Thomas Ba. 13 years ago
parent d2f7140c1a
commit 4b3f26701e

@ -19,9 +19,22 @@
pdfkeywords={Algorithmen},
pdfborder={0 0 0}
]{hyperref}
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@ -34,12 +47,11 @@
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@ -28,5 +28,42 @@ dann Def. \( f(X,Y) = \sum\limits_{x\in X} \sum\limits_{y\in Y} f(x,y) \)
Gegeben: \( G=(V,E), \ s,t \in V, c \)\\
Gesucht: Fluss $f$, so dass \(\|f\|\) maximal ist.
Ist \((u,v) \not\in E\) und \((v,u) \not\in E\), dann ist \( c(u,v) = c(v,u) = 0 \).\\
Für \(f\) muss also gelten:
\begin{itemize}
\item \( f(u,v) \le 0 \) und \( f(v,u) \le 0 \)
\item \( f(u,v) = -f(v,u) \Rightarrow f(u,v) = f(v,u) = 0 \)
\end{itemize}
\subsection{Restnetzwerk}
Sei $f$ ein Fluss in $G$. Die \underline{Restkapazität der Kante \((u,v)\)} ist \( c_f(u,v) = c(u,v) - f(u,v) \). Das \underline{Restnetzwerk von $G$ bzgl. $f$} ist \( G_f = (V,E_f) \) mit \( E_f = \{ (u,v) \in V\times V \mid c_f(u,v) > 0 \}\).
\bsp\\
\includegraphics{img/restnetzwerk.pdf}
Sei $p$ Weg von $s$ nach $t$ in $G_f$ die \underline{Restkapazität von $p$ in $G_f$}\\
\( c_f(p) = min \{ c_f(u,v) \mid (u,v) \in p \} \)\\
Im \bsp \( c_f(p) = 4 \)
\begin{mydef}
Definiere \(f_p\):\\
\begin{math}
f_p(u,v) = \begin{cases}
c_f(p) &,\text{ falls } (u,v) \in p\\
-c_f(p) &,\text{ falls } (v,u) \in p\\
0 &,\text{ sonst}
\end{cases}
\end{math}
\end{mydef}
\(f_p\) ist Fluss:
\begin{enumerate}
\item Kapazitätsbedingung:\\
Für \( (u,v) \in p \) ist \(f_p(u,v) = c_f(p) \le c_f(u,v) \) nach Definition von \(c_f(p) \)
\item Symmetrie gilt nach Definition
\item Kirchhoffsches Gestz: Sei $u$ Knoten auf $p$\\
\includegraphics{img/restnetzwerk_p.pdf}
\end{enumerate}
% vim: ft=tex :

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