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\documentclass[11pt]{scrartcl}
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\usepackage[utf8]{inputenc}
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\usepackage[ngerman]{babel}
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\usepackage{amsmath}
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\usepackage{enumerate}
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\usepackage{latexsym}
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%\usepackage{amssymb}
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\usepackage{multicol}
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%\usepackage{booktabs}
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%\usepackage{pstricks}
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%\usepackage{pst-node}
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\usepackage[paper=a4paper,left=30mm,right=20mm,top=20mm,bottom =25mm]{geometry}
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\usepackage[
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pdftitle={Uebungsblatt 5: Graphentheorie II - AlgoDS2},
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pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 5: Graphentheorie II},
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pdfauthor={Thomas Battermann},
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pdfkeywords={Uebungsblatt 5: Graphentheorie II - AlgoDS2},
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pdfborder={0 0 0}
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]{hyperref}
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\usepackage{listings}
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\lstset{
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language=Java,
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basicstyle=\ttfamily,
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tabsize=4,
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captionpos=b
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}
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\usepackage{tabularx}
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\usepackage{graphicx}
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\usepackage[usenames,dvipsnames]{color}
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\usepackage{lastpage}
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\usepackage{fancyhdr}
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\setlength{\parindent}{0ex}
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\setlength{\parskip}{2ex}
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\setcounter{secnumdepth}{4}
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\setcounter{tocdepth}{4}
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\usepackage{multirow}
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\usepackage[table]{xcolor}
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\pagestyle{fancy} %eigener Seitenstil
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\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
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\fancyhead[L]{Übungsblatt 5\\Graphentheorie II} %Kopfzeile links
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\fancyhead[C]{} %zentrierte Kopfzeile
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\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\26. Januar 2011} %Kopfzeile rechts
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\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie
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\fancyfoot[C]{Seite \thepage\ von \pageref{LastPage}}
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\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie
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\title{Übung Graphentheorie II -- AlgoDS2}
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\author{Thomas Battermann}
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\date{26. Januar 2011}
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\begin{document}
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\pagestyle{fancy}
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\section*{Aufgabe 1}
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\addcontentsline{toc}{section}{Aufgabe 1}
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\subsection*{a)}
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\addcontentsline{toc}{subsection}{a)}
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\begin{tabular}{rlc}
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\textbf{Kante} & \textbf{Knoten} & (Gewicht)\\
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& \(\{a\},\{b\},\{d\},\{f\},\{g\},\{h\},\{k\}\) & \\
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\((g,k)\) & \(\{a\},\{b\},\{d\},\{f\},\{g,k\},\{h\}\) & 12\\
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\((g,h)\) & \(\{a\},\{b\},\{d\},\{f\},\{g,k,h\}\) & 15\\
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\((a,b)\) & \(\{a,b\},\{d\},\{f\},\{g,k,h\}\) & 15\\
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\((a,h)\) & Bereits in einer Menge. & 18\\
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\((a,f)\) & \(\{a,b,f\},\{d\},\{g,k,h\}\) & 20\\
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\((a,d)\) & \(\{a,b,d,f\},\{g,k,h\}\) & 22\\
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\((d,g)\) & \(\{a,b,d,f,g,k,h\}\) & 23\\
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\end{tabular}
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\subsection*{b)}
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\addcontentsline{toc}{subsection}{b)}
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\begin{tabular}{c||c|c|c|c|c|c|c}
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\( d \) & \( a \) & \( b \) & \( d \) & \( f \) & \( g \) & \( h \) & \( k \) \\\hline
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\( \pi(d) \) & \( - \) & \( a \) & \( a \) & \( a \) & \( d \) & \( g \) & \( g \)
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\end{tabular}
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\subsection*{c)}
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\addcontentsline{toc}{subsection}{c)}
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\begin{tabular}{c|c||c|c|c|c|c|c|c}
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& & \( a \) & \( b \) & \( d \) & \( f \) & \( g \) & \( h \) & \( k \) \\
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\hline
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& \( \pi \) & \cellcolor{green} \( - \) & \( - \) & \( - \) & \( - \) & \( - \) & \( - \) & \( - \)\\
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& \( d \) & \cellcolor{green} \( 0 \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \) & \( \infty \)\\
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\hline
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\multirow{2}{*}{a} & \( \pi \) & \( - \) & \cellcolor{green} \( a \) & \cellcolor{green} \( a \) & \cellcolor{green}\( a \) & \cellcolor{green} \( a \) & \( - \) & \( - \)\\
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& \( d \) & \( 0 \) & \cellcolor{green} \( 16 \) & \cellcolor{green} \( 22 \) & \cellcolor{green}\( 20 \) & \cellcolor{green} \( 29 \) & \( \infty \) & \( \infty \)\\
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\hline
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\multirow{2}{*}{b} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \cellcolor{green} \( b \) & \( - \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \cellcolor{green} \( 47 \) & \( \infty \)\\
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\hline
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\multirow{2}{*}{f} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( b \) & \cellcolor{green} \( f \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 47 \) & \cellcolor{green} \( 45 \)\\
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\hline
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\multirow{2}{*}{d} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( b \) & \( f \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 47 \) & \( 45 \)\\
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\hline
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\multirow{2}{*}{g} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \cellcolor{green} \( g \) & \cellcolor{green} \( g \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \cellcolor{green} \( 44 \) & \cellcolor{green} \( 41 \)\\
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\hline
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\multirow{2}{*}{k} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( g \) & \( g \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 44 \) & \( 41 \)\\
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\hline
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\multirow{2}{*}{h} & \( \pi \) & \( - \) & \( a \) & \( a \) & \( a \) & \( a \) & \( g \) & \( g \)\\
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& \( d \) & \( 0 \) & \( 16 \) & \( 22 \) & \( 20 \) & \( 29 \) & \( 44 \) & \( 41 \)\\
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\hline
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\hline
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\end{tabular}
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\section*{Aufgabe 2}
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\addcontentsline{toc}{section}{Aufgabe 2}
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\begin{tabular}{cc|c|c|c|c}
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\multicolumn{2}{c|}{Runde} & 1 & 2 & 3 & 4 \\
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\hline
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\((s,t)\) & \( 6\) & \(t: 6/s\) & \(-\) & \(-\) & \(-\) \\
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\((s,y)\) & \( 7\) & \(y: 7/s\) & \(-\) & \(-\) & \(-\) \\
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\((t,x)\) & \( 5\) & \(x: 11/t\) & \(-\) & \(-\) & \(-\) \\
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\((t,y)\) & \( 8\) & \(-\) & \(-\) & \(-\) & \(-\) \\
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\((t,z)\) & \(-4\) & \(z: 2/t\) & \(-\) & \(z: 2/t\) & \(-\) \\
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\((x,t)\) & \(-2\) & \(-\) & \(t: 2/x\) & \(-\) & \(-\) \\
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\((y,z)\) & \( 9\) & \(-\) & \(-\) & \(-\) & \(-\) \\
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\((y,x)\) & \(-3\) & \(x: 4/a\) & \(-\) & \(-\) & \(-\) \\
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\((z,s)\) & \( 2\) & \(-\) & \(-\) & \(-\) & \(-\) \\
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\((z,x)\) & \( 7\) & \(-\) & \(-\) & \(-\) & \(-\) \\
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\end{tabular}
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\((t: d[t]/\pi[t])\)
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\textbf{Rundenergebnisse:}
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\begin{tabular}{cc|ccccc}
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&& \(s\) & \(t\) & \(x\) & \(y\) & \(z\) \\
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\hline
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\multirow{2}{*}{Runde 1} & d & \( 0 \) & \( 6 \) & \( 4 \) & \( 7 \) & \( 2 \) \\
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& \(\pi\) & \( - \) & \( s \) & \( y \) & \( s \) & \( t \) \\
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\hline
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\multirow{2}{*}{Runde 2} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( 2 \) \\
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& \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\
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\hline
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\multirow{2}{*}{Runde 3} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( -2 \) \\
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& \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\
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\hline
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\multirow{2}{*}{Runde 4} & d & \( 0 \) & \( 2 \) & \( 4 \) & \( 7 \) & \( -2 \) \\
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& \(\pi\) & \( - \) & \( x \) & \( y \) & \( s \) & \( t \) \\
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\hline
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\end{tabular}
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Runde 3 ist identisch mit Runde 4
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\end{document}
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