
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<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
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|
<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
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<tiling name="falling" angle="-60" step="4" width="1"/>
|
||||||
|
<tiling name="rising" angle="30" step="4" width="1"/>
|
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|
</ipestyle>
|
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<ipestyle name="ngslides">
|
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<symbol name="arrow/ng-fnormal(spx)">
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-0.7 0 m
|
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-1.7 0.333 l
|
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-1.7 -0.333 l
|
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|
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|
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-0.7 0 m
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-1.7 0.333 l
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-1.7 -0.333 l
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|
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-0.7 0 m
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-1.7 0.333 m
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-0.7 0 l
|
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-1.7 -0.333 l
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</path>
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</symbol>
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-0.7 0 m
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-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
-1.7 0 m
|
||||||
|
-2.7 0.333 l
|
||||||
|
-2.7 -0.333 l
|
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|
h
|
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-0.7 0 m
|
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-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
-1.7 0 m
|
||||||
|
-2.7 0.333 l
|
||||||
|
-2.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
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</symbol>
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<symbol name="decoration/remark">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
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|
</symbol>
|
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|
<symbol name="decoration/theorem">
|
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|
<path stroke="dark blue" fill="light blue">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
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|
</symbol>
|
||||||
|
<symbol name="decoration/problem">
|
||||||
|
<path stroke="dark green" fill="light green">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/definition">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
96 96 m
|
||||||
|
304 96 l
|
||||||
|
304 204 l
|
||||||
|
96 204 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/algorithm">
|
||||||
|
<path stroke="dark gray" fill="light green" dash="dashed">
|
||||||
|
96 96 m
|
||||||
|
304 96 l
|
||||||
|
304 204 l
|
||||||
|
96 204 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/graph">
|
||||||
|
<path stroke="dark gray">
|
||||||
|
92 92 m
|
||||||
|
308 92 l
|
||||||
|
308 208 l
|
||||||
|
92 208 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-bl">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-bm">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
186 96 s
|
||||||
|
186 92
|
||||||
|
182 88 s
|
||||||
|
192 92
|
||||||
|
194 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-br">
|
||||||
|
<path matrix="-1 0 0 1 400 0" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tl">
|
||||||
|
<path matrix="1 0 0 -1 0 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tm">
|
||||||
|
<path matrix="1 0 0 -1 0 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
186 96 s
|
||||||
|
186 92
|
||||||
|
182 88 s
|
||||||
|
192 92
|
||||||
|
194 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tr">
|
||||||
|
<path matrix="-1 0 0 -1 400 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<color name="dark blue" value="0.121 0.47 0.705"/>
|
||||||
|
<color name="dark brown" value="0.651 0.337 0.157"/>
|
||||||
|
<color name="dark cyan" value="0.106 0.62 0.467"/>
|
||||||
|
<color name="dark gray" value="0.5"/>
|
||||||
|
<color name="dark green" value="0.2 0.627 0.172"/>
|
||||||
|
<color name="dark orange" value="1 0.498 0"/>
|
||||||
|
<color name="dark pink" value="0.969 0.506 0.749"/>
|
||||||
|
<color name="dark purple" value="0.415 0.239 0.603"/>
|
||||||
|
<color name="dark red" value="0.89 0.102 0.109"/>
|
||||||
|
<color name="dark yellow" value="1 1 0.2"/>
|
||||||
|
<color name="light blue" value="0.651 0.807 0.89"/>
|
||||||
|
<color name="light brown" value="0.898 0.847 0.741"/>
|
||||||
|
<color name="light cyan" value="0.553 0.827 0.78"/>
|
||||||
|
<color name="light gray" value="0.8"/>
|
||||||
|
<color name="light green" value="0.698 0.874 0.541"/>
|
||||||
|
<color name="light orange" value="0.992 0.749 0.435"/>
|
||||||
|
<color name="light pink" value="0.992 0.855 0.925"/>
|
||||||
|
<color name="light purple" value="0.792 0.698 0.839"/>
|
||||||
|
<color name="light red" value="0.984 0.603 0.6"/>
|
||||||
|
<color name="light yellow" value="1 1 0.8"/>
|
||||||
|
<color name="pagenumber" value="0.5"/>
|
||||||
|
<color name="title" value="0.247 0.282 0.392"/>
|
||||||
|
<pen name="fat" value="3.6"/>
|
||||||
|
<pen name="heavier" value="2.4"/>
|
||||||
|
<pen name="normal" value="1.2"/>
|
||||||
|
<pen name="ultrafat" value="6"/>
|
||||||
|
<symbolsize name="large" value="15"/>
|
||||||
|
<symbolsize name="normal" value="9"/>
|
||||||
|
<symbolsize name="small" value="6"/>
|
||||||
|
<symbolsize name="tiny" value="3.3"/>
|
||||||
|
<textstretch name="Huge" value="2.4"/>
|
||||||
|
<textstretch name="LARGE" value="2.4"/>
|
||||||
|
<textstretch name="Large" value="2.4"/>
|
||||||
|
<textstretch name="footnote" value="2.4"/>
|
||||||
|
<textstretch name="huge" value="2.4"/>
|
||||||
|
<textstretch name="large" value="2.4"/>
|
||||||
|
<textstretch name="normal" value="2.4"/>
|
||||||
|
<textstretch name="script" value="2.4"/>
|
||||||
|
<textstretch name="small" value="2.4"/>
|
||||||
|
<textstretch name="tiny" value="2.4"/>
|
||||||
|
<textstyle name="algorithmic" begin="\begin{algorithmic}" end="\end{algorithmic}"/>
|
||||||
|
<textstyle name="definition" begin="\begin{definition}" end="\end{definition}"/>
|
||||||
|
<textstyle name="enumerate" begin="\begin{enumerate}" end="\end{enumerate}"/>
|
||||||
|
<textstyle name="example" begin="\begin{example}" end="\end{example}"/>
|
||||||
|
<textstyle name="lemma" begin="\begin{lemma}" end="\end{lemma}"/>
|
||||||
|
<textstyle name="proof" begin="\begin{proof}" end="\end{proof}"/>
|
||||||
|
<textstyle name="theorem" begin="\begin{theorem}" end="\end{theorem}"/>
|
||||||
|
<preamble>
|
||||||
|
\usepackage[ngerman]{babel}
|
||||||
|
\usepackage[sansdefault]{fontsetup}
|
||||||
|
\usepackage{ngutils}
|
||||||
|
|
||||||
|
\renewcommand{\theoremfont}{\normalfont\bfseries\color{dark blue}}
|
||||||
|
\renewcommand{\definitionfont}{\normalfont\bfseries\color{dark gray}}
|
||||||
|
|
||||||
|
\newcommand{\labelfont}{\normalfont\bfseries\color{dark blue}}
|
||||||
|
|
||||||
|
%\setlength{\leftmargini}{1.33em}
|
||||||
|
%\setlength{\leftmarginii}{1.33em}
|
||||||
|
%\setlength{\leftmarginiii}{1.33em}
|
||||||
|
%\setlength{\leftmarginvi}{1.33em}
|
||||||
|
|
||||||
|
\renewcommand{\labelenumi}{\labelfont\arabic{enumi}.}
|
||||||
|
\renewcommand{\labelenumii}{\labelfont\alph{enumii})}
|
||||||
|
\renewcommand{\labelenumiii}{\labelfont\roman{enumiii}.}
|
||||||
|
\renewcommand{\labelenumiv}{\labelfont\Alph{enumiv}.}
|
||||||
|
|
||||||
|
\renewcommand{\labelitemi}{\labelfont $\blockfull$}
|
||||||
|
\renewcommand{\labelitemii}{\labelfont $\blackpointerright$}
|
||||||
|
\renewcommand{\labelitemiii}{\labelfont $\bullet$}
|
||||||
|
\renewcommand{\labelitemiv}{\labelfont $\bullet$}
|
||||||
|
|
||||||
|
\setlength{\fboxsep}{1pt}
|
||||||
|
|
||||||
|
\newcommand{\bk}{\color{black}}
|
||||||
|
\newcommand{\wt}{\color{white}}
|
||||||
|
\newcommand{\bu}{\color{dark blue}}
|
||||||
|
\newcommand{\bn}{\color{dark brown}}
|
||||||
|
\newcommand{\cn}{\color{dark cyan}}
|
||||||
|
\newcommand{\gy}{\color{dark gray}}
|
||||||
|
\newcommand{\gn}{\color{dark green}}
|
||||||
|
\newcommand{\og}{\color{dark orange}}
|
||||||
|
\newcommand{\pk}{\color{dark pink}}
|
||||||
|
\newcommand{\pl}{\color{dark purple}}
|
||||||
|
\newcommand{\rd}{\color{dark red}}
|
||||||
|
\newcommand{\yo}{\color{dark yellow}}
|
||||||
|
\newcommand{\tl}{\color{title}}
|
||||||
|
</preamble>
|
||||||
|
<layout paper="1024 576" origin="16 16" frame="992 496" skip="16" crop="no"/>
|
||||||
|
<pagenumberstyle pos="1000 552" color="pagenumber" size="tiny" halign="right" valign="top"></pagenumberstyle>
|
||||||
|
<titlestyle pos="0 544" size="Large" color="title" valign="top"/>
|
||||||
|
</ipestyle>
|
||||||
|
<page>
|
||||||
|
<layer name="knoten"/>
|
||||||
|
<layer name="kanten"/>
|
||||||
|
<layer name="msb"/>
|
||||||
|
<layer name="odddeg"/>
|
||||||
|
<layer name="matching"/>
|
||||||
|
<layer name="tsp1"/>
|
||||||
|
<layer name="tsp2"/>
|
||||||
|
<layer name="tsp3"/>
|
||||||
|
<layer name="tsp4"/>
|
||||||
|
<layer name="tsp5"/>
|
||||||
|
<layer name="kanteweird"/>
|
||||||
|
<layer name="msbweird"/>
|
||||||
|
<layer name="kanteweirdalt"/>
|
||||||
|
<layer name="msbweirdalt"/>
|
||||||
|
<view layers="knoten kanten kanteweird" active="knoten"/>
|
||||||
|
<view layers="knoten kanten msb kanteweird msbweird" active="knoten"/>
|
||||||
|
<view layers="knoten kanten msb odddeg kanteweird msbweird" active="knoten"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching kanteweird msbweird" active="knoten"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching tsp1 msbweird kanteweirdalt" active="kanteweirdalt"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching tsp1 tsp2 msbweird kanteweirdalt" active="tsp1"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching tsp1 tsp2 tsp3 msbweird kanteweirdalt" active="knoten"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching tsp1 tsp2 tsp3 tsp4 kanteweirdalt msbweirdalt" active="msbweirdalt"/>
|
||||||
|
<view layers="knoten kanten msb odddeg matching tsp1 tsp2 tsp3 tsp4 tsp5 kanteweirdalt msbweirdalt" active="msbweirdalt"/>
|
||||||
|
<path layer="kanteweird" stroke="light gray">
|
||||||
|
688 448 m
|
||||||
|
48 0 0 48 640 448 640 496 a
|
||||||
|
256 448 m
|
||||||
|
48 0 0 -48 304 448 304 496 a
|
||||||
|
304 496 m
|
||||||
|
640 496 l
|
||||||
|
688 448 m
|
||||||
|
688 256
|
||||||
|
640 64 c
|
||||||
|
</path>
|
||||||
|
<path layer="kanten" stroke="light gray">
|
||||||
|
640 448 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path layer="msb" stroke="dark purple" pen="heavier">
|
||||||
|
640 448 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="dark purple" pen="heavier">
|
||||||
|
640 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path layer="matching" stroke="dark green" pen="heavier">
|
||||||
|
256 448 m
|
||||||
|
352 160
|
||||||
|
640 64 c
|
||||||
|
</path>
|
||||||
|
<path stroke="dark green" pen="heavier">
|
||||||
|
640 448 m
|
||||||
|
352 352
|
||||||
|
256 64 c
|
||||||
|
</path>
|
||||||
|
<path layer="tsp3" stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
448 256 m
|
||||||
|
640 448 l
|
||||||
|
</path>
|
||||||
|
<path layer="tsp2" stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
640 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path layer="kanten" stroke="light gray">
|
||||||
|
688 64 m
|
||||||
|
48 0 0 -48 640 64 640 16 a
|
||||||
|
256 64 m
|
||||||
|
48 0 0 48 304 64 304 16 a
|
||||||
|
304 16 m
|
||||||
|
640 16 l
|
||||||
|
688 64 m
|
||||||
|
688 256
|
||||||
|
640 448 c
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 448 m
|
||||||
|
256 64 l
|
||||||
|
</path>
|
||||||
|
<path layer="tsp5" stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
256 64 m
|
||||||
|
256 448 l
|
||||||
|
</path>
|
||||||
|
<path layer="kanten" stroke="light gray">
|
||||||
|
256 64 m
|
||||||
|
640 64 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 64 m
|
||||||
|
640 448 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 448 m
|
||||||
|
256 448 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 448 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path layer="tsp4" stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
640 448 m
|
||||||
|
352 352
|
||||||
|
256 64 c
|
||||||
|
</path>
|
||||||
|
<text layer="kanten" transformations="translations" pos="448 464" stroke="light gray" type="label" width="30.5568" height="16.224" depth="2.976" halign="center" valign="baseline">3,5</text>
|
||||||
|
<text transformations="translations" pos="240 256" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" halign="right" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="352 368" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="544 368" stroke="light gray" type="label" width="11.9544" height="16.212" depth="0" halign="right" valign="baseline">2</text>
|
||||||
|
<text transformations="translations" pos="624 256" stroke="light gray" type="label" width="30.5568" height="15.6984" depth="2.976" halign="right" valign="baseline">4,5</text>
|
||||||
|
<text transformations="translations" pos="352 176" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" halign="right" valign="baseline">3</text>
|
||||||
|
<text matrix="1 0 0 1 0 -16" transformations="translations" pos="560 176" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" valign="baseline">3</text>
|
||||||
|
<text transformations="translations" pos="448 80" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" halign="center" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="448 32" stroke="light gray" type="label" width="11.9544" height="16.212" depth="0" halign="center" valign="baseline">1</text>
|
||||||
|
<text transformations="translations" pos="448 512" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" halign="center" valign="baseline">6</text>
|
||||||
|
<path layer="msb" stroke="dark purple" pen="heavier">
|
||||||
|
640 448 m
|
||||||
|
256 448 l
|
||||||
|
</path>
|
||||||
|
<path layer="tsp1" stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
256 448 m
|
||||||
|
352 160
|
||||||
|
640 64 c
|
||||||
|
</path>
|
||||||
|
<path layer="kanteweirdalt" stroke="light gray" dash="dashed">
|
||||||
|
688 448 m
|
||||||
|
48 0 0 48 640 448 640 496 a
|
||||||
|
256 448 m
|
||||||
|
48 0 0 -48 304 448 304 496 a
|
||||||
|
304 496 m
|
||||||
|
640 496 l
|
||||||
|
688 448 m
|
||||||
|
688 256
|
||||||
|
640 64 c
|
||||||
|
</path>
|
||||||
|
<use layer="knoten" matrix="1 0 0 1 -64 64" name="mark/disk(sx)" pos="320 384" size="normal" stroke="black"/>
|
||||||
|
<path layer="msbweird" stroke="dark purple" pen="heavier">
|
||||||
|
688 64 m
|
||||||
|
48 0 0 -48 640 64 640 16 a
|
||||||
|
256 64 m
|
||||||
|
48 0 0 48 304 64 304 16 a
|
||||||
|
304 16 m
|
||||||
|
640 16 l
|
||||||
|
688 64 m
|
||||||
|
688 256
|
||||||
|
640 448 c
|
||||||
|
</path>
|
||||||
|
<path layer="msbweirdalt" stroke="dark purple" dash="dashed" pen="heavier">
|
||||||
|
688 64 m
|
||||||
|
48 0 0 -48 640 64 640 16 a
|
||||||
|
256 64 m
|
||||||
|
48 0 0 48 304 64 304 16 a
|
||||||
|
304 16 m
|
||||||
|
640 16 l
|
||||||
|
688 64 m
|
||||||
|
688 256
|
||||||
|
640 448 c
|
||||||
|
</path>
|
||||||
|
<use layer="knoten" matrix="1 0 0 1 -64 -64" name="mark/disk(sx)" pos="320 128" size="normal" stroke="black"/>
|
||||||
|
<use matrix="1 0 0 1 64 -64" name="mark/disk(sx)" pos="576 128" size="normal" stroke="black"/>
|
||||||
|
<use matrix="1 0 0 1 64 64" name="mark/disk(sx)" pos="576 384" size="normal" stroke="black"/>
|
||||||
|
<use name="mark/disk(sx)" pos="448 256" size="normal" stroke="black"/>
|
||||||
|
<use layer="odddeg" name="mark/circle(sx)" pos="256 448" size="large" stroke="dark red"/>
|
||||||
|
<use name="mark/circle(sx)" pos="640 448" size="large" stroke="dark red"/>
|
||||||
|
<use name="mark/circle(sx)" pos="640 64" size="large" stroke="dark red"/>
|
||||||
|
<use name="mark/circle(sx)" pos="256 64" size="large" stroke="dark red"/>
|
||||||
|
</page>
|
||||||
|
<page>
|
||||||
|
<layer name="alpha"/>
|
||||||
|
<view layers="alpha" active="alpha"/>
|
||||||
|
<path layer="alpha" stroke="light gray">
|
||||||
|
688 448 m
|
||||||
|
48 0 0 48 640 448 640 496 a
|
||||||
|
256 448 m
|
||||||
|
48 0 0 -48 304 448 304 496 a
|
||||||
|
304 496 m
|
||||||
|
640 496 l
|
||||||
|
688 448 m
|
||||||
|
688 256
|
||||||
|
640 64 c
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 448 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
688 64 m
|
||||||
|
48 0 0 -48 640 64 640 16 a
|
||||||
|
256 64 m
|
||||||
|
48 0 0 48 304 64 304 16 a
|
||||||
|
304 16 m
|
||||||
|
640 16 l
|
||||||
|
688 64 m
|
||||||
|
688 256
|
||||||
|
640 448 c
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 448 m
|
||||||
|
256 64 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 64 m
|
||||||
|
640 64 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 64 m
|
||||||
|
640 448 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
640 448 m
|
||||||
|
256 448 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 448 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<path stroke="light gray">
|
||||||
|
256 64 m
|
||||||
|
448 256 l
|
||||||
|
</path>
|
||||||
|
<text transformations="translations" pos="448 464" stroke="light gray" type="label" width="30.5568" height="16.224" depth="2.976" halign="center" valign="baseline">3,5</text>
|
||||||
|
<text transformations="translations" pos="240 256" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" halign="right" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="352 368" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="544 368" stroke="light gray" type="label" width="11.9544" height="16.212" depth="0" halign="right" valign="baseline">2</text>
|
||||||
|
<text transformations="translations" pos="624 256" stroke="light gray" type="label" width="30.5568" height="15.6984" depth="2.976" halign="right" valign="baseline">4,5</text>
|
||||||
|
<text transformations="translations" pos="352 176" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" halign="right" valign="baseline">3</text>
|
||||||
|
<text matrix="1 0 0 1 0 -16" transformations="translations" pos="560 176" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" valign="baseline">3</text>
|
||||||
|
<text transformations="translations" pos="448 80" stroke="light gray" type="label" width="11.9544" height="15.684" depth="0" halign="center" valign="baseline">4</text>
|
||||||
|
<text transformations="translations" pos="448 32" stroke="light gray" type="label" width="11.9544" height="16.212" depth="0" halign="center" valign="baseline">1</text>
|
||||||
|
<text transformations="translations" pos="448 512" stroke="light gray" type="label" width="11.9544" height="16.2336" depth="0.504" halign="center" valign="baseline">6</text>
|
||||||
|
<use matrix="1 0 0 1 -64 64" name="mark/disk(sx)" pos="320 384" size="normal" stroke="black"/>
|
||||||
|
<use matrix="1 0 0 1 -64 -64" name="mark/disk(sx)" pos="320 128" size="normal" stroke="black"/>
|
||||||
|
<use matrix="1 0 0 1 64 -64" name="mark/disk(sx)" pos="576 128" size="normal" stroke="black"/>
|
||||||
|
<use matrix="1 0 0 1 64 64" name="mark/disk(sx)" pos="576 384" size="normal" stroke="black"/>
|
||||||
|
<use name="mark/disk(sx)" pos="448 256" size="normal" stroke="black"/>
|
||||||
|
<path stroke="light purple" pen="ultrafat" arrow="ng-normal/normal">
|
||||||
|
640 448 m
|
||||||
|
256 448 l
|
||||||
|
</path>
|
||||||
|
</page>
|
||||||
|
</ipe>
|
||||||
Loading…
Add table
Add a link
Reference in a new issue