From 7bb41f24d8014db48aa32a599ef73ab84014f4f2 Mon Sep 17 00:00:00 2001 From: Thomas Ba Date: Tue, 31 Jan 2012 20:55:33 +0100 Subject: [PATCH] =?UTF-8?q?AlgoDS2=20=C3=9Cbungsblatt=205=20-=20L=C3=B6sun?= =?UTF-8?q?gen?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Uebung_5/Uebung_5.tex | 174 ++++++++++++++++++ 1 file changed, 174 insertions(+) create mode 100644 Algorithmen_und_Datenstrukturen_2/Uebung_5/Uebung_5.tex diff --git a/Algorithmen_und_Datenstrukturen_2/Uebung_5/Uebung_5.tex b/Algorithmen_und_Datenstrukturen_2/Uebung_5/Uebung_5.tex new file mode 100644 index 0000000..9a4b297 --- /dev/null +++ b/Algorithmen_und_Datenstrukturen_2/Uebung_5/Uebung_5.tex @@ -0,0 +1,174 @@ +\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 5: Graphentheorie II - AlgoDS2}, + pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 5: Graphentheorie II}, + pdfauthor={Thomas Battermann}, + pdfkeywords={Uebungsblatt 5: Graphentheorie II - 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} +\usepackage{lastpage} +\usepackage{fancyhdr} +\setlength{\parindent}{0ex} +\setlength{\parskip}{2ex} +\setcounter{secnumdepth}{4} +\setcounter{tocdepth}{4} +\usepackage{multirow} +\usepackage[table]{xcolor} + +\pagestyle{fancy} %eigener Seitenstil +\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen +\fancyhead[L]{Übungsblatt 5\\Graphentheorie II} %Kopfzeile links +\fancyhead[C]{} %zentrierte Kopfzeile +\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\26. Januar 2011} %Kopfzeile rechts +\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie +\fancyfoot[C]{Seite \thepage\ von \pageref{LastPage}} +\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie + + +\title{Übung Graphentheorie II -- AlgoDS2} +\author{Thomas Battermann} +\date{26. Januar 2011} + +\begin{document} +\pagestyle{fancy} + +\section*{Aufgabe 1} +\addcontentsline{toc}{section}{Aufgabe 1} + +\subsection*{a)} +\addcontentsline{toc}{subsection}{a)} + +\begin{tabular}{rlc} + \textbf{Kante} & \textbf{Knoten} & (Gewicht)\\ + & \(\{a\},\{b\},\{d\},\{f\},\{g\},\{h\},\{k\}\) & \\ + \((g,k)\) & \(\{a\},\{b\},\{d\},\{f\},\{g,k\},\{h\}\) & 12\\ + \((g,h)\) & \(\{a\},\{b\},\{d\},\{f\},\{g,k,h\}\) & 15\\ + \((a,b)\) & \(\{a,b\},\{d\},\{f\},\{g,k,h\}\) & 15\\ + \((a,h)\) & Bereits in einer Menge. & 18\\ + \((a,f)\) & \(\{a,b,f\},\{d\},\{g,k,h\}\) & 20\\ + \((a,d)\) & \(\{a,b,d,f\},\{g,k,h\}\) & 22\\ + \((d,g)\) & \(\{a,b,d,f,g,k,h\}\) & 23\\ +\end{tabular} + +\subsection*{b)} +\addcontentsline{toc}{subsection}{b)} + +\begin{tabular}{c||c|c|c|c|c|c|c} + \( d \) & \( a \) & \( b \) & \( d \) & \( f \) & \( g \) & \( h \) & \( k \) \\\hline + \( \pi(d) \) & \( - \) & \( a \) & \( a \) & \( a \) & \( d \) & \( g \) & \( g \) +\end{tabular} + +\subsection*{c)} +\addcontentsline{toc}{subsection}{c)} + +\begin{tabular}{c|c||c|c|c|c|c|c|c} + & & \( a \) & \( b \) & \( d \) & \( f \) & \( g \) & \( h \) & \( k \) \\ + \hline + & \( \pi \) & \cellcolor{green} \( - \) & \( - \) & \( - \) & \( - \) & \( - \) & \( - \) & \( - \)\\ + & \( d \) & \cellcolor{green} \( 0 \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \)\\ + \hline + \multirow{2}{*}{a} & \( \pi \) & \( - \) & \cellcolor{green} \( a \) & \cellcolor{green} \( a \) & \cellcolor{green}\( a \) & \cellcolor{green} \( a \) & \( - \) & \( - \)\\ + & \( d \) & \( 0 \) & \cellcolor{green} \( 16 \) & \cellcolor{green} \( 22 \) & \cellcolor{green}\( 20 \) & \cellcolor{green} \( 29 \) & \( \infty \) & \( \infty \)\\ + \hline + \multirow{2}{*}{b} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \cellcolor{green} \( b \) & \( - \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \cellcolor{green} \( 47 \) & \( \infty \)\\ + \hline + \multirow{2}{*}{f} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( b \) & \cellcolor{green} \( f \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 47 \) & \cellcolor{green} \( 45 \)\\ + \hline + \multirow{2}{*}{d} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( b \) & \( f \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 47 \) & \( 45 \)\\ + \hline + \multirow{2}{*}{g} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \cellcolor{green} \( g \) & \cellcolor{green} \( g \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \cellcolor{green} \( 44 \) & \cellcolor{green} \( 41 \)\\ + \hline + \multirow{2}{*}{k} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( g \) & \( g \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 44 \) & \( 41 \)\\ + \hline + \multirow{2}{*}{h} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( g \) & \( g \)\\ + & \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 44 \) & \( 41 \)\\ + \hline + \hline + +\end{tabular} + + +\section*{Aufgabe 2} +\addcontentsline{toc}{section}{Aufgabe 2} + +\begin{tabular}{cc|c|c|c|c} + \multicolumn{2}{c|}{Runde} & 1 & 2 & 3 & 4 \\ + \hline + \((s,t)\) & \( 6\) & \(t: 6/s\) & \(-\) & \(-\) & \(-\) \\ + \((s,y)\) & \( 7\) & \(y: 7/s\) & \(-\) & \(-\) & \(-\) \\ + \((t,x)\) & \( 5\) & \(x: 11/t\) & \(-\) & \(-\) & \(-\) \\ + \((t,y)\) & \( 8\) & \(-\) & \(-\) & \(-\) & \(-\) \\ + \((t,z)\) & \(-4\) & \(z: 2/t\) & \(-\) & \(z: 2/t\) & \(-\) \\ + \((x,t)\) & \(-2\) & \(-\) & \(t: 2/x\) & \(-\) & \(-\) \\ + \((y,z)\) & \( 9\) & \(-\) & \(-\) & \(-\) & \(-\) \\ + \((y,x)\) & \(-3\) & \(x: 4/a\) & \(-\) & \(-\) & \(-\) \\ + \((z,s)\) & \( 2\) & \(-\) & \(-\) & \(-\) & \(-\) \\ + \((z,x)\) & \( 7\) & \(-\) & \(-\) & \(-\) & \(-\) \\ +\end{tabular} + +\((t: d[t]/\pi[t])\) + +\textbf{Rundenergebnisse:} + +\begin{tabular}{cc|ccccc} + && \(s\) & \(t\) & \(x\) & \(y\) & \(z\) \\ + \hline + \multirow{2}{*}{Runde 1} & d & \( 0 \) & \( 6 \) & \( 4 \) & \( 7 \) & \( 2 \) \\ + & \(\pi\) & \( - \) & \( s \) & \( y \) & \( s \) & \( t \) \\ + \hline + \multirow{2}{*}{Runde 2} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( 2 \) \\ + & \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\ + \hline + \multirow{2}{*}{Runde 3} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( -2 \) \\ + & \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\ + \hline + \multirow{2}{*}{Runde 4} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( -2 \) \\ + & \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\ + \hline +\end{tabular} + +Runde 3 ist identisch mit Runde 4 + +\end{document}