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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{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{listings}
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\lstset{
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language=Java,
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basicstyle=\ttfamily,
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keywordstyle=\color{red},
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commentstyle=\color{blue},
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stringstyle=\color{darkgreen},
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backgroundcolor=\color{lightgray},
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morecomment=[s][\color{blue}]{/*}{*/},
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emphstyle=\textbf,
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frame=single,
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showstringspaces=true,
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breaklines=true,
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numberstyle=\textcolor{grey},
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numbers=left,
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morekeywords={Merge,ExtractMinimum,Union,NIL,return,and},
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tabsize=4,
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captionpos=b
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}
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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 1: Priority Queues - AlgoDS2},
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pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 1: Priority Queues},
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pdfauthor={Thomas Battermann},
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pdfkeywords={Uebungsblatt 1: Priority Queues - AlgoDS2},
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pdfborder={0 0 0}
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]{hyperref}
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\usepackage{tabularx}
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\usepackage{graphicx}
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\usepackage[usenames,dvipsnames]{color}
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\definecolor{grey}{rgb}{0.5,0.5,0.5}
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\definecolor{lightgray}{rgb}{0.9,0.9,0.9}
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\definecolor{darkgreen}{rgb}{0.0,0.4,0.0}
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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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\definecolor{darkgreen}{rgb}{0,0.5,0}
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\definecolor{darkblue}{rgb}{0,0,0.5}
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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 1\\Priority Queues} %Kopfzeile links
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\fancyhead[C]{} %zentrierte Kopfzeile
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\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\14. November 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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\newcommand{\spa}{\hspace*{4mm}}
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\newcommand{\defin}{\textcolor{darkgreen}{\textbf{Def.: }}}
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\newcommand{\rrfloor}{\right\rfloor}
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\newcommand{\llfloor}{\left\lfloor}
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\title{Übung Priority Queues -- AlgoDS2}
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\author{Thomas Battermann}
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\date{14. November 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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%\includegraphics{bilder/aufgabe_1.eps}
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\begin{lstlisting}[frame=single]
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Merge(H1,H2):
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head = NIL
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l1 = head(H1)
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l2 = head(H2)
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while ( l1 != NIL and l2 != NIL ) do
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if ( degree(l1) < degree(l2) ) then
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y = l1
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h = sibling(l1)
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else
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y = l2
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l2 = sibling(l2)
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if ( head == NIL ) then
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head = y
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x = y
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if ( l1 != NIL ) then
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if ( head == NIL ) then
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head = l1
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else
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sibling(x) = h
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if ( l2 != NIL ) then
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if ( head != NIL ) then
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head = l2
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else
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sibling(x) = l2
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return head;
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\end{lstlisting}
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\newpage
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\section*{Aufgabe 2}
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\addcontentsline{toc}{section}{Aufgabe 2}
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\begin{lstlisting}[frame=single]
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ExtractMinimum(H):
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x = head(H)
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prev_x = NIL
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min = x
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prev_min = NIL
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if ( x == NIL ) then
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return NIL
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prev_x = x
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x = sibling(x)
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while ( x != NIL ) do
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if ( key(x) < key(min) ) then
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min = x
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prev_min = prev_x
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prev_x = x
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x = sibling(x)
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if ( prev_min != NIL ) then
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sibling(prev_min) = sibling(min)
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else
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head = sibling(min)
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if ( child(min) != NIL ) then
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H2 = InvertChildren(min)
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Union(head,H2)
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sibling(min) = NIL
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parent(min) = NIL
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child(min) = NIL
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return min;
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\end{lstlisting}
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\section*{Aufgabe 3}
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\addcontentsline{toc}{section}{Aufgabe 3}
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Leider keine Lösung vorhanden :-(
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\end{document}
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@ -0,0 +1,159 @@
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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{ulem} % \sout
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\usepackage{marvosym} % \Lightning
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\usepackage{mathtools}
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\usepackage{multicol}
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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 2: Hashing - AlgoDS2},
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pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 2: Hashing},
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pdfauthor={Thomas Battermann},
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pdfkeywords={Uebungsblatt 2: Hashing - AlgoDS2},
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pdfborder={0 0 0}
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]{hyperref}
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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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\pagestyle{fancy} %eigener Seitenstil
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\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
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\fancyhead[L]{Übungsblatt 2\\Hashing}%Kopfzeile links
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\fancyhead[C]{} %zentrierte Kopfzeile
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\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\1. Dezember 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 Hashing -- AlgoDS2}
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\author{Thomas Battermann}
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\date{1. Dezember 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*{Mit der Formel}
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\addcontentsline{toc}{subsection}{Mit der Formel}
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\( m = 2^{12} = 4096; \quad A = \frac{\pi-2}{3} \)
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\subsubsection*{a)}
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\addcontentsline{toc}{subsubsection}{a)}
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\begin{align*}
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k &= 1000\\[4mm]
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h &= \left( 1000 * \frac{\pi-2}{3} - \lfloor 1000 * \frac{\pi-2}{3} \rfloor \right) * 4096\\
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&= 2174
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\end{align*}
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\subsubsection*{b)}
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\addcontentsline{toc}{subsubsection}{b)}
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\begin{align*}
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k &= 2500\\[4mm]
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h &= \left( 2500 * \frac{\pi-2}{3} - \lfloor 2500 * \frac{\pi-2}{3} \rfloor \right) * 4096\\
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&= 1340
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\end{align*}
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\subsection*{Mit Ganzzahlarithmetik}
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\addcontentsline{toc}{subsection}{Mit Ganzzahlarithmetik}
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\textbf{Approximation}
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\begin{align*}
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&\left| \frac{S}{2^{16}} - A \right| &\text{ minimal}\\
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&\frac{S}{2^{16}} - A = 0\\
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&S = \lceil A * 2^{16} \rceil = 24939
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\end{align*}
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\subsubsection*{a)}
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\addcontentsline{toc}{subsubsection}{a)}
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\begin{align*}
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k &= 1000\\[4mm]
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r &= s * k \mod 2^{16}\\
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&= 2493900 \mod 65536\\
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&= 35320\\[4mm]
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&\Rightarrow \underbracket{1000\,1001\,1111}\,1000\\
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h(1000) &= 2207 = \lfloor \frac{35320}{2^4} \rfloor
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\end{align*}
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\subsubsection*{b)}
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\addcontentsline{toc}{subsubsection}{b)}
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\begin{align*}
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k &= 2500\\[4mm]
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r &= s * k \mod 2^{16}\\
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&= 62347500 \mod 65536\\
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&= 22764\\[4mm]
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&\Rightarrow \underbracket{0101\,1000\,1110}\,1100\\
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H(2500) &= 1422 = \lfloor \frac{22764}{2^4} \rfloor
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\end{align*}
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\section*{Aufgabe 2}
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\addcontentsline{toc}{section}{Aufgabe 2}
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\subsection*{a)}
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\addcontentsline{toc}{subsection}{a)}
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\includegraphics{bilder/aufgabe_2a.eps}
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\subsection*{b)}
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\addcontentsline{toc}{subsection}{b)}
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\includegraphics{bilder/aufgabe_2b.eps}
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\subsection*{c)}
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\addcontentsline{toc}{subsection}{c)}
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\includegraphics{bilder/aufgabe_2c.eps}
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\begin{align*}
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h(k,i) &= (k+\frac12i+\frac12i^2) \mod 16
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\end{align*}
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\subsection*{d)}
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\addcontentsline{toc}{subsection}{d)}
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\includegraphics{bilder/aufgabe_2d.eps}
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\begin{align*}
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h_1(k) &= k \mod 11\\
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h_2(k) &= 1 + ( k \mod 10 )\\
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h(k,i) &= \Big( h_1(k) + i * h_2(k) \Big) \mod 11
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\end{align*}
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\textbf{Sondierungsfrequenz für k=31}
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\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
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$i$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
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\hline
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$h(k,i)$ & 9 & 0 & 2 & 4 & 6 & 8 & 10 & 1 & 3 & 5 & 7
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\end{tabular}
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$<9,0,2,4,6,8,10,1,3,5,7>$
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\section*{Aufgabe 3}
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\addcontentsline{toc}{section}{Aufgabe 3}
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Für alle $ 0 \le j < m $ gilt:\\
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\[ \frac{n-j}{m-j} \le \frac n m \]
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Annahme: Es gibt ein $j$ mit:\\
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\begin{align*}
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\frac{n-j}{m-j} &> \frac n m & | * (m-j) * m \\
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(n-j)*m &> n * (m-j) \\
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\sout{n*m} -j * m &> \sout{n*m} - j * m &| *\frac{1}{-j}\\[4mm]
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m &< n & \text{\huge\color{Red}\Lightning}
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\end{align*}
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\end{document}
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@ -0,0 +1,506 @@
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80 736
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144 736
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168 752 m
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176 736
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144 720
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160 720
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176 720
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184 736 m
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192 720
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<path stroke="black">
|
||||
128 768 m
|
||||
128 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
144 768 m
|
||||
144 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
160 768 m
|
||||
160 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
176 768 m
|
||||
176 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
192 768 m
|
||||
192 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
208 768 m
|
||||
208 752 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
224 768 m
|
||||
224 752 l
|
||||
</path>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="72 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$0$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="88 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$1$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="104 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$2$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="120 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$3$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="136 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$4$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="152 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$5$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="168 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$6$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="184 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$7$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="200 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$8$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="216 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">$9$</text>
|
||||
<text matrix="1 0 0 1 0 12" transformations="translations" pos="232 760" stroke="black" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">$10$</text>
|
||||
<text transformations="translations" pos="72 760" stroke="black" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">22</text>
|
||||
<text transformations="translations" pos="104 760" stroke="red" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">59</text>
|
||||
<text transformations="translations" pos="120 760" stroke="darkorange" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">17</text>
|
||||
<text transformations="translations" pos="136 760" stroke="black" type="label" width="4.981" height="6.42" depth="0" halign="center" valign="center">4</text>
|
||||
<text transformations="translations" pos="152 760" stroke="turquoise" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">15</text>
|
||||
<text transformations="translations" pos="168 760" stroke="black" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">28</text>
|
||||
<text transformations="translations" pos="184 760" stroke="seagreen" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">88</text>
|
||||
<text transformations="translations" pos="216 760" stroke="black" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">31</text>
|
||||
<text transformations="translations" pos="232 760" stroke="black" type="label" width="9.963" height="6.42" depth="0" halign="center" valign="center">10</text>
|
||||
<path stroke="darkorange" arrow="normal/normal">
|
||||
168 752 m
|
||||
152 736
|
||||
136 736
|
||||
120 752 s
|
||||
</path>
|
||||
<path stroke="seagreen" arrow="normal/normal">
|
||||
216 752 m
|
||||
200 728
|
||||
184 752 s
|
||||
</path>
|
||||
<path stroke="seagreen" arrow="normal/normal">
|
||||
72 752 m
|
||||
88 720
|
||||
204 720
|
||||
220 752 s
|
||||
</path>
|
||||
<path stroke="turquoise" arrow="normal/normal">
|
||||
136 752 m
|
||||
152 736
|
||||
216 736
|
||||
232 752 s
|
||||
</path>
|
||||
<path matrix="1 0 0 1 0 -8" stroke="turquoise" arrow="normal/normal">
|
||||
232 784 m
|
||||
216 800
|
||||
168 800
|
||||
152 784 s
|
||||
</path>
|
||||
<path stroke="red" arrow="normal/normal">
|
||||
136 776 m
|
||||
128 792
|
||||
120 776 s
|
||||
</path>
|
||||
<path stroke="red" arrow="normal/normal">
|
||||
120 776 m
|
||||
112 792
|
||||
104 776 s
|
||||
</path>
|
||||
</page>
|
||||
</ipe>
|
||||
Loading…
Reference in new issue