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\usepackage[
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pdftitle={Algorithmen},
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pdfsubject={Mitschrift der Vorlesung "Algorithmen" an der Hochschule Aalen, bei Herrn Thierauf.},
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pdfauthor={Thomas Battermann},
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pdfkeywords={Algorithmen},
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]{hyperref}
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\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
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\fancyhead[L]{Algorithmen} %Kopfzeile rechts
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\fancyhead[C]{} %zentrierte Kopfzeile
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\fancyhead[R]{\rightmark} %Kopfzeile links
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\newcommand{\bsp}{\textcolor{lightgreen}{\textbf{Bsp.: }}}
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\newcommand{\beh}{\textcolor{lightblue}{\textbf{Beh.:}}}
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\newcommand{\lemma}{\textbf{Lemma:}}
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\title{Algorithmen}
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\author{Mitschrift von Thomas Battermann\\Dozent: Prof. Thomas Thierauf}
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\date{4. Semester}
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\begin{document}
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\input{flussnetzwerke}
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\end{document}
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@ -0,0 +1,32 @@
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\section{Flüsse in Netzwerken}
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\includegraphics{img/flussnetzwerk.pdf}
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Ein \underline{Flussnetzwerk} besteht aus einem gerichteten Graph \(G=(V,E)\) und einer \underline{Kapazitätsfunktion} \(c: V\times V \to \mathbb R\) mit \(c(u,v) \ge 0 \forall u,v \in V\) und \(c(u,v) = 0\), falls \((u,v) \not\in E\).\\
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Außerdem seien \(s\ne t \in V\) 2 ausgezeichnete Knoten. Dabei liege jeder Knoten auf einem Weg von \(s\) nach \(t\).
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Ein \underline{Fluss (im Netzwerk)} ist eine Funktion \( f: V\times V \to \mathbb R\) mit folgenden Eigenschaften:
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\begin{enumerate}
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\item \underline{Kapazitätsbedingung}: \(f(u,v) \le c(u,v) \forall u,v \in V\)
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\item \underline{Symmetriebedingung}: \(f(u,v) = -f(v,u) \forall u,v \in V\)\\
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Ein Fluss von z.\,B. 12 von $u$ nach $v$ ist auch ein Fluss von \(-12\) von $v$ nach $u$.
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\item \underline{Kirchhoffsches Gesetz}:\\
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\(\forall u\in V - \{s,t\}\)\\
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\(\sum\limits_{v\in V} f(u,v) = 0\)\\
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Bsp.: \(v: 12+(-11)+(-1) = 0\)
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\end{enumerate}
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\subsection{Wert des Flusses}
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Der Wert des Flusses $f$ ist \( \|f\| = \sum\limits_{v\in V} f(s,v) \)\\
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Kurzschreibweise sei \(X,Y \subseteq V\)\\
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dann Def. \( f(X,Y) = \sum\limits_{x\in X} \sum\limits_{y\in Y} f(x,y) \)
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\underline{Maximaler Fluss} berechnen:
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Gegeben: \( G=(V,E), \ s,t \in V, c \)\\
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Gesucht: Fluss $f$, so dass \(\|f\|\) maximal ist.
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% vim: ft=tex :
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<gridsize name="32 pts (~12 mm)" value="32"/>
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<gridsize name="10 pts (~3.5 mm)" value="10"/>
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<gridsize name="20 pts (~7 mm)" value="20"/>
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<gridsize name="14 pts (~5 mm)" value="14"/>
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<gridsize name="28 pts (~10 mm)" value="28"/>
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<gridsize name="56 pts (~20 mm)" value="56"/>
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<anglesize name="60 deg" value="60"/>
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<anglesize name="45 deg" value="45"/>
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208 760 l
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64 712 m
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64 696 m
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216 768 m
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136 768 m
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232 768 l
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</path>
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<path matrix="1 0 0 1 0 4" transformations="rigid" stroke="black" arrow="normal/normal">
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136 640 m
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232 640 l
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</path>
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<path transformations="rigid" stroke="black" arrow="normal/normal">
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124 760 m
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124 648 l
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</path>
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<path transformations="rigid" stroke="black" arrow="normal/normal">
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132 648 m
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132 760 l
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</path>
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120 760 m
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72 712 l
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</path>
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<path stroke="black" arrow="normal/normal">
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120 648 m
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72 696 l
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</path>
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<path stroke="black" arrow="normal/normal">
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228 760 m
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140 652 l
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</path>
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<path stroke="black" arrow="normal/normal">
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144 648 m
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232 756 l
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</path>
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<path stroke="black" arrow="normal/normal">
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296 712 m
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248 760 l
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</path>
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<path stroke="black" arrow="normal/normal">
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232 636 m
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136 636 l
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</path>
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</page>
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</ipe>
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@ -0,0 +1,13 @@
|
||||
#!/bin/bash
|
||||
|
||||
for FILE in $(find img/ -iname '*.xml') ; do
|
||||
if [ ! -f "${FILE%.xml}.pdf" -o "$FILE" -nt "${FILE%.xml}.pdf" ] ; then
|
||||
ipetoipe -pdf -runlatex $FILE ${FILE%.xml}.pdf
|
||||
fi
|
||||
done
|
||||
|
||||
if [ ! -f "Algorithmen.toc" ] ; then
|
||||
# Run it twice to generate TOC
|
||||
pdflatex Algorithmen.tex
|
||||
fi
|
||||
pdflatex Algorithmen.tex
|
||||
Loading…
Reference in new issue