misc

presentations/Diskrete-Mathematik/LaTeX/Grundlagen.tex

@@ -31,6 +31,7 @@ Kantenmenge bezeichnet.
 \framedgraphic{Modellierung, Flüsse, Netzwerke}{../images/Unit_disk_graph.png}
 \framedgraphic{Karten}{../images/map.png}
 \framedgraphic{Good Will Hunting}{../images/good-will-hunting.jpg}
+\framedgraphic{Graham's Number}{../images/hypercube.png}
 
 \begin{frame}{Isomorphe Graphen}
 \begin{center}

presentations/Diskrete-Mathematik/Plan/Makefile

@@ -1,7 +0,0 @@
-SOURCE = Plan
-make:
-	pdflatex $(SOURCE).tex -output-format=pdf
-	make clean
-
-clean:
-	rm -rf  $(TARGET) *.class *.html *.log *.aux *.out

presentations/Diskrete-Mathematik/Plan/Plan.tex

@@ -1,121 +0,0 @@
-\documentclass[a4paper,9pt]{scrartcl}
-\usepackage{amssymb, amsmath} % needed for math
-\usepackage[utf8]{inputenc} % this is needed for umlauts
-\usepackage[ngerman]{babel} % this is needed for umlauts
-\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
-\usepackage[margin=2.5cm]{geometry} %layout
-\usepackage{hyperref}   % links im text
-\usepackage{color}
-\usepackage{framed}
-\usepackage{enumerate}  % for advanced numbering of lists
-\clubpenalty  = 10000   % Schusterjungen verhindern
-\widowpenalty = 10000   % Hurenkinder verhindern
-
-\hypersetup{ 
-  pdfauthor   = {Martin Thoma}, 
-  pdfkeywords = {Diskrete Mathematik}, 
-  pdftitle    = {Vortrag Graphentheorie I: Tafelbild + Text} 
-} 
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Custom definition style, by                                       %
-% http://mathoverflow.net/questions/46583/what-is-a-satisfactory-way-to-format-definitions-in-latex/58164#58164
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\makeatletter
-\newdimen\errorsize \errorsize=0.2pt
-% Frame with a label at top
-\newcommand\LabFrame[2]{%
-    \fboxrule=\FrameRule
-    \fboxsep=-\errorsize
-    \textcolor{FrameColor}{%
-    \fbox{%
-      \vbox{\nobreak
-      \advance\FrameSep\errorsize
-      \begingroup
-        \advance\baselineskip\FrameSep
-        \hrule height \baselineskip
-        \nobreak
-        \vskip-\baselineskip
-      \endgroup
-      \vskip 0.5\FrameSep
-      \hbox{\hskip\FrameSep \strut
-        \textcolor{TitleColor}{\textbf{#1}}}%
-      \nobreak \nointerlineskip
-      \vskip 1.3\FrameSep
-      \hbox{\hskip\FrameSep
-        {\normalcolor#2}%
-        \hskip\FrameSep}%
-      \vskip\FrameSep
-    }}%
-}}
-\definecolor{FrameColor}{rgb}{0.25,0.25,1.0}
-\definecolor{TitleColor}{rgb}{1.0,1.0,1.0}
-
-\newenvironment{contlabelframe}[2][\Frame@Lab\ (cont.)]{% 
-  % Optional continuation label defaults to the first label plus
-  \def\Frame@Lab{#2}%
-  \def\FrameCommand{\LabFrame{#2}}%
-  \def\FirstFrameCommand{\LabFrame{#2}}%
-  \def\MidFrameCommand{\LabFrame{#1}}%
-  \def\LastFrameCommand{\LabFrame{#1}}%
-  \MakeFramed{\advance\hsize-\width \FrameRestore} 
-}{\endMakeFramed} 
-\newcounter{definition}
-\newenvironment{definition}[1]{%
-  \par
-  \refstepcounter{definition}%
-  \begin{contlabelframe}{Definition \thedefinition:\quad #1}
- \noindent\ignorespaces}
-{\end{contlabelframe}} 
-\makeatother
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Begin document                                                    %
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{document}
-\section{Königsberger Brückenproblem}
-\subsection{Beweis des Satzes von Euler}
-Tafelbild:
-
-Sie $G = (E, K)$ ein eulerscher Graph, $K$ ein Eulerkreis durch $G$ und 
-$e \in E$ eine beliebige Kante.
-Dann geht $K$ durch $e$. Nun sei $a$ die Anzahl, wie häufig $K$ durch $e$ geht.
-Offensichtlich geht der Kreis sowohl in den Knoten hinein, als auch hinaus.
-$\Rightarrow e$ hat mindestens den Knotengrad $2a$. Es kann keine weitere
-Kante geben, da jeder Eulerkreis zu $G$ alle Kanten von $G$ beinhaltet.
-$\Rightarrow e$ hat den Knotengrad $2a \Rightarrow$ Jede Ecke von $G$ hat geraden
-Grad. $\blacksquare$
-
-\subsection{Rückrichtung}
-Hat jede Ecke in einem zusammenhängendem Graphen $G$ geraden Grad, so ist $G$ eulerisch.
-
-Beweis durch Induktion über die Anzahl $m$ der Kanten.
-
-\textbf{I.A.}:
-\begin{itemize}
-    \item $m=0 \rightarrow$ trivial
-    \item $m = 1$: nicht möglich
-    \item $m = 2$: Da $G$ zusammenhängend ist, können in diesem Fall nur zwei
-Ecken zweifach miteinander verbunden sein $\Rightarrow$ auch eulersch
-\end{itemize}
-
-\textbf{I.V.}: Sei $m \in \mathbb{N}_{\geq 2}$ die Anzahl der Kanten eines 
-Graphs $G$ und jeder zusammenhängende Graph mit weniger als $m$ Kanten und 
-ausschließlich Knoten geraden Grades sei eulerisch.
-
-\textbf{I.S.} 
-
-Jeder Knoten hat mindestens Grad 2 (zusammenhängend + gerader Grad)
-$\Rightarrow$ es gibt einen Kreis in $G$. TODO
-
-Sei nun $C$ ein Kreis in $G$ mit maximaler Länge.
-
-Annahme: $C$ ist kein Eulerkreis
-
-Wir entfernen alle Kanten in $C$ aus $G$ und nennen das Ergebnis $G^*$.
-Dann hat jeder Zusammenhängende Teilgraph in $G^*$ nur Knoten geraden Grades
-und hat daher einen Eulerkreis. Dieser Eulerkreis hat keine Kante, die in $C$
-enthalten ist und könnte deshalb zu $C$ hinzugefügt werden, wodurch $C$ Länger
-werden würde $\Rightarrow$ Widerspruch $\Rightarrow C$ ist ein Eulerkreis
-$\Rightarrow G$ ist eulersch  $\blacksquare$
- 
-\end{document}

source-code/Pseudocode/Vertex-coloring/Makefile

@@ -0,0 +1,36 @@
+SOURCE = Vertex-coloring-brute-force
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	make
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg
+	rsvg-convert -a -w $(WIDTH) -f svg $(SOURCE).svg -o $(SOURCE)2.svg
+	inkscape $(SOURCE)2.svg --export-plain-svg=$(SOURCE).svg
+	rm $(SOURCE)2.svg

source-code/Pseudocode/Vertex-coloring/Vertex-coloring-brute-force.tex

@@ -0,0 +1,38 @@
+\documentclass{article}
+\usepackage[pdftex,active,tightpage]{preview}
+\setlength\PreviewBorder{2mm}
+
+\usepackage[utf8]{inputenc} % this is needed for umlauts
+\usepackage[ngerman]{babel} % this is needed for umlauts
+\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
+\usepackage{amssymb,amsmath,amsfonts} % nice math rendering
+\usepackage{braket} % needed for \Set
+\usepackage{algorithm,algpseudocode}
+\renewcommand{\thealgorithm}{3} %disable numbers for algorithm
+
+\begin{document}
+\begin{preview}
+    \begin{algorithm}[H]
+        \begin{algorithmic}
+            \Require $G = (V, E)$ an undirected graph
+            \State $n \gets |V|$
+            \State Give all vertices an index $1 \leq i \leq n$ that defines an order
+            \For{$i \in 1, \dots, n$}
+                \State $v_i$.color $\gets 0$
+            \EndFor
+            \\
+            \If{$n==1$}
+                \State \Return
+            \Else
+                \For{$maxColors \in 2, \dots, n$}
+                    \While{$G$ is not properly colored and not all vertices have color $(maxColors-1)$}
+                        \State $(v_1 v_2 \dots v_n) \gets (v_1 v_2 \dots v_n) + 1$ \Comment{count up in base $maxColor$}
+                    \EndWhile
+                \EndFor
+            \EndIf
+        \end{algorithmic}
+    \caption{Find a vertex coloring for $G$ with brute force}
+    \label{alg:vertexColoring}
+    \end{algorithm}
+\end{preview}
+\end{document}

source-code/Pseudocode/Vertex-coloring/Vertex-coloring-first-wrong.tex

@@ -0,0 +1,36 @@
+\documentclass{article}
+\usepackage[pdftex,active,tightpage]{preview}
+\setlength\PreviewBorder{2mm}
+
+\usepackage[utf8]{inputenc} % this is needed for umlauts
+\usepackage[ngerman]{babel} % this is needed for umlauts
+\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
+\usepackage{amssymb,amsmath,amsfonts} % nice math rendering
+\usepackage{braket} % needed for \Set
+\usepackage{algorithm,algpseudocode}
+
+\begin{document}
+\begin{preview}
+    \begin{algorithm}[H]
+        \begin{algorithmic}
+            \Require $G = (V, E)$ an undirected graph
+            \State $n \gets |V|$
+            \State Give all vertices an index $1 \leq i \leq n$ that defines an order
+            
+            \For{$i \in 1, \dots, n$}
+                \State $v_i$.color $\gets 1$
+            \EndFor
+            \\
+            \For{$i \in 1, \dots, n$}
+                \For{$j \in i+1, \dots, n$}
+                    \If{$\Set{v_i, v_j} \in E \land v_i.\text{color} = v_j.\text{color}$}
+                        \State $v_j.color \gets v_j.color + 1$
+                    \EndIf
+                \EndFor
+            \EndFor
+        \end{algorithmic}
+    \caption{Find a vertex coloring for $G$}
+    \label{alg:vertexColoring}
+    \end{algorithm}
+\end{preview}
+\end{document}

source-code/Pseudocode/Vertex-coloring/Vertex-coloring-second-wrong.tex

@@ -0,0 +1,39 @@
+\documentclass{article}
+\usepackage[pdftex,active,tightpage]{preview}
+\setlength\PreviewBorder{2mm}
+
+\usepackage[utf8]{inputenc} % this is needed for umlauts
+\usepackage[ngerman]{babel} % this is needed for umlauts
+\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
+\usepackage{amssymb,amsmath,amsfonts} % nice math rendering
+\usepackage{braket} % needed for \Set
+\usepackage{algorithm,algpseudocode}
+\renewcommand{\thealgorithm}{2} %disable numbers for algorithm
+
+\begin{document}
+\begin{preview}
+    \begin{algorithm}[H]
+        \begin{algorithmic}
+            \Require $G = (V, E)$ an undirected graph
+            \State $n \gets |V|$
+            \State Give all vertices an index $1 \leq i \leq n$ that defines an order
+            
+            \For{$i \in 1, \dots, n$}
+                \State $v_i$.color $\gets 1$
+            \EndFor
+            \\
+            \For{$i \in 1, \dots, n$}
+                \State $possible \gets \Set{1, \dots, n}$
+                \For{$j \in i+1, \dots, n$}
+                    \If{$\Set{v_i, v_j} \in E$}
+                        \State $possible \gets possible \setminus \Set{v_j.\text{color}}$
+                    \EndIf
+                \EndFor
+                \State $v_i$.color $\gets \min(possible)$
+            \EndFor
+        \end{algorithmic}
+    \caption{Find a vertex coloring for $G$}
+    \label{alg:vertexColoring}
+    \end{algorithm}
+\end{preview}
+\end{document}

tikz/graph-triangles/graph-triangles.tex

@@ -1,16 +1,12 @@
-\documentclass{article}
-\usepackage[pdftex,active,tightpage]{preview}
-\setlength\PreviewBorder{2mm}
-
+\documentclass[varwidth=true, border=2pt]{standalone}
 \usepackage{ifthen}
 \usepackage{tikz}
 \usetikzlibrary{calc} 
+
+\begin{document}
 \tikzstyle{vertex}=[draw,red,fill=red,circle,
 minimum size=10pt,inner sep=0pt]
 \tikzstyle{edge}=[red, very thick]
-
-\begin{document}
-\begin{preview}
 \begin{tikzpicture}
     \newcommand{\n}{10}
     \foreach \y in {0, ..., \n}{
@@ -28,5 +24,4 @@ minimum size=10pt,inner sep=0pt]
         }
     }
 \end{tikzpicture}
-\end{preview}
 \end{document}

tikz/graph-v6-e8/Makefile

@@ -0,0 +1,35 @@
+SOURCE = graph-v6-e8
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	make
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg
+	rsvg-convert -a -w $(WIDTH) -f svg $(SOURCE).svg -o $(SOURCE)2.svg
+	inkscape $(SOURCE)2.svg --export-plain-svg=$(SOURCE).svg
+	rm $(SOURCE)2.svg

tikz/graph-v6-e8/README.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](graph-v6-e8.png)

tikz/graph-v6-e8/graph-v6-e8.tex

@@ -0,0 +1,21 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{ifthen}
+\usepackage{tikz}
+\usetikzlibrary{calc} 
+
+\begin{document}
+\tikzstyle{vertex}=[draw,black,fill=blue,circle,minimum size=10pt,inner sep=0pt]
+\tikzstyle{edge}=[very thick]
+\begin{tikzpicture}
+    \node (a)[vertex,fill=lime] at (1,0) {};
+    \node (b)[vertex,fill=red]  at (0,1) {};
+    \node (c)[vertex,fill=lime] at (1,2) {};
+    \node (d)[vertex,fill=blue] at (2,2) {};
+    \node (e)[vertex,fill=red]  at (3,1) {};
+    \node (f)[vertex,fill=blue] at (2,0) {};
+
+    \draw[edge] (a) -- (b) -- (c) -- (d) -- (e) -- (f) -- (a) -- cycle;
+    \draw[edge] (b) -- (f);
+    \draw[edge] (b) -- (d);
+\end{tikzpicture}
+\end{document}