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@@ -7,6 +7,7 @@
 \usepackage{hyperref} %Links
 \usepackage{ifthen} %ifthenelse
 \usepackage{enumerate}
+\usepackage{stmaryrd}
 
 \usepackage{color}
 \usepackage{algpseudocode} %Pseudocode
@@ -87,24 +88,17 @@
 \header{6}{}{2015-12-02}{Kommunikationsnetze}{\textit{Jonas Jaszkowic, 3592719}\\\textit{Jan-Peter Hohloch, 3908712}}{WS 15/16}{4}
 \vspace{1cm}
 \Aufgabe{Banyan Switches}{20}\\
-We assume $N=2^k,\ k\in\mathds{Z}$, otherwise it is possible to construct a blocking configuration.\\[0.2cm]
-The lower half of the whole switch is non-blocking if the upper half isn't (symmetric, we can exchange the labels $a_i$ and $b_i$).\\We have to show $a_i\geq N \oplus b_{N-i+1}\geq N$ which means only one of them has $1$ as most significant bit, the other $0$.\\
-Given $a_i < a_{i+1},\ b_i<b_{i+1}$ we know that $a_1\le N$ and $b_N\geq N-1$. We distinguish two cases:
-\begin{enumerate}[c1:]
-    \item Assume $a_1<N$\\
-        $\Rightarrow b_N\geq N$ since it is the largest of the $b_i$ and there are no other addresses left.\checkmark
-    \item Assume $a_1=N$\\
-        $\Rightarrow b_N=N-1$ since $a_1$ has to be the smallest of the $a_i$\checkmark
+We assume $N=2^k,\ k\in\mathds{N}_0$, otherwise there are cases where the first stage is blocking.\\[0.2cm]
+We have $2N$ port values, $N$ are $< N$, $N$ are $\geq N$.\\
+Assuming there might be an $a_i,b_{N-i+1}$ where both have the same most significant bit (here we use the assumption from above) we distinguish two cases:
+\begin{enumerate}[c1]
+    \item $a_i\geq N \wedge b_{N-i+1}\geq N$\\
+        To assign numbers to all ports $a_k$ we need an additional amount of $N-i$ numbers $\geq N$ ($k>i$). For the ports $b_k$ we need $i$ ($k\geq N-i+1$). In sum that are $1+N-i+i=N+1>N\ \lightning$
+    \item $a_i< N \wedge b_{N-i+1}< N$\\
+        To assign numbers to all ports $a_k$ we need $i$ numbers $<N$ ($k\leq i$). For the ports $b_k$ we need $N-i+1$ ($k<N-i+1$). In sum that are $N-i+1+i=N+1>N\ \lightning$
 \end{enumerate}
-So the switch for $a_1,b_N$ is non-blocking.\\
-Similar we can handle the switches $a_i,b_{N-i+1}$:\\
-We know that $i-1\le a_i < N+i\forall 1\le i\le N$, same for $b_i$.\\
-For the current switch we show:\\ $x\geq N \oplus y\geq N$, where $i-1\le x < N+i,\ N-i< y \le 2N-i$\\
-We distinguish two cases:
-\begin{enumerate}[c1:]
-    \item Assume $i-1\le x<N$\\%TODO
-    \item Assume $N\le x < N+i$\\
-\end{enumerate}
+So all possibilities where there is blocking in first stage aren't possible which means the first stage is non-blocking.\qed
+\\[0.5cm]
 \Aufgabe{Flow Control and ARQ Protocols}{22}
 \begin{enumerate}
     \item Flow Control means the controlling of the transmission of data. This is necessary for example if the receiver is slower or has higher traffic than the sender and can't keep up. For flow control ACK message can be used to acknowledge received messages (used in the examples from he lecture).