From 37322fac086890d9d51d77f05d51db63005ec432 Mon Sep 17 00:00:00 2001
From: Thomas Ba <git@thomasba.de>
Date: Sun, 25 Dec 2011 01:14:19 +0100
Subject: [PATCH] =?UTF-8?q?Aufgabenbl=C3=A4tter=201=20&=202=20hinzugef?=
 =?UTF-8?q?=C3=BCgt?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../Uebung_1/Uebung_1.tex                     | 151 ++++++
 .../Uebung_2/Uebung_2.tex                     | 159 ++++++
 .../Uebung_2/bilder/aufgabe_2a.xml            | 506 ++++++++++++++++++
 .../Uebung_2/bilder/aufgabe_2b.xml            | 338 ++++++++++++
 .../Uebung_2/bilder/aufgabe_2c.xml            | 334 ++++++++++++
 .../Uebung_2/bilder/aufgabe_2d.xml            | 342 ++++++++++++
 6 files changed, 1830 insertions(+)
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_1/Uebung_1.tex
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_2/Uebung_2.tex
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2a.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2b.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2c.xml
 create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2d.xml

diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_1/Uebung_1.tex b/Algorithmen_und_Datenstrukturen_2/Uebung_1/Uebung_1.tex
new file mode 100644
index 0000000..9b6cfcc
--- /dev/null
+++ b/Algorithmen_und_Datenstrukturen_2/Uebung_1/Uebung_1.tex
@@ -0,0 +1,151 @@
+\documentclass[11pt]{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage[ngerman]{babel}
+\usepackage{amsmath}
+%\usepackage{amssymb}
+\usepackage{multicol}
+%\usepackage{booktabs}
+%\usepackage{pstricks}
+%\usepackage{pst-node}
+
+\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},
+  tabsize=4,
+  captionpos=b
+}
+
+
+\usepackage[paper=a4paper,left=30mm,right=20mm,top=20mm,bottom =25mm]{geometry}
+\usepackage[
+ pdftitle={Uebungsblatt 1: Priority Queues - AlgoDS2},
+ pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 1: Priority Queues},
+ pdfauthor={Thomas Battermann},
+ pdfkeywords={Uebungsblatt 1: Priority Queues - AlgoDS2},
+ pdfborder={0 0 0}
+]{hyperref}
+\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 1\\Priority Queues} %Kopfzeile links
+\fancyhead[C]{} %zentrierte Kopfzeile
+\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\14. November 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 Priority Queues -- AlgoDS2}
+\author{Thomas Battermann}
+\date{14. November 2011}
+
+\begin{document}
+\pagestyle{fancy}
+
+\section*{Aufgabe 1}
+\addcontentsline{toc}{section}{Aufgabe 1}
+
+%\includegraphics{bilder/aufgabe_1.eps}
+
+\begin{lstlisting}[frame=single]
+Merge(H1,H2):
+	head = NIL
+	l1 = head(H1)
+	l2 = head(H2)
+	while ( l1 != NIL and l2 != NIL ) do
+		if ( degree(l1) < degree(l2) ) then
+			y = l1
+			h = sibling(l1)
+		else
+			y = l2
+			l2 = sibling(l2)
+		if ( head == NIL ) then
+			head = y
+			x = y
+	if ( l1 != NIL ) then
+		if ( head == NIL ) then
+			head = l1
+		else
+			sibling(x) = h
+	if ( l2 != NIL ) then
+		if ( head != NIL ) then
+			head = l2
+		else
+			sibling(x) = l2
+	return head;
+\end{lstlisting}
+
+\newpage
+\section*{Aufgabe 2}
+\addcontentsline{toc}{section}{Aufgabe 2}
+
+\begin{lstlisting}[frame=single]
+ExtractMinimum(H):
+	x = head(H)
+	prev_x = NIL
+	min = x
+	prev_min = NIL
+	if ( x == NIL ) then
+		return NIL
+	prev_x = x
+	x = sibling(x)
+	while ( x != NIL ) do
+		if ( key(x) < key(min) ) then
+			min = x
+			prev_min = prev_x
+		prev_x = x
+		x = sibling(x)
+	if ( prev_min != NIL ) then
+		sibling(prev_min) = sibling(min)
+	else
+		head = sibling(min)
+	if ( child(min) != NIL ) then
+		H2 = InvertChildren(min)
+		Union(head,H2)
+	sibling(min) = NIL
+	parent(min) = NIL
+	child(min) = NIL
+	return min;
+\end{lstlisting}
+
+
+\section*{Aufgabe 3}
+\addcontentsline{toc}{section}{Aufgabe 3}
+
+Leider keine Lösung vorhanden :-(
+
+\end{document}
diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_2/Uebung_2.tex b/Algorithmen_und_Datenstrukturen_2/Uebung_2/Uebung_2.tex
new file mode 100644
index 0000000..9d88d7f
--- /dev/null
+++ b/Algorithmen_und_Datenstrukturen_2/Uebung_2/Uebung_2.tex
@@ -0,0 +1,159 @@
+\documentclass[11pt]{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage[ngerman]{babel}
+\usepackage{ulem} % \sout
+\usepackage{marvosym} % \Lightning
+\usepackage{mathtools}
+\usepackage{multicol}
+\usepackage[paper=a4paper,left=30mm,right=20mm,top=20mm,bottom =25mm]{geometry}
+\usepackage[
+ pdftitle={Uebungsblatt 2: Hashing - AlgoDS2},
+ pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 2: Hashing},
+ pdfauthor={Thomas Battermann},
+ pdfkeywords={Uebungsblatt 2: Hashing - AlgoDS2},
+ pdfborder={0 0 0}
+]{hyperref}
+\usepackage{tabularx}
+\usepackage{graphicx}
+\usepackage[usenames,dvipsnames]{color}
+\usepackage{lastpage}
+\usepackage{fancyhdr}
+\setlength{\parindent}{0ex}
+\setlength{\parskip}{2ex}
+
+\pagestyle{fancy} %eigener Seitenstil
+\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
+\fancyhead[L]{Übungsblatt 2\\Hashing}%Kopfzeile links
+\fancyhead[C]{} %zentrierte Kopfzeile
+\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\1. Dezember 2011} %Kopfzeile rechts
+\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie
+\fancyfoot[C]{Seite \thepage\ von \pageref{LastPage}}
+\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie
+
+\title{Übung Hashing -- AlgoDS2}
+\author{Thomas Battermann}
+\date{1. Dezember 2011}
+
+\begin{document}
+\pagestyle{fancy}
+
+\section*{Aufgabe 1}
+\addcontentsline{toc}{section}{Aufgabe 1}
+
+	\subsection*{Mit der Formel}
+	\addcontentsline{toc}{subsection}{Mit der Formel}
+
+		\( m = 2^{12} = 4096; \quad A = \frac{\pi-2}{3} \)
+
+		\subsubsection*{a)}
+		\addcontentsline{toc}{subsubsection}{a)}
+
+			\begin{align*}
+				k &= 1000\\[4mm]
+				h &= \left( 1000 * \frac{\pi-2}{3} - \lfloor 1000 * \frac{\pi-2}{3} \rfloor \right) * 4096\\
+				  &= 2174
+			\end{align*}
+
+		\subsubsection*{b)}
+		\addcontentsline{toc}{subsubsection}{b)}
+
+			\begin{align*}
+				k &= 2500\\[4mm]
+				h &= \left( 2500 * \frac{\pi-2}{3} - \lfloor 2500 * \frac{\pi-2}{3} \rfloor \right) * 4096\\
+				  &= 1340
+			\end{align*}
+
+	\subsection*{Mit Ganzzahlarithmetik}
+	\addcontentsline{toc}{subsection}{Mit Ganzzahlarithmetik}
+
+		\textbf{Approximation}
+
+		\begin{align*}
+			&\left| \frac{S}{2^{16}} - A \right| &\text{ minimal}\\
+			&\frac{S}{2^{16}} - A = 0\\
+			&S = \lceil A * 2^{16} \rceil = 24939
+		\end{align*}
+
+		\subsubsection*{a)}
+		\addcontentsline{toc}{subsubsection}{a)}
+
+			\begin{align*}
+				k &= 1000\\[4mm]
+				r &= s * k \mod 2^{16}\\
+				  &= 2493900 \mod 65536\\
+				  &= 35320\\[4mm]
+				  &\Rightarrow \underbracket{1000\,1001\,1111}\,1000\\
+				  h(1000) &= 2207 = \lfloor \frac{35320}{2^4} \rfloor
+			\end{align*}
+
+		\subsubsection*{b)}
+		\addcontentsline{toc}{subsubsection}{b)}
+
+			\begin{align*}
+				k &= 2500\\[4mm]
+				r &= s * k \mod 2^{16}\\
+				  &= 62347500 \mod 65536\\
+				  &= 22764\\[4mm]
+				  &\Rightarrow \underbracket{0101\,1000\,1110}\,1100\\
+				  H(2500) &= 1422 = \lfloor \frac{22764}{2^4} \rfloor
+			\end{align*}
+
+
+\section*{Aufgabe 2}
+\addcontentsline{toc}{section}{Aufgabe 2}
+
+	\subsection*{a)}
+	\addcontentsline{toc}{subsection}{a)}
+	
+		\includegraphics{bilder/aufgabe_2a.eps}
+	
+	\subsection*{b)}
+	\addcontentsline{toc}{subsection}{b)}
+	
+		\includegraphics{bilder/aufgabe_2b.eps}
+	
+	\subsection*{c)}
+	\addcontentsline{toc}{subsection}{c)}
+	
+		\includegraphics{bilder/aufgabe_2c.eps}
+	
+		\begin{align*}
+			h(k,i) &= (k+\frac12i+\frac12i^2) \mod 16
+		\end{align*}
+	
+	\subsection*{d)}
+	\addcontentsline{toc}{subsection}{d)}
+	
+		\includegraphics{bilder/aufgabe_2d.eps}
+
+		\begin{align*}
+			h_1(k) &= k \mod 11\\
+			h_2(k) &= 1 + ( k \mod 10 )\\
+			h(k,i) &= \Big( h_1(k) + i * h_2(k) \Big) \mod 11
+		\end{align*}
+
+		\textbf{Sondierungsfrequenz für k=31}
+
+		\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
+			$i$      &  0 &  1 &  2 &  3 &  4 &  5 &  6 &  7 &  8 &  9 & 10\\
+			\hline
+			$h(k,i)$ &  9 &  0 &  2 &  4 &  6 &  8 & 10 &  1 &  3 &  5 &  7
+		\end{tabular}
+
+		$<9,0,2,4,6,8,10,1,3,5,7>$
+
+\section*{Aufgabe 3}
+\addcontentsline{toc}{section}{Aufgabe 3}
+
+	Für alle $ 0 \le j < m $ gilt:\\
+	\[ \frac{n-j}{m-j} \le \frac n m \]
+
+	Annahme: Es gibt ein $j$ mit:\\
+	\begin{align*}
+		\frac{n-j}{m-j} &> \frac n m & | * (m-j) * m \\
+		(n-j)*m &> n * (m-j) \\
+		\sout{n*m} -j * m &> \sout{n*m} - j * m &| *\frac{1}{-j}\\[4mm]
+		m &< n & \text{\huge\color{Red}\Lightning}
+	\end{align*}
+
+\end{document}
diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2a.xml b/Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2a.xml
new file mode 100644
index 0000000..965642c
--- /dev/null
+++ b/Algorithmen_und_Datenstrukturen_2/Uebung_2/bilder/aufgabe_2a.xml
@@ -0,0 +1,506 @@
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new file mode 100644
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