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3.14 +\textsc{Peter~Widmayer}, Thomas~O.:
3.15 +\newblock \emph{Algorithmen und Datenstrukturen}.
3.16 +\newblock 4. Auflage.
3.17 +\newblock Spektrum Akademischer Verlag, 2002. --
3.18 +\newblock ISBN 978--3827410290
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3.21 +\textsc{Thomas H.~Cormen}, Robert L. Rivest und Cliford~S. Charles
3.22 + E.~Leiserson~L. Charles E.~Leiserson:
3.23 +\newblock \emph{Introduction to Algorithms}.
3.24 +\newblock 2nd.
3.25 +\newblock Prentice Hall India, 2001. --
3.26 +\newblock ISBN 978--8120321410
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8.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
8.2 +++ b/exzerpt.tex Mon Jan 10 00:16:59 2011 +0100
8.3 @@ -0,0 +1,194 @@
8.4 +\documentclass[a4paper,11pt,twoside]{article}
8.5 +
8.6 +\usepackage[ngerman]{babel}
8.7 +\usepackage[T1]{fontenc}
8.8 +\usepackage[latin1]{inputenc}
8.9 +\usepackage[colorlinks=true,linkcolor=black]{hyperref}
8.10 +
8.11 +\usepackage{vaucanson-g}
8.12 +\usepackage{hyperref}
8.13 +\usepackage{hypcap}
8.14 +\usepackage{fancyhdr}
8.15 +\usepackage{graphicx}
8.16 +\usepackage{lastpage}
8.17 +\usepackage[cm]{fullpage}
8.18 +\usepackage{amsthm, amsmath, amsfonts}
8.19 +\usepackage{float}
8.20 +\usepackage{paralist}
8.21 +\usepackage{listings}
8.22 +\usepackage{color}
8.23 +\usepackage{bookmark}
8.24 +
8.25 +% Festlegung Art der Zitierung - Havardmethode: Abkuerzung Autor + Jahr
8.26 +\bibliographystyle{alphadin}
8.27 +
8.28 +\title{\underline{Exzerpt Algorithmentheorie WS10/11}}
8.29 +\author{Markus Lindenmann}
8.30 +\date{\today{} }
8.31 +
8.32 +\begin{document}
8.33 + \maketitle
8.34 +
8.35 + \tableofcontents
8.36 + \newpage
8.37 +
8.38 + \section{Einleitung}\label{sec:intro}
8.39 + Dozent: Matthias Westermann\\
8.40 + Website: lak.informatik.uni-freiburg.de\\
8.41 + Klausur: 09.03.2011\\
8.42 + Hilfsmittel: Ein auf beiden Seiten handbeschriebenes DIN-A4-Blatt
8.43 + \subsection{Themen:}
8.44 + \begin{itemize}
8.45 + \item[$\bullet$] Divide and Conquer
8.46 + \item[$\bullet$] Greedy Prinzip
8.47 + \item[$\bullet$] Dynamische Programmierung
8.48 + \item[$\bullet$] Randomisierung
8.49 + \item[$\bullet$] Amortisierte Analyse
8.50 + \end{itemize}
8.51 + \subsection{Problem-/Anwendungsbereiche:}
8.52 + \begin{itemize}
8.53 + \item[$\bullet$] Geometrische Algorithmen
8.54 + \item[$\bullet$] Algebraische Algorithmen
8.55 + \item[$\bullet$] Graphenalgorithmen
8.56 + \item[$\bullet$] Datenstrukturen
8.57 + \item[$\bullet$] Internet-Algorithmen
8.58 + \item[$\bullet$] ptimierungs-Verfahren
8.59 + \item[$\bullet$] Zeichenkettenverarbeitung
8.60 + \end{itemize}
8.61 + \subsection{Literatur:}
8.62 + \bibliography{literatur}
8.63 + \cite{1}\cite{2}
8.64 +
8.65 + \newpage
8.66 + \section{Divide and Conquer}\label{sec:div&conq}
8.67 + \begin{itemize}
8.68 + \item[$\bullet$] Quicksort
8.69 + \item[$\bullet$] Formulierung und Analyse des Prinzips
8.70 + \item[$\bullet$] Geometrisches Devide and Conquer
8.71 + \begin{itemize}
8.72 + \item[-] Closest-Pair
8.73 + \item[-] Segmentschnitt
8.74 + \end{itemize}
8.75 + \end{itemize}
8.76 +
8.77 + \subsection{Formulierung des Divide and Conquer Prinzips}
8.78 + D\&C Verfahren zur Lösung eines Problems der Größe $n$
8.79 + \begin{itemize}
8.80 + \item[1.] Divide\\$n>2$: Teile das Problem in $k$ Teilprobleme der Größe $n_1,\ldots n_k$ ($k\geq2$)\\$n\leq c:$ Löse das Problem direkt
8.81 + \item[2.] Conquer\\Löse die $k$ Teilprobleme auf dieselbe Art (rekursiv)
8.82 + \item[3.] Merge\\Füge die berechneten Teillösungen zu einer Gesamtlösung zusammen
8.83 + \end{itemize}
8.84 +
8.85 + \subsection{Quicksort}
8.86 + Wähle beliebiges Pivot Element $v$ und sortiere alle Elemente aus der Menge, die größer sind als $v$ rechts von $v$ ein, die anderen links. Verfahre im selben Muster mit den neuen Teilmengen links und rechts des Pivot Elements. Wenn die zu sortierende Folge $F$ nur ein Element beinhaltet, gebe $F$ zurück.
8.87 +
8.88 + \subsubsection{Analyse}
8.89 + $T(n)$ - maximale Anzahl von Schritten, um ein Problem der Größe $n$ zu lösen.
8.90 + \[T(n)=\begin{cases}n\neq c&a \\ n>c&T(n_1+\ldots+T(n_k)+\text{Divide- und Mergeaufwand}\end{cases}\]
8.91 + Best Case: $k=2, n_1=n_2=\frac{n}{2}$\\
8.92 + Divide- und Mergeaufwand: $DM(n)$\\
8.93 + $T(1)=a$\\
8.94 + $T(n)=2T(\frac{n}{2})+DM(n)$
8.95 +
8.96 + \subsection{Nächste Paare}
8.97 + Eingabe: $n$ Punkte in Ebene\\
8.98 + Ausgabe: Punkt-Paar $q,r$ mit geringstem Abstand\\
8.99 + Abstand: $d(q,r)$ bechreibt Abstand zwischen Punkten $q$ und $r$
8.100 + \subsubsection{Algorithmus}
8.101 + \begin{itemize}
8.102 + \item[$\bullet$] sortiere Punktmenge nach x-Koordinate
8.103 + \item[$\bullet$] Teile Punktmenge in der Mitte L in die Mengen Q und R
8.104 + \item[$\bullet$] Löse rekursiv auf Q und R
8.105 + \item[$\bullet$] Sortiere Punkte nach y-Koordinate
8.106 + \item[$\bullet$] Teste alle Paare, deren Abstand in der Sortierung kleiner als 16 ist (siehe unten)
8.107 + \item[$\bullet$] Füge zusammen
8.108 + \end{itemize}
8.109 +
8.110 + $\delta=$ Minimum der Teillösungen\\
8.111 + Gibt es $q\in Q$ und $r\in R$ mit $d(q,r)<\delta$, dann sind sowohl $q$ als auch $r$ höchstens $\delta$ von $L$ (Mitte) entfernt.\\
8.112 + \textbf{Problem:} alle Punkte können innerhalb der $\delta$-Zone liegen\\
8.113 + \textbf{Lösung:}\begin{itemize}
8.114 + \item[$\bullet$] Sortiere Punkte in $\delta$ Zone nach $y$-Abstand
8.115 + \item[$\bullet$] Liegen in dieser Sortierung mehr als $c$ Punkte zwischen $q$ und $r$, dann kann $(q,r)$ nicht nächstes Paar sein
8.116 + \end{itemize}
8.117 +
8.118 + Dies kann für $c=15$ bewiesen werden (vgl. Abbildung \ref{cl_pair}):
8.119 + \begin{itemize}
8.120 + \item[$\bullet$] Seien min. 15 Punkte zwischen $q$ und $r$
8.121 + \item[$\bullet$] Dann sind mindestens 3 Reihen Quadrate zwischen $q,r$, weil in jedem Quadrat höchstens ein Punkt ist.
8.122 + \item[$\rightarrow$] Dann muss $d(q,r)$ mindestens $\frac{3}{2}\delta > \delta$ sein.
8.123 + \end{itemize}
8.124 +
8.125 + \begin{figure}[H]
8.126 + \centering
8.127 + \includegraphics[scale=0.5]{img/cl_pair}
8.128 + \caption{Nächste Paare: Prüf Distanz}
8.129 + \label{cl_pair}
8.130 + \end{figure}
8.131 +
8.132 + \subsubsection{Analyse}
8.133 + \[T(n)=\begin{cases}n\leq3&a \\ n>3&2T(\frac{n}{2})+an\end{cases}\]
8.134 + \[T(n)\leq an \log n \]
8.135 +
8.136 + \subsection{Segmentschnitt}
8.137 + Eingabe: Menge S bestehend aus vertikalen Segmenten und Endpunkten vn horizontalen Segmenten\\
8.138 + Ausgabe: Alle Schnittpunkte von vertikalen Segmenten mit horizontalen Segmenten, von denen mindesens ein Endpunkt in S ist
8.139 +
8.140 + \subsubsection{Algorithmus}
8.141 + \textbf{Divide \& Conquer:}\\Fallunterscheidung
8.142 + \begin{itemize}
8.143 + \item[1.] $\left| S\right| > 1$:\\ Teile S mittels einer vertikalen Gerade G in 2 gleichgroße Mengen $S_1$ (links von G) und $S_2$ (rechts von G). Führe rekursiv auf beiden Hälften aus.
8.144 + \item[2.] else: S enthält keine Schnitte
8.145 + \end{itemize}
8.146 +
8.147 + \textbf{Merge}\\
8.148 + Bilde L(S), R(S) und V(S) (Definition am Beispiel $i=1,2$ berechnet: $S=S_1\cup S_2$):
8.149 + \begin{itemize}
8.150 + \item[$\bullet$] L(S): y-Koordinate aller linken Endpunkte in S, deren rechter Partner nicht in S\\$=L(S_2)\cup(L(S_1)-R(S_2))$
8.151 + \item[$\bullet$] R(S): y-Koordinate aller rechten Endpunkte in S, deren linker Partner nicht in S\\$=R(S_1)\cup(R(S_2)-L(S_1))$
8.152 + \item[$\bullet$] V(S): y-Intervalle der vertikalen Segmente in S\\$=V(S_1)\cup V(S_2)$
8.153 + \end{itemize}
8.154 + L,R sortiert nach steigender y-Koordinate\\
8.155 + V sortiert nach steigendem unteren Endpunkt\\
8.156 + \medskip
8.157 +
8.158 + \textbf{Basisfälle}\\
8.159 + S enthält nur ein Element s. Fallunterscheidung:
8.160 + \begin{itemize}
8.161 + \item[1.] $s=(x,y)$ ist ein linker Endpunkt:\\$L(S)=\{y\},\qquad R(S)=\emptyset,\qquad V(S)=\emptyset$
8.162 + \item[2.] $s=(x,y)$ ist ein rechter Endpunkt:\\$L(S)=\emptyset,\qquad R(S)=\{y\},\qquad V(S)=\emptyset$
8.163 + \item[3.] $s=(x,y_1,y_2)$ ist ein vertikales Segment:\\$L(S)=\emptyset,\qquad R(S)=\emptyset,\qquad V(S)=\{[y_1,y_2]\}$
8.164 + \end{itemize}
8.165 +
8.166 + \subsubsection{Analyse}
8.167 + Eingabe wird Anfangs sortiert und in Array gespeichert
8.168 + \[T(n)=2T(\frac{n}{2}+an+\text{Größe der Ausgabe}\]
8.169 + \[T(1)=O(1)\]
8.170 + \[O(n \log n + k \mid k=\#Schnittpunkte\]
8.171 +
8.172 + \subsection{Polynomprodukt und Fast Fourier-Transformation}
8.173 +
8.174 +
8.175 + \newpage
8.176 + \section{Randomisierung}
8.177 + \subsection{Hashing}
8.178 +
8.179 + \newpage
8.180 + \section{Amortisierte Analyse}
8.181 + \subsection{Binomial Heaps}
8.182 + \subsection{Fibonacci Heaps}
8.183 + \subsection{Union Find}
8.184 +
8.185 + \newpage
8.186 + \section{Greedy}
8.187 +
8.188 + \newpage
8.189 + \section{Anhang}
8.190 + \subsection{Landau-Symbole}
8.191 + \begin{figure}[H]
8.192 + \centering
8.193 + \includegraphics[scale=0.45]{img/landau_sym}
8.194 + \caption{Definition der Landau Symbole}
8.195 + \label{fig:landau_sym}
8.196 + \end{figure}
8.197 +\end{document}
8.198 \ No newline at end of file
9.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
9.2 +++ b/exzerpt.toc Mon Jan 10 00:16:59 2011 +0100
9.3 @@ -0,0 +1,25 @@
9.4 +\select@language {ngerman}
9.5 +\contentsline {section}{\numberline {1}Einleitung}{2}{section.1}
9.6 +\contentsline {subsection}{\numberline {1.1}Themen:}{2}{subsection.1.1}
9.7 +\contentsline {subsection}{\numberline {1.2}Problem-/Anwendungsbereiche:}{2}{subsection.1.2}
9.8 +\contentsline {subsection}{\numberline {1.3}Literatur:}{2}{subsection.1.3}
9.9 +\contentsline {section}{\numberline {2}Divide and Conquer}{3}{section.2}
9.10 +\contentsline {subsection}{\numberline {2.1}Formulierung des Divide and Conquer Prinzips}{3}{subsection.2.1}
9.11 +\contentsline {subsection}{\numberline {2.2}Quicksort}{3}{subsection.2.2}
9.12 +\contentsline {subsubsection}{\numberline {2.2.1}Analyse}{3}{subsubsection.2.2.1}
9.13 +\contentsline {subsection}{\numberline {2.3}N\"achste Paare}{3}{subsection.2.3}
9.14 +\contentsline {subsubsection}{\numberline {2.3.1}Algorithmus}{3}{subsubsection.2.3.1}
9.15 +\contentsline {subsubsection}{\numberline {2.3.2}Analyse}{4}{subsubsection.2.3.2}
9.16 +\contentsline {subsection}{\numberline {2.4}Segmentschnitt}{4}{subsection.2.4}
9.17 +\contentsline {subsubsection}{\numberline {2.4.1}Algorithmus}{5}{subsubsection.2.4.1}
9.18 +\contentsline {subsubsection}{\numberline {2.4.2}Analyse}{5}{subsubsection.2.4.2}
9.19 +\contentsline {subsection}{\numberline {2.5}Fast Fourier-Transformation}{5}{subsection.2.5}
9.20 +\contentsline {section}{\numberline {3}Randomisierung}{6}{section.3}
9.21 +\contentsline {subsection}{\numberline {3.1}Hashing}{6}{subsection.3.1}
9.22 +\contentsline {section}{\numberline {4}Amortisierte Analyse}{7}{section.4}
9.23 +\contentsline {subsection}{\numberline {4.1}Binomial Heaps}{7}{subsection.4.1}
9.24 +\contentsline {subsection}{\numberline {4.2}Fibonacci Heaps}{7}{subsection.4.2}
9.25 +\contentsline {subsection}{\numberline {4.3}Union Find}{7}{subsection.4.3}
9.26 +\contentsline {section}{\numberline {5}Greedy}{8}{section.5}
9.27 +\contentsline {section}{\numberline {6}Anhang}{9}{section.6}
9.28 +\contentsline {subsection}{\numberline {6.1}Landau-Symbole}{9}{subsection.6.1}
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13.321 +end restore showpage
13.322 +%%Trailer
13.323 +%%EOF
14.1 Binary file img/landau_sym.png has changed
15.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
15.2 +++ b/literatur.bib Mon Jan 10 00:16:59 2011 +0100
15.3 @@ -0,0 +1,17 @@
15.4 +@BOOK{1,
15.5 + AUTHOR="Thomas H. Cormen, Charles E. Leiserson, Robert L. Rivest und Cliford Stein",
15.6 + TITLE="Introduction to Algorithms",
15.7 + PUBLISHER="Prentice Hall India",
15.8 + YEAR="2001",
15.9 + ISBN="978-8120321410",
15.10 + EDITION="2nd"
15.11 +}
15.12 +
15.13 +@BOOK{2,
15.14 + AUTHOR="Thomas Ottmann und Peter Widmayer",
15.15 + TITLE="Algorithmen und Datenstrukturen",
15.16 + PUBLISHER="Spektrum Akademischer Verlag",
15.17 + YEAR="2002",
15.18 + ISBN="978-3827410290",
15.19 + EDITION="4. Auflage"
15.20 +}
15.21 \ No newline at end of file
