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output out of 10000 (max. 100000000) + diff --git a/übung_2/agt_übung_2.pdf b/übung_2/agt_übung_2.pdf new file mode 100644 index 0000000..dabedcd Binary files /dev/null and b/übung_2/agt_übung_2.pdf differ diff --git a/übung_2/agt_übung_2.synctex.gz b/übung_2/agt_übung_2.synctex.gz new file mode 100644 index 0000000..90eedf5 Binary files /dev/null and b/übung_2/agt_übung_2.synctex.gz differ diff --git a/übung_2/agt_übung_2.tex b/übung_2/agt_übung_2.tex index bcc8f1c..2a27d84 100644 --- a/übung_2/agt_übung_2.tex +++ b/übung_2/agt_übung_2.tex @@ -1,5 +1,4 @@ -\documentclass{../ngexrcs} -\usepackage{../ngutils} +\documentclass{ngexrcs} \usepackage{hyperref} \setkeys{Gin}{pagebox=artbox, width=0.2\textwidth} @@ -14,6 +13,5 @@ \input{aufgabe_1.tex} \input{aufgabe_2.tex} \input{aufgabe_3.tex} -\input{aufgabe_4.tex} \end{document} diff --git a/übung_2/aufgabe_1.tex b/übung_2/aufgabe_1.tex index b214c4b..d549493 100644 --- a/übung_2/aufgabe_1.tex +++ b/übung_2/aufgabe_1.tex @@ -1,47 +1 @@ -\section{Spannbäume \& Breitensuche} -Sei $G = \tup{V, E}$ ein zusammenhängender Graph mit Kantengewichten $w: E \to \NN$ -und $s \in V$ ein ausgezeichneter Knoten. - -\begin{tasks} - -\item -\begin{quote} -Wenn $w(e) = 1$ für alle $e \in E$, dann ist der Breitensuchbaum mit -Quelle $s$ ein minimaler Spannbaum. -\end{quote} -Die Breitensuche berechnet in diesem -Fall den kürzesten Weg von jedem Knoten zum Knoten $s$, also den -Breitensuchbaum. Dieser spannt also einen minimalen Spannbaum auf. -\points{2} - -\item -\begin{quote} -Wenn $w(e) = 1$ für alle $e \in E$, dann ist jeder minimale Spannbaum -von $G$ ein Breitensuchbaum mit Quelle $s$. -\end{quote} -Falsch, siehe \autoref{fig:msb}. -\points{2} -\begin{figure}[h] - \centering - \includegraphics[page=1]{figures.pdf} - \caption{$\pi$-Zeiger des Breitensuchbaums und MSB blau hinterlegt.} - \label{fig:msb} -\end{figure} - -\item -\begin{quote} -Wenn $w(e) \in \set{1, 2, 3}$ für alle $e \in E$, dann ist jeder minimale -Spannbaum von $G$ ein Tiefensuchbaum mit Quelle $s$. -\end{quote} -Falsch, siehe \autoref{fig:dfs}. Der Minimale Spannbaum kann kein Tiefensuchbaum -sein. -\points{2} -\begin{figure}[h] - \centering - \includegraphics[page=2]{figures} - \caption{Tiefensuchbaum in orange und MSB blau hinterlegt.} - \label{fig:dfs} -\end{figure} - -\end{tasks} - +\section{Kleinste Knotenüberdeckung} \ No newline at end of file diff --git a/übung_2/aufgabe_2.tex b/übung_2/aufgabe_2.tex index 266a448..d0e8cc4 100644 --- a/übung_2/aufgabe_2.tex +++ b/übung_2/aufgabe_2.tex @@ -1,78 +1 @@ -\section{Kreissuche} -\begin{tasks} -\item - Wähle Startknoten $s$ und füge ihn in eine neue Queue $Q$ ein. - - Nimm den vordersten Knoten $u$ aus $Q$ und füge seine noch nicht entdeckten (weißen) - Nachbarn ein und färbe sie grau. Wenn einer der Nachbarn schon entdeckt wurde, - also grau ist, gibt es einen Kreis in $G$. - Markiere den Knoten $u$ als abgeschlossen ($black$) und entferne ihn aus $Q$ - - Wiederhole den Schritt solange bis $Q$ leer ist. - -% \begin{pseudocode}[caption=Hello World Algorithmische] -% EinfacherKreis(|Graph| $G$, |Vertex| $s$) -% Initialize($G$, $s$) -% $Q \gets$ new Queue -% $Q.$Enqueue($s$) -% $t_1 \gets nil$ -% $t_2 \gets s$ -% // Comment -% while $Q \neq \emptyset$ do -% $u \gets Q.$Dequeue -% $t_1 \gets t_2$ -% $t_2 \gets u$ -% foreach $v \in Adj[u]$ do -% if $v.color = white$ then -% $Q.$Enqueue($v$) -% $u.color \gets black$ -% if $t_1 = t_2$ then -% return $true$ -% else -% return $false$ -% \end{pseudocode} - \begin{algorithm}[h] - \centering - \begin{algorithmic} - \alg{EinfacherKreis}{Graph $G$, Vertex $s$} \+ \\ - \alg{Initialize}{$G$, $s$} \com{So wie in der Breitensuche} \\ - $Q \gets$ new \alg{Queue}{} \\ - $Q.$\alg{Enqueue}{$s$} \\ - while $Q \neq \emptyset$ do \+ \\ - $u \gets Q.$\alg{Dequeue}{} \\ - $u.color \gets gray$ \\ - foreach $v \in Adj[u]$ do \+ \\ - \com{Füge alle noch nicht entdeckten Knoten ein} \\ - if $v.color = white$ then \+ \\ - $v.color \gets gray$ \\ - $Q.$\alg{Enqueue}{$v$} \- \\ - \com{Wenn ein Knoten schon entdeckt wurde, gibt es einen Kreis} \\ - else return $true$ \- \\ - $u.color \gets black$ \- \\ - return $false$ - \end{algorithmic} - \end{algorithm} - \points{4} - -\item - Dadurch, dass wir nur Knoten - einfügen, die noch nicht entdeckt wurden, können wir nie auf dem selben Pfad - zu einem Knoten kommen. Das heißt, wenn wir einen schon entdeckten Knoten - finden, haben wir einen Kreis im Graphen gefunden. - - Der Algorithmus kann aufgrund der Struktur des Graphens (kein Multigraph, - keine Selbstkanten) nur Kreise der Länge mindestens 3 finden. - - Jeder Knoten wird nur einmal in die Queue eingefügt und nur einmal herausgenommen. - Somit läuft der Algorithmus in $\Oh(\abs{V})$. - \points{2} - -\item - Solange es weiße Knoten im Graphen gibt, wählen wir einen neuen Startknoten - für diese Zusammenhangskomponente. Als Ausgabe geben wir ein Array von - Tupeln $\tup{s_i, c_i}$ zurück, wobei $s_i$ der Startknoten einer - Zusammenhangskomponente und $c_i \in \set{true, false}$ der Wahrheitswert, ob - ein Kreis in der Komponente existiert. - \points{1} -\end{tasks} - +\section{Straßenreparatur mittels Linearer Programmierung und Flüssen} \ No newline at end of file diff --git a/übung_2/aufgabe_3.tex b/übung_2/aufgabe_3.tex index ef594c1..ce91cf4 100644 --- a/übung_2/aufgabe_3.tex +++ b/übung_2/aufgabe_3.tex @@ -1,26 +1 @@ -\section{Eulerwege} -\begin{quote} -Sei $G = \tup{V, E}$ ein ungerichteter, zusammenhängender Graph. Dann gilt: $G$ hat genau -dann einen Eulerweg, wenn die Anzahl an Knoten $v \in V$, für die gilt, dass $deg(v)$ -ungerade ist, genau $0$ oder $2$ ist. -\end{quote} - -\begin{itemize} - \item[$\seilpmi$] - 1. Fall: $0$ Knoten mit ungeradem Grad. Nach dem Satz in der Vorlesung - gibt es einen Eulerkreis. Im Eulerweg sind also Start- und Endknoten - identisch. - - 2. Fall: $2$ Knoten mit ungeradem Grad. Die beiden Knoten bilden den Start- - und Endknoten des Eulerwegs. Die Kante die den Eulerkreis schließen würde - braucht genau zwei Knoten, zu denen sie inzident ist. Nehmen wir diese Kante - weg, ergibt sich eine ungerader Grad an diesen beiden Knoten. - \item[$\implies$] - Ein Graph mit ungerader Anzahl an Knoten mit ungeradem Grad kann nicht - existieren, da die Summe aller Knoten mit ungeradem Grad gerade ist. - - Für alle anderen Fälle gilt, wenn ein Knoten ungeraden Grad hat, dann gibt - es keinen Weg aus dem Knoten heraus, wenn man hineingelaufen ist. -\end{itemize} -\points{4} - +\section{Flüsse finden} \ No newline at end of file diff --git a/übung_2/ngexrcs.cls b/übung_2/ngexrcs.cls new file mode 100644 index 0000000..c7218dc --- /dev/null +++ b/übung_2/ngexrcs.cls @@ -0,0 +1,75 @@ +% vim: set filetype:tex + +% Identification % +\NeedsTeXFormat{LaTeX2e} +\ProvidesClass{ngexrcs}[2021/12/21 ADS Exercise class] + +% Handle options % +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}} +\ProcessOptions\relax + +% More declarations % +\LoadClass{scrartcl} + +\KOMAoptions{mpinclude=true} +\recalctypearea + +\RequirePackage{enumitem} +\RequirePackage{ngutils} + +% define fonts +\RequirePackage{fontspec} +%\RequirePackage{tgpagella} +%\RequirePackage{tgheros} +%\RequirePackage{euler-math} +\RequirePackage[math-style=upright]{unicode-math} +\setmainfont{TeX Gyre Pagella} +\setsansfont{TeX Gyre Heros} +\setmonofont{TeX Gyre Cursor} +\setmathfont{Euler Math} + +\setlength{\marginparwidth}{1.5\marginparwidth} +\setlength{\fboxrule}{\heavyrulewidth} + + +% +% \renewcommand\theenumi{\@alph\c@enumi} +% \renewcommand\theenumii{\@arabic\c@enumii} +%\renewcommand\theenumiii{\@roman\c@enumiii} +%\renewcommand\theenumiv{\@Alph\c@enumiv} + +% \renewcommand\labelenumi{\theenumi)} +% \renewcommand\labelenumii{\theenumii.} +%\renewcommand\labelenumiii{\theenumiii.} +%\renewcommand\labelenumiv{\theenumiv.} + +% Use sansfont for all title elements +\addtokomafont{titlehead}{\sffamily} +\addtokomafont{subject}{\normalfont \sffamily \itshape} +\addtokomafont{title}{\sffamily} +\addtokomafont{subtitle}{\sffamily} +\addtokomafont{author}{\sffamily} +\addtokomafont{date}{\sffamily} +\addtokomafont{publishers}{\sffamily} + +% Use serif font for headings +\addtokomafont{disposition}{\rmfamily} + +% Let sections be formated as in: Aufgabe 1 -- Section title +\renewcommand*{\sectionformat}{Aufgabe \thesection\autodot\enskip--\enskip} + +% Let points of an exercise be printed as in: [__ / 2] +\newkomafont{points}{\sffamily} +\newcommand\points[2][\quad]{\marginline{\framebox{{\usekomafont{points}#1 \textbf{/} #2}}}} + +\newenvironment{tasks}[1][1em] +{ +\setlist{leftmargin=*} +\renewcommand\theenumi{\@alph\c@enumi} +\renewcommand\theenumii{\@arabic\c@enumii} +\renewcommand\labelenumi{\theenumi)} +\renewcommand\labelenumii{\theenumii.} +\begin{enumerate} +}{% +\end{enumerate} +} diff --git a/übung_2/ngutils.sty b/übung_2/ngutils.sty new file mode 100644 index 0000000..2dd431e --- /dev/null +++ b/übung_2/ngutils.sty @@ -0,0 +1,150 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{ngutils}[2025/12/24 Never's LaTeX utils] + +\RequirePackage[ngerman]{babel} +\RequirePackage{graphicx} +\RequirePackage{tabularx} +\RequirePackage{booktabs} +\RequirePackage{listings} + + +\lstnewenvironment{pseudocode}[1][] %defines the algorithm listing environment +{ + \renewcommand{\lstlistingname}{Algorithmus} + \lstset{ %this is the stype + mathescape=true, + columns=fullflexible, + basicstyle=\normalfont, + identifierstyle=\scshape, + keywordstyle=\bfseries, + keywords={, new, if, then, else, while, for, in, to, up, down, foreach, do, return}, + commentstyle=\itshape, + comment=[l]//, + delim=[is][\normalfont]{|}{|}, + tabsize=3, + frame=tb, + framerule=1pt, + #1 % this is to add specific settings to an usage of this environment (for instnce, the caption and referable label) + } +} +{} + +\newcommand{\seilpmi}{\ensuremath{\Longleftarrow}} + +\newcommand\NN{\ensuremath{\mathbb{N}}} +\newcommand\ZZ{\ensuremath{\mathbb{Z}}} +\newcommand\QQ{\ensuremath{\mathbb{Q}}} +\newcommand\RR{\ensuremath{\mathbb{R}}} +\newcommand\CC{\ensuremath{\mathbb{C}}} +\newcommand\PP{\ensuremath{\mathbb{P}}} + +\newcommand\Oh{\ensuremath{\mathcal{O}}} +\newcommand\oh{\ensuremath{\scriptstyle{\mathcal{O}}}} +\newcommand\Eh{\ensuremath{\mathrm{E}}} +\newcommand\Var{\ensuremath{\mathrm{Var}}} + +\newcommand\parens[1]{\ensuremath{\left(#1\right)}} +\newcommand\brackets[1]{\ensuremath{\left[#1\right]}} +\newcommand\braces[1]{\ensuremath{\left\{#1\right\}}} +\newcommand\angled[1]{\ensuremath{\left\langle#1\right\rangle}} + +\newcommand\tup[1]{\parens{#1}} +\newcommand\ivl[1]{\brackets{#1}} +\newcommand\set[1]{\braces{#1}} +\newcommand\arr[1]{\angled{#1}} +\newcommand\abs[1]{\ensuremath{\lvert#1\rvert}} +\newcommand\norm[1]{\ensuremath{\lVert#1\rVert}} +\newcommand\ceil[1]{\ensuremath{\lceil#1\rceil}} +\newcommand\floor[1]{\ensuremath{\lfloor#1\rfloor}} + +\newcommand\alg[2]{{\normalfont\scshape#1}{\normalfont(#2)}} +\newcommand\algt[1]{{\normalfont\scshape#1}} +\newcommand\com[1]{{\normalfont\itshape/\!\!/ #1}} + +\newenvironment{algorithmic}[1][1em]{ + \begin{minipage}[t]{\dimexpr\linewidth} + \bfseries + \begin{tabbing} + \hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\kill + }{% + \end{tabbing} + \end{minipage} +} + +% \newenvironment{algorithm}[1][1em]{ +% \begin{center} +% \begin{minipage}[t]{\dimexpr\linewidth} +% \bfseries +% \begin{tabbing} +% \hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\kill +% }{% +% \end{tabbing} +% \end{minipage} +% \end{center} +% } + +\makeatletter +\newcounter{algorithm}% the counter +\renewcommand*\thealgorithm{\arabic{algorithm}}% the default format for the counter +\newcommand*\fps@algorithm{tbp}% default floating options +\newcommand*\ftype@algorithm{4}% float type number, needs to be a power of 2; % \ftype@figure=1, \ftype@table=2 +\newcommand*\ext@algorithm{loa}% extension of file for the list of schemes +\newcommand*\fnum@algorithm{\algorithmname\nobreakspace\thealgorithm}% used in caption +\newcommand*\algorithmname{Algorithmus}% the name +\newcommand*\listalgorithmname{Algorithmenverzeichnis}% the name of the list +\newenvironment{algorithm}{\@float{algorithm}}{\end@float}% the environment +\newenvironment{algorithm*}{\@dblfloat{algorithm}}{\end@dblfloat}% the starred % version for twocolumn documents +\newcommand*\listofalgorithms{% the list of schemes + \section*{\listalgorithmname} + \@mkboth + {\MakeUppercase\listalgorithmname} + {\MakeUppercase\listalgorithmname} + \@starttoc{\ext@algorithm} } + \let\l@algorithm\l@figure % layout of list is the same as for figures +\makeatother + +\newlength{\theoremskip} +\setlength{\theoremskip}{6em} + +\newcommand{\theoremfont}{\normalfont\bfseries} +\newcommand{\definitionfont}{\normalfont\bfseries} + +\newenvironment{theoremic}[2][\theoremfont]{ + {#1 #2} + \hfill + \begin{minipage}[t]{\the\dimexpr\linewidth-\theoremskip\relax} + }{ + \end{minipage} +} + +\newenvironment{theorem}{ + \begin{theoremic}{Satz.} + }{ + \end{theoremic} +} + +\newenvironment{lemma}{ + \begin{theoremic}{Lemma.} + }{ + \end{theoremic} +} + +\newenvironment{proof}{ + \begin{theoremic}{Beweis.} + }{ + \end{theoremic} +} + +\newenvironment{definition}{ + \begin{theoremic}[\definitionfont]{Definition.} + }{ + \end{theoremic} +} + +\newenvironment{example}{ + \begin{theoremic}[\definitionfont]{Beispiel.} + }{ + \end{theoremic} +} + +