From 21b6c6d6fe8fa614199f8c3771a0087927b7d5d9 Mon Sep 17 00:00:00 2001
From: Peter Gerwinski <peter.gerwinski@hs-bochum.de>
Date: Thu, 19 Dec 2019 15:08:40 +0100
Subject: [PATCH] Vortragsfolien 19.12.2019

---
 20191219/Tower_of_Hanoi.jpeg                  |   1 +
 20191219/hp-20191219.pdf                      | Bin 0 -> 333758 bytes
 20191219/hp-20191219.tex                      | 721 ++++++++++++++++++
 20191219/landau-symbols.pdf                   |   1 +
 .../logo-hochschule-bochum-cvh-text-v2.pdf    | Bin 0 -> 11990 bytes
 20191219/logo-hochschule-bochum.pdf           | Bin 0 -> 31581 bytes
 20191219/pgscript.sty                         |   1 +
 7 files changed, 724 insertions(+)
 create mode 120000 20191219/Tower_of_Hanoi.jpeg
 create mode 100644 20191219/hp-20191219.pdf
 create mode 100644 20191219/hp-20191219.tex
 create mode 120000 20191219/landau-symbols.pdf
 create mode 100644 20191219/logo-hochschule-bochum-cvh-text-v2.pdf
 create mode 100644 20191219/logo-hochschule-bochum.pdf
 create mode 120000 20191219/pgscript.sty

diff --git a/20191219/Tower_of_Hanoi.jpeg b/20191219/Tower_of_Hanoi.jpeg
new file mode 120000
index 0000000..a1a794a
--- /dev/null
+++ b/20191219/Tower_of_Hanoi.jpeg
@@ -0,0 +1 @@
+../common/Tower_of_Hanoi.jpeg
\ No newline at end of file
diff --git a/20191219/hp-20191219.pdf b/20191219/hp-20191219.pdf
new file mode 100644
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diff --git a/20191219/hp-20191219.tex b/20191219/hp-20191219.tex
new file mode 100644
index 0000000..c5ac855
--- /dev/null
+++ b/20191219/hp-20191219.tex
@@ -0,0 +1,721 @@
+% hp-20191219.pdf - Lecture Slides on Low-Level Programming
+% Copyright (C) 2012, 2013, 2015, 2016, 2017, 2018, 2019  Peter Gerwinski
+%
+% This document is free software: you can redistribute it and/or
+% modify it either under the terms of the Creative Commons
+% Attribution-ShareAlike 3.0 License, or under the terms of the
+% GNU General Public License as published by the Free Software
+% Foundation, either version 3 of the License, or (at your option)
+% any later version.
+%
+% This document is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+% GNU General Public License for more details.
+%
+% You should have received a copy of the GNU General Public License
+% along with this document.  If not, see <http://www.gnu.org/licenses/>.
+%
+% You should have received a copy of the Creative Commons
+% Attribution-ShareAlike 3.0 Unported License along with this
+% document.  If not, see <http://creativecommons.org/licenses/>.
+
+% README: Binärdarstellung negativer Zahlen, Speicherausrichtung - Alignment
+
+\documentclass[10pt,t]{beamer}
+
+\usepackage{pgslides}
+\usepackage{pdftricks}
+\usepackage{tikz}
+
+\begin{psinputs}
+  \usepackage[utf8]{inputenc}
+  \usepackage[german]{babel}
+  \usepackage[T1]{fontenc}
+  \usepackage{helvet}
+  \renewcommand*\familydefault{\sfdefault}
+  \usepackage{pstricks,pst-grad}
+\end{psinputs}
+
+\title{Hardwarenahe Programmierung}
+\author{Prof.\ Dr.\ rer.\ nat.\ Peter Gerwinski}
+\date{19.\ Dezember 2019}
+
+\begin{document}
+
+\maketitleframe
+
+\nosectionnonumber{\inserttitle}
+
+\begin{frame}
+
+  \shownosectionnonumber
+
+  \begin{itemize}
+    \item[\textbf{1}] \textbf{Einführung}
+      \hfill\makebox(0,0)[br]{\raisebox{2.25ex}{\url{https://gitlab.cvh-server.de/pgerwinski/hp}}}
+    \item[\textbf{2}] \textbf{Einführung in C}
+    \item[\textbf{3}] \textbf{Bibliotheken}
+      \begin{itemize}
+        \item[3.1] Der Präprozessor
+        \item[3.2] Bibliotheken einbinden
+        \item[3.3] Bibliotheken verwenden
+        \color{medgreen}
+        \item[3.4] Projekt organisieren: make
+      \end{itemize}
+    \vspace*{-\smallskipamount}
+    \item[\textbf{4}] \textbf{Hardwarenahe Programmierung}
+      \begin{itemize}
+        \vspace*{-0.1cm}
+        \item[\dots]
+%        \item[4.3] Interrupts
+        \item[4.4] volatile-Variable
+        \color{medgreen}
+        \item[4.6] Byte-Reihenfolge -- Endianness
+        \item[4.7] Binärdarstellung negativer Zahlen
+        \item[4.8] Speicherausrichtung -- Alignment
+      \end{itemize}
+    \item[\textbf{5}] \textbf{Algorithmen}
+      \begin{itemize}
+        \item[5.1] Differentialgleichungen
+        \vspace*{-0.1cm}
+        \item[\dots]
+      \end{itemize}
+    \vspace*{-\smallskipamount}
+    \item[\textbf{\dots}]
+  \end{itemize}
+
+\end{frame}
+
+\setcounter{section}{2}
+\section{Bibliotheken}
+\setcounter{subsection}{3}
+\subsection{Projekt organisieren: make}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+  \begin{itemize}
+    \item
+      \only<4->{explizite und implizite} Regeln
+      \begin{onlyenv}<2>
+        \smallskip
+        \begin{lstlisting}[language=make,gobble=10]
+          hello-6: hello-6.o pruzzel.o
+                  gcc hello-6.o pruzzel.o -o hello-6
+
+          pruzzel.o: pruzzel.c pruzzel.h
+                  gcc -Wall -O pruzzel.c -c
+
+          hello-6.o: hello-6.c pruzzel.h
+                  gcc -Wall -O hello-6.c -c
+        \end{lstlisting}
+      \end{onlyenv}
+      \begin{onlyenv}<4>
+        \smallskip
+        \begin{lstlisting}[language=make,gobble=10]
+          TARGET = hello-6
+          OBJECTS = hello-6.o pruzzel.o
+          HEADERS = pruzzel.h
+          CFLAGS = -Wall -O
+
+          $(TARGET): $(OBJECTS)
+                  gcc $(OBJECTS) -o $(TARGET)
+
+          %.o: %.c $(HEADERS)
+                  gcc $(CFLAGS) $< -c
+
+          clean:
+                  rm -f $(OBJECTS) $(TARGET)
+        \end{lstlisting}
+      \end{onlyenv}
+    \item
+      Makros
+      \begin{onlyenv}<3>
+        \smallskip
+        \begin{lstlisting}[language=make,gobble=10]
+          TARGET = hello-6
+          OBJECTS = hello-6.o pruzzel.o
+          HEADERS = pruzzel.h
+          CFLAGS = -Wall -O
+
+          $(TARGET): $(OBJECTS)
+                  gcc $(OBJECTS) -o $(TARGET)
+
+          pruzzel.o: pruzzel.c $(HEADERS)
+                  gcc $(CFLAGS) pruzzel.c -c
+
+          hello-6.o: hello-6.c $(HEADERS)
+                  gcc $(CFLAGS) hello-6.c -c
+
+          clean:
+                  rm -f $(OBJECTS) $(TARGET)
+        \end{lstlisting}
+        \vspace*{-1cm}
+      \end{onlyenv}
+    \begin{onlyenv}<5->
+      \smallskip
+      \arrowitem
+        3 Sprachen: C, Präprozessor, make
+    \end{onlyenv}
+  \end{itemize}
+
+\end{frame}
+
+\setcounter{section}{3}
+\section{Hardwarenahe Programmierung}
+\setcounter{subsection}{3}
+\subsection{volatile-Variable}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  Was ist eigentlich \lstinline{PORTD}?
+
+  \bigskip
+%  \pause
+
+  \lstinline[style=cmd]{avr-gcc -Wall -Os -mmcu=atmega328p blink-3.c -E}
+
+  \bigskip
+%  \pause
+  \lstinline{PORTD = 0x01;}\\
+  \textarrow\quad
+  \lstinline[style=terminal]{(*(volatile uint8_t *)((0x0B) + 0x20)) = 0x01;}\\
+%  \pause
+  \begin{picture}(0,2)(0,-1.7)
+    \color{red}
+    \put(5.75,0.3){$\underbrace{\rule{2.95cm}{0pt}}_{\mbox{Zahl: \lstinline|0x2B|}}$}
+%    \pause
+    \put(1.55,0.3){$\underbrace{\rule{4.0cm}{0pt}}_{\mbox{\shortstack[t]{Umwandlung in Zeiger\\
+      auf \lstinline|volatile uint8_t|}}}$}
+%    \pause
+    \put(1.32,-1){\makebox(0,0)[b]{\tikz{\draw[-latex](0,0)--(0,1.3)}}}
+    \put(1.12,-1.1){\makebox(0,0)[tl]{Dereferenzierung des Zeigers}}
+  \end{picture}
+
+%  \pause
+  \textarrow\quad
+  \lstinline|volatile uint8_t|-Variable an Speicheradresse \lstinline|0x2B|
+
+%  \pause
+  \bigskip
+  \bigskip
+
+  \textarrow\quad
+  \lstinline|PORTA = PORTB = PORTC = PORTD = 0| ist eine schlechte Idee.
+
+\end{frame}
+
+\subsection{Byte-Reihenfolge -- Endianness}
+\subsubsection{Konzept}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+  \showsubsubsection
+
+  Eine Zahl geht über mehrere Speicherzellen.\\
+  Beispiel: 16-Bit-Zahl in 2 8-Bit-Speicherzellen
+
+  \smallskip
+
+  Welche Bits liegen wo?
+
+%  \pause
+  \bigskip
+
+  $1027 = 1024 + 2 + 1 = 0000\,0100\,0000\,0011_2 = 0403_{16}$
+
+%  \pause
+  \bigskip
+  Speicherzellen:
+
+  \medskip
+  \begin{tabular}{|c|c|l}\cline{1-2}
+    \raisebox{-0.25ex}{04} & \raisebox{-0.25ex}{03} & \strut Big-Endian "`großes Ende zuerst"' \\\cline{1-2}
+    \multicolumn{2}{c}{} & für Menschen leichter lesbar \\
+    \multicolumn{3}{c}{} \\[-5pt]\cline{1-2}
+    \raisebox{-0.25ex}{03} & \raisebox{-0.25ex}{04} & \strut Little-Endian "`kleines Ende zuerst"' \\\cline{1-2}
+    \multicolumn{2}{c}{} & bei Additionen effizienter
+  \end{tabular}
+
+%  \pause
+  \medskip
+  \textarrow\ Geschmackssache
+%  \pause\\
+  \quad\textbf{\dots\ außer bei Datenaustausch!}
+
+%  \pause
+%  \bigskip
+%
+%  Aber: nicht verwechseln! \qquad $0304_{16} = 772$
+
+\end{frame}
+
+\begin{frame}
+
+  \showsubsection
+  \showsubsubsection
+
+  Eine Zahl geht über mehrere Speicherzellen.\\
+  Beispiel: 16-Bit-Zahl in 2 8-Bit-Speicherzellen
+
+  \smallskip
+
+  Welche Bits liegen wo?
+
+  \medskip
+
+  \textarrow\ Geschmackssache\\
+  \textbf{\dots\ außer bei Datenaustausch!}
+
+  \begin{itemize}
+    \item
+      Dateiformate
+    \item
+      Datenübertragung
+  \end{itemize}
+
+\end{frame}
+
+\subsubsection{Dateiformate}
+
+\begin{frame}
+
+  \showsubsection
+  \showsubsubsection
+
+  Audio-Formate: Reihenfolge der Bytes in 16- und 32-Bit-Zahlen
+  \begin{itemize}
+    \item
+      RIFF-WAVE-Dateien (\file{.wav}): Little-Endian
+    \item
+      Au-Dateien (\file{.au}): Big-Endian
+%    \pause
+    \item
+      ältere AIFF-Dateien (\file{.aiff}): Big-Endian
+    \item
+      neuere AIFF-Dateien (\file{.aiff}): Little-Endian
+  \end{itemize}
+
+%  \pause
+  \bigskip
+
+  Grafik-Formate: Reihenfolge der Bits in den Bytes
+  \begin{itemize}
+    \item
+      PBM-Dateien: Big-Endian\only<1->{, MSB first}
+    \item
+      XBM-Dateien: Little-Endian\only<1->{, LSB first}
+  \end{itemize}
+  \only<1->{MSB/LSB = most/least significant bit}
+
+\end{frame}
+
+\subsubsection{Datenübertragung}
+
+\begin{frame}
+
+  \showsubsection
+  \showsubsubsection
+
+  \begin{itemize}
+    \item
+      RS-232 (serielle Schnittstelle): LSB first
+    \item
+      I$^2$C: MSB first
+    \item
+      USB: beides
+%    \pause
+    \medskip
+    \item
+      Ethernet: LSB first
+    \item
+      TCP/IP (Internet): Big-Endian
+  \end{itemize}
+
+\end{frame}
+
+\subsection{Binärdarstellung negativer Zahlen}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  Speicher ist begrenzt!\\
+  \textarrow\ feste Anzahl von Bits
+
+  \medskip
+
+  8-Bit-Zahlen ohne Vorzeichen: \lstinline{uint8_t}\\
+  \textarrow\ Zahlenwerte von \lstinline{0x00} bis \lstinline{0xff} = 0 bis 255\\
+  \pause
+  \textarrow\ 255 + 1 = 0
+
+  \pause
+  \medskip
+
+  8-Bit-Zahlen mit Vorzeichen: \lstinline{int8_t}\\
+  \lstinline{0xff} = 255 ist die "`natürliche"' Schreibweise für $-1$.\\
+  \pause
+  \textarrow\ Zweierkomplement
+
+  \pause
+  \medskip
+
+  Oberstes Bit = 1: negativ\\
+  Oberstes Bit = 0: positiv\\
+  \textarrow\ 127 + 1 = $-128$
+
+\end{frame}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  Speicher ist begrenzt!\\
+  \textarrow\ feste Anzahl von Bits
+
+  \medskip
+
+  16-Bit-Zahlen ohne Vorzeichen:
+  \lstinline{uint16_t}\hfill\lstinline{uint8_t}\\
+  \textarrow\ Zahlenwerte von \lstinline{0x0000} bis \lstinline{0xffff}
+  = 0 bis 65535\hfill 0 bis 255\\
+  \textarrow\ 65535 + 1 = 0\hfill 255 + 1 = 0
+
+  \medskip
+
+  16-Bit-Zahlen mit Vorzeichen:
+  \lstinline{int16_t}\hfill\lstinline{int8_t}\\
+  \lstinline{0xffff} = 66535 ist die "`natürliche"' Schreibweise für $-1$.\hfill
+  \lstinline{0xff} = 255 = $-1$\\
+  \textarrow\ Zweierkomplement
+
+  \medskip
+
+  Oberstes Bit = 1: negativ\\
+  Oberstes Bit = 0: positiv\\
+  \textarrow\ 32767 + 1 = $-32768$
+
+  \bigskip
+  Literatur: \url{http://xkcd.com/571/}
+
+\end{frame}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  Frage: \emph{Für welche Zahl steht der Speicherinhalt\,
+  \raisebox{2pt}{%
+    \tabcolsep0.25em
+    \begin{tabular}{|c|c|}\hline
+      \rule{0pt}{11pt}a3 & 90 \\\hline
+    \end{tabular}}
+  (hexadezimal)?}
+  
+  \pause
+  \smallskip
+  Antwort: \emph{Das kommt darauf an.} ;--)
+
+  \pause
+  \medskip
+  Little-Endian:
+
+  \smallskip
+
+  \begin{tabular}{lrl}
+    als \lstinline,int8_t,: & $-93$ & (nur erstes Byte)\\
+    als \lstinline,uint8_t,: & $163$ & (nur erstes Byte)\\
+    als \lstinline,int16_t,: & $-28509$\\
+    als \lstinline,uint16_t,: & $37027$\\
+    \lstinline,int32_t, oder größer: & $37027$
+      & (zusätzliche Bytes mit Nullen aufgefüllt)
+  \end{tabular}
+
+  \pause
+  \medskip
+  Big-Endian:
+
+  \smallskip
+
+  \begin{tabular}{lrl}
+    als \lstinline,int8_t,:   & $-93$ & (nur erstes Byte)\\
+    als \lstinline,uint8_t,:  & $163$ & (nur erstes Byte)\\
+    als \lstinline,int16_t,:  & $-23664$\\
+    als \lstinline,uint16_t,: & $41872$\\
+    als \lstinline,int32_t,:  & $-1550843904$ & (zusätzliche Bytes\\
+    als \lstinline,uint32_t,: & $2744123392$  & mit Nullen aufgefüllt)\\
+    als \lstinline,int64_t,:  & $-6660823848880963584$\\
+    als \lstinline,uint64_t,: & $11785920224828588032$\\
+  \end{tabular}
+
+  \vspace*{-1cm}
+
+\end{frame}
+
+\subsection{Speicherausrichtung -- Alignment}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  \begin{lstlisting}
+    #include <stdint.h>
+
+    uint8_t a;
+    uint16_t b;
+    uint8_t c;
+  \end{lstlisting}
+
+  \pause
+  \bigskip
+
+  Speicheradresse durch 2 teilbar -- "`16-Bit-Alignment"'
+  \begin{itemize}
+    \item
+      2-Byte-Operation: effizienter
+    \pause
+    \item
+      \dots\ oder sogar nur dann erlaubt
+    \pause
+    \arrowitem
+      Compiler optimiert Speicherausrichtung
+  \end{itemize}
+
+  \medskip
+
+  \pause
+  \begin{minipage}{3cm}
+    \begin{lstlisting}[gobble=6]
+      ¡uint8_t a;
+      uint8_t dummy;
+      uint16_t b;
+      uint8_t c;¿
+    \end{lstlisting}
+  \end{minipage}
+  \pause
+  \begin{minipage}{3cm}
+    \begin{lstlisting}[gobble=6]
+      ¡uint8_t a;
+      uint8_t c;
+      uint16_t b;¿
+    \end{lstlisting}
+  \end{minipage}
+
+  \pause
+  \vspace{-1.75cm}
+  \strut\hfill
+  \begin{minipage}{6.5cm}
+    Fazit:
+    \begin{itemize}
+      \item
+        \textbf{Adressen von Variablen\\
+        sind systemabhängig}
+      \item
+        Bei Definition von Datenformaten\\
+        Alignment beachten \textarrow\ effizienter
+    \end{itemize}
+  \end{minipage}
+
+\end{frame}
+
+\section{Algorithmen}
+\setcounter{subsection}{1}
+\subsection{Rekursion}
+
+\begin{frame}[fragile]
+
+  \showsubsection
+
+  Vollständige Induktion:
+  \vspace*{-0.725cm}
+  \begin{displaymath}
+    \hspace*{4cm}
+    \left.
+      \begin{array}{r}
+        \mbox{Aussage gilt für $n = 1$}\\[2pt]
+        \mbox{Schluß von $n - 1$ auf $n$}
+      \end{array}
+    \right\}
+    \mbox{Aussage gilt für alle $n\in\mathbb{N}$}
+  \end{displaymath}
+  \vspace*{-0.5cm}
+
+  \pause
+
+  Türme von Hanoi
+
+  \begin{onlyenv}<2>
+    \begin{center}
+      \includegraphics[width=12.2cm]{Tower_of_Hanoi.jpeg}
+    \end{center}
+  \end{onlyenv}
+
+  \begin{onlyenv}<3->
+    \begin{itemize}
+      \item
+        64 Scheiben, 3 Plätze,
+        \only<3-4>{\hfill\makebox(0,0)[rt]{\includegraphics[width=6cm]{Tower_of_Hanoi.jpeg}}}\\
+        immer 1 Scheibe verschieben
+      \item
+        Ziel: Turm verschieben
+      \item
+        Es dürfen nur kleinere Scheiben\\
+        auf größeren liegen.
+      \bigskip
+      \pause
+      \pause
+      \item
+        $n = 1$ Scheibe: fertig
+      \item
+        Wenn $n - 1$ Scheiben verschiebbar:\\
+        schiebe $n - 1$ Scheiben auf Hilfsplatz,\\
+        verschiebe die darunterliegende,\\
+        hole $n - 1$ Scheiben von Hilfsplatz
+    \end{itemize}
+    \begin{onlyenv}<5>
+      \vspace{-4.3cm}
+      \begin{lstlisting}[gobble=8,xleftmargin=6.4cm]
+        void move (int from, int to, int disks)
+        {
+          if (disks == 1)
+            move_one_disk (from, to);
+          else
+            {
+              int help = 0 + 1 + 2 - from - to;
+              move (from, help, disks - 1);
+              move (from, to, 1);
+              move (help, to, disks - 1);
+            }
+        }
+      \end{lstlisting}
+    \end{onlyenv}
+    \begin{onlyenv}<6->
+      \vspace{-5.0cm}
+      \hspace*{7.4cm}\begin{minipage}[t]{5cm}
+        32 Scheiben:
+        \begin{lstlisting}[gobble=10,style=terminal]
+          $ ¡time ./hanoi-9a¿
+          ...
+          real    0m32,712s
+          user    0m32,708s
+          sys     0m0,000s
+        \end{lstlisting}
+        \pause[7]
+        \begin{itemize}
+          \arrowitem
+            etwas über 1 Minute\\
+            für 64 Scheiben
+        \end{itemize}
+        \pause
+        \vspace*{-0.5cm}
+        \begin{picture}(0,0)
+          \color{red}
+          \put(0,0){\makebox(0,0)[bl]{\tikz[line width=1pt]{\draw(0,0)--(4,0.8);}}}
+          \put(0,0.8){\makebox(0,0)[tl]{\tikz[line width=1pt]{\draw(0,0)--(4,-0.8);}}}
+        \end{picture}
+
+        Für jede zusätzliche Scheibe\\verdoppelt sich die Rechenzeit!
+        % 32.712 * 2^32 / 3600 / 24 / 365.25 = 4452.08032888280477602859
+        \begin{itemize}
+          \arrowitem
+            $\frac{32,712\,\text{s}\,\cdot\,2^{32}}{3600\,\cdot\,24\,\cdot\,365,25} \approx 4452$
+            Jahre\\[\smallskipamount]
+            für 64 Scheiben
+        \end{itemize}
+      \end{minipage}
+    \end{onlyenv}
+  \end{onlyenv}
+
+\end{frame}
+
+\subsection{Aufwandsabschätzungen \protect\color{gray}-- Komplexitätsanalyse}
+
+\begin{frame}[fragile]
+
+  \newcommand{\w}{\hspace*{0.75pt}}
+
+  \showsubsection
+
+  \begin{picture}(0,0)
+    \put(7.8,-0.3){%
+      \begin{minipage}[t]{5.3cm}
+        \vspace*{-1.0cm}\includegraphics{landau-symbols.pdf}
+%        \vspace*{-1.0cm}\alt<2->{\includegraphics{landau-symbols-2.pdf}}%
+%                                {\includegraphics{landau-symbols.pdf}}
+        \small
+        \begin{description}\itemsep0pt\leftskip-0.5cm
+          \item[$n$:] Eingabedaten
+          \item[$g(n)$:] Rechenzeit
+        \end{description}
+      \end{minipage}}
+  \end{picture}
+
+  \vspace*{-1.5\bigskipamount}
+
+  \begin{itemize}
+    \item
+      Türme von Hanoi: $\mathcal{O}(2^n)$
+      \begin{onlyenv}<1>
+        \par\medskip
+        Für jede zusätzliche Scheibe\\verdoppelt sich die Rechenzeit!
+        % 32.712 * 2^32 / 3600 / 24 / 365.25 = 4452.08032888280477602859
+        \begin{itemize}
+          \arrowitem
+            $\frac{32,712\,\text{s}\,\cdot\,2^{32}}{3600\,\cdot\,24\,\cdot\,365,25} \approx 4452$
+            Jahre\\[\smallskipamount]
+            für 64 Scheiben
+        \end{itemize}
+      \end{onlyenv}
+  \end{itemize}
+
+  \pause
+
+  Beispiel: Sortieralgorithmen
+
+  \begin{itemize}
+    \item
+      Minimum suchen: \alt<4->{$\mathcal{O}(n)$}{$\mathcal{O}(\textbf{\color{red}?})$}
+    \pause
+    \item
+      \dots\ mit Schummeln: $\mathcal{O}(1)$
+      \pause
+      \pause
+      \begin{picture}(0,0)
+        \put(-4.742,-3.7){\begin{minipage}[t]{12cm}
+          Faustregel:\\Schachtelung der Schleifen zählen\\
+          $x$ Schleifen \textarrow\ $\mathcal{O}(n\w^x)$
+%          \pause
+%          \begin{tabbing}
+%            Verschlüsselung brechen (Primfaktorzerlegung):~\=\kill
+%            \textbf{\color{red}RSA}: Schlüsselerzeugung (Berechnung von $d\/$): 
+%              \> \color{red}$\mathcal{O}\bigl((\log n)^2\bigr)$,\\[\smallskipamount]
+%            Ver- und Entschlüsselung (Exponentiation):
+%              \> \color{red}$\mathcal{O}\kern0.5pt(n\log n)$,\\[0.5\smallskipamount]
+%            Verschlüsselung brechen (Primfaktorzerlegung):
+%              \> \color{red}$\mathcal{O}\bigl(2^{\sqrt{\log n\,\cdot\,\log\log n}}\bigr)$
+%          \end{tabbing}
+        \end{minipage}}
+      \end{picture}
+    \pause
+    \item
+      Minimum an den Anfang tauschen,\\
+      nächstes Minimum suchen:\\
+      \textarrow\ Selectionsort\pause: $\mathcal{O}(n\w^2)$
+%    \pause
+%    \item
+%      Während Minimumsuche prüfen\\und abbrechen, falls schon sortiert\\
+%      \textarrow\ Bubblesort\pause: $\mathcal{O}(n)$ bis $\mathcal{O}(n\w^2)$
+%    \pause
+%    \item
+%      Rekursiv sortieren\\
+%      \textarrow\ Quicksort\pause: $\mathcal{O}(n\log n)$ bis $\mathcal{O}(n\w^2)$\hfill
+  \end{itemize}
+
+\end{frame}
+
+\end{document}
diff --git a/20191219/landau-symbols.pdf b/20191219/landau-symbols.pdf
new file mode 120000
index 0000000..ca14542
--- /dev/null
+++ b/20191219/landau-symbols.pdf
@@ -0,0 +1 @@
+../common/landau-symbols.pdf
\ No newline at end of file
diff --git a/20191219/logo-hochschule-bochum-cvh-text-v2.pdf b/20191219/logo-hochschule-bochum-cvh-text-v2.pdf
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diff --git a/20191219/pgscript.sty b/20191219/pgscript.sty
new file mode 120000
index 0000000..95c8884
--- /dev/null
+++ b/20191219/pgscript.sty
@@ -0,0 +1 @@
+../common/pgscript.sty
\ No newline at end of file
-- 
GitLab