From 9213a8ba54ec8e00cb138710540ff4a394a2eed6 Mon Sep 17 00:00:00 2001
From: Thomas Ba <git@thomasba.de>
Date: Mon, 16 Jan 2012 15:19:27 +0100
Subject: [PATCH] =?UTF-8?q?AlgoDS2=20L=C3=B6sungen=20f=C3=BCr=20Aufgabenbl?=
 =?UTF-8?q?att=204=20hinzugef=C3=BCgt?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../Uebung_4/Uebung_4.tex                     | 246 ++++++++++++
 .../Uebung_4/bilder/1a.xml                    | 339 +++++++++++++++++
 .../Uebung_4/bilder/1b.xml                    | 296 +++++++++++++++
 .../Uebung_4/bilder/1c.xml                    | 349 +++++++++++++++++
 .../Uebung_4/bilder/2b.xml                    | 279 ++++++++++++++
 .../Uebung_4/bilder/2c.xml                    | 359 ++++++++++++++++++
 .../Uebung_4/bilder/4.xml                     | 253 ++++++++++++
 .../Uebung_4/bilder/6.xml                     | 260 +++++++++++++
 8 files changed, 2381 insertions(+)
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/Uebung_4.tex
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1a.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1b.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1c.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/2b.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/2c.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/4.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/6.xml

diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_4/Uebung_4.tex b/Algorithmen_und_Datenstrukturen_2/Uebung_4/Uebung_4.tex
new file mode 100644
index 0000000..5c0e4be
--- /dev/null
+++ b/Algorithmen_und_Datenstrukturen_2/Uebung_4/Uebung_4.tex
@@ -0,0 +1,246 @@
+\documentclass[11pt]{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage[ngerman]{babel}
+\usepackage{amsmath}
+\usepackage{enumerate}
+\usepackage{latexsym}
+%\usepackage{amssymb}
+\usepackage{multicol}
+%\usepackage{booktabs}
+%\usepackage{pstricks}
+%\usepackage{pst-node}
+\usepackage[paper=a4paper,left=30mm,right=20mm,top=20mm,bottom =25mm]{geometry}
+\usepackage[
+ pdftitle={Uebungsblatt 4: Graphentheorie - AlgoDS2},
+ pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 4: Graphentheorie},
+ pdfauthor={Thomas Battermann},
+ pdfkeywords={Uebungsblatt 4: Graphentheorie - AlgoDS2},
+ pdfborder={0 0 0}
+]{hyperref}
+
+\usepackage{listings}
+
+
+\lstset{
+  language=Java,
+  basicstyle=\ttfamily,
+  keywordstyle=\color{red},
+  commentstyle=\color{blue},
+  stringstyle=\color{darkgreen},
+  backgroundcolor=\color{lightgray},
+  morecomment=[s][\color{blue}]{/*}{*/},
+  emphstyle=\textbf,
+  frame=single,
+  showstringspaces=true,
+  breaklines=true,
+  numberstyle=\textcolor{grey},
+  numbers=left,
+  morekeywords={Merge,ExtractMinimum,Union,NIL,return,and,InvertChildren,then,to},
+  tabsize=4,
+  captionpos=b
+}
+
+
+\usepackage{tabularx}
+\usepackage{graphicx}
+\usepackage[usenames,dvipsnames]{color}
+\definecolor{grey}{rgb}{0.5,0.5,0.5}
+\definecolor{lightgray}{rgb}{0.9,0.9,0.9}
+\definecolor{darkgreen}{rgb}{0.0,0.4,0.0}
+\usepackage{lastpage}
+\usepackage{fancyhdr}
+\setlength{\parindent}{0ex}
+\setlength{\parskip}{2ex}
+\setcounter{secnumdepth}{4}
+\setcounter{tocdepth}{4}
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+\definecolor{darkblue}{rgb}{0,0,0.5}
+
+\pagestyle{fancy} %eigener Seitenstil
+\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
+\fancyhead[L]{Übungsblatt 4\\Graphentheorie} %Kopfzeile links
+\fancyhead[C]{} %zentrierte Kopfzeile
+\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\12. Januar 2011} %Kopfzeile rechts
+\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie
+\fancyfoot[C]{Seite \thepage\ von \pageref{LastPage}}
+\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie
+
+\newcommand{\spa}{\hspace*{4mm}}
+\newcommand{\defin}{\textcolor{darkgreen}{\textbf{Def.: }}}
+\newcommand{\rrfloor}{\right\rfloor}
+\newcommand{\llfloor}{\left\lfloor}
+
+
+\title{Übung Graphentheorie 1 -- AlgoDS2}
+\author{Thomas Battermann}
+\date{12. Januar 2011}
+
+\begin{document}
+\pagestyle{fancy}
+
+\section*{Aufgabe 1}
+\addcontentsline{toc}{section}{Aufgabe 1}
+
+\subsection*{a)}
+\addcontentsline{toc}{subsection}{a)}
+
+\includegraphics{bilder/1a.eps}
+
+\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
+	a&q&s&t&w&v&x&y&z&r&u\\\hline
+	$d(a)$ & 0&1&1&1&2&2&2&3&$\infty$&$\infty$\\\hline
+  $\pi(a)$ & -&q&q&q&s&t&t&x&-&-
+\end{tabular}
+
+\subsection*{b)}
+\addcontentsline{toc}{subsection}{b)}
+
+Vorgängergraph:\\
+\includegraphics{bilder/1b.eps}
+
+\subsection*{c)}
+\addcontentsline{toc}{subsection}{c)}
+
+\includegraphics{bilder/1c.eps}
+
+
+\section*{Aufgabe 2}
+\addcontentsline{toc}{section}{Aufgabe 2}
+
+\subsection*{a)}
+\addcontentsline{toc}{subsection}{a)}
+
+Kante \((u,v) \in E \)
+
+Angenommen $(u,v)$ ist eine TreeEdge (\textcolor{Red}{ForwardEdge})\\
+\begin{itemize}
+	\item[$\Rightarrow$] In der Tiefensuche wird $v$ von $u$ aus entdeckt.
+	\item[$\Rightarrow$] $v$ ist ein Nachkomme von $u$ im Tiefensuchebaum
+	\item[$\Rightarrow$] Verschachtelungssatz:\\
+		\( d[u] < d[v] < f[v] < f[u] \)
+\end{itemize}
+
+\('\Rightarrow'\) Angenommen \( d[u] < d[v] < f[v] < f[u] \)
+\begin{itemize}
+	\item[$\Rightarrow$] Verschachtelungssatz:\\
+		$v$ ist ein Nachkomme von $u$ um selben Tiefensuchebaum.\\
+		\begin{itemize}
+			\item[Fall 1:]
+				$v$ wurde von $u$ aus entdeckt $\Rightarrow (u,v)$ TreeEdge
+			\item[Fall 2:]
+				$v$ wurde nicht von $u$ aus entdeckt $\Rightarrow (u,v)$ ForwardEdge
+		\end{itemize}
+\end{itemize}
+
+\subsection*{b)}
+\addcontentsline{toc}{subsection}{b)}
+
+$'\Rightarrow'$ Angenommen $(u,v)$ ist eine BackEdge\\
+d.\,h. $u$ ist ein Nachfolger von $v$ im selben Tiefesuchebaum\\
+$\Leftrightarrow$ Verschachtelungssatz \( d[v] < d[u] < f[u] y f[v] \)
+
+\includegraphics{bilder/2b.eps}
+
+\subsection*{c)}
+\addcontentsline{toc}{subsection}{c)}
+
+\includegraphics{bilder/2c.eps}
+
+$'\Rightarrow': (u,v)$ CrossEdge
+
+Also sind $u$ und $v$ in verschiedenen Tiefensuchebäumen.
+
+\(\Rightarrow\) $v$ muss vor $u$ in der Tiefensuche entdeckt worden sein.\\
+\(\Rightarrow d[v] < f[v] < d[u] < f[u] \)\\
+\(\Leftarrow\) Angenommen: \( d[v] < f[v] < d[u] < f[u] \)\\
+\(\Rightarrow\) Verschachtelungssatz: $u$ und $v$ sind in zwei verschiedenen TS-Bäumen\\
+\(\Rightarrow (u,v) \) ist CrossEdge
+
+\section*{Aufgabe 3}
+\addcontentsline{toc}{section}{Aufgabe 3}
+
+Laufzeit: \( O(\|\|V\|\|^2) \)\\
+sowohl Breitensuche und Tiefensuche
+
+$\Rightarrow$ Es müssen alle Spalten/Zeilen durchgegangen werden.
+
+\section*{Aufgabe 4}
+\addcontentsline{toc}{section}{Aufgabe 4}
+
+Annahme: \includegraphics{bilder/4.eps}
+
+Behauptung: Jede Tiefensuche liefert $d[v] < f[u]$ als Ergebnis.
+
+Betrachte eine beliebige Tiefensuche zu dem Zeitpunkt, an dem $v$ entdeckt wird.
+
+\underline{Fall 1:} $u$ wurde vor $v$ entdeckt\\
+Weißer Pfadsatz: $v$ ist ein Nachfolger von u \(\to d[v] < f[u]\) 
+
+\underline{Fall 2:} $u$ wurde noch nicht entdeckt:\\
+\begin{enumerate}[a)]
+	\item \(v\overset{P}{\leadsto}u \Rightarrow u\) Nachkomme von $v$\\
+		Verschachtelungssatz: \( d[v] < d[u] < f[u] < f[v] \)
+	\item $u$ ist von $v$ aus nicht erreichbar.\\
+		Also wird $v$ komplett abgearbeitet, bevor $u$ entdeckt wird:\\
+		\( d[v] < f[v] < d[u] < f[u] \)
+\end{enumerate}
+
+
+\section*{Aufgabe 5}
+\addcontentsline{toc}{section}{Aufgabe 5}
+
+Graph $G=(V,E)$ (gerichtet)\\
+$E^T = \{ (v,u) \mid (u,v) \in E \}$
+
+\begin{lstlisting}[frame=single]
+MatrixTranspose(a):
+	n = size(a)
+	for i:= 1 to n do
+		for j:= 1 to n do
+			B[i][j] = A[j][i]
+	return B
+\end{lstlisting}
+
+Laufzeit: \( \Theta(n^2)\)
+
+\begin{lstlisting}[frame=single]
+ListTranspose(G=(V,E)):
+	Erzeuge einen leeren Graphen G2 mit der Knotenmenge V
+	for jeden Knoten u in V do
+		for jeden Knoten v in Adj[u] do
+			Fuege Kante (v,u) zu G2 hinzu
+	return G2
+\end{lstlisting}
+
+Laufzeit: \( O(\|V\| + \|E\| ) \)
+
+
+\section*{Aufgabe 6}
+\addcontentsline{toc}{section}{Aufgabe 6}
+
+\includegraphics{bilder/6.eps}
+
+\underline{Ansatz:} Führe Tiefensuche durch und speichere die benutzten Kanten in einer Liste.
+
+\begin{lstlisting}[frame=single]
+Search(G,s):
+	for each v in V do
+		color[v] := WHITE
+	Initialisiere die Liste p
+	Visit(G,s,p)
+\end{lstlisting}
+
+\begin{lstlisting}[frame=single]
+Visit(G,v,p):
+	color[v] := GRAY
+	Haenge v an das Ende von p
+	for each u in Neighbour(v) do
+		if color[u] = WHITE then
+			Visit(G,u,p)
+			Haenge v an das Ende von P
+		else if color[u] = GRAY then
+			Haenge (u,v) an das Ende von p
+	color[v] := BLACK
+\end{lstlisting}
+
+\end{document}
diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1a.xml b/Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1a.xml
new file mode 100644
index 0000000..6eae2ee
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+++ b/Algorithmen_und_Datenstrukturen_2/Uebung_4/bilder/1a.xml
@@ -0,0 +1,339 @@
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index 0000000..01fce54
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