From 9d19f3cd198d7be44ddf5142befd07699f3ff2b1 Mon Sep 17 00:00:00 2001 From: Never Gude Date: Sun, 19 Apr 2026 18:21:32 +0200 Subject: [PATCH] Aufgabe 2 und 3 fertig --- übung_1/ads_übung_1.pdf | Bin 37194 -> 0 bytes übung_1/ads_übung_1.synctex.gz | Bin 10657 -> 0 bytes übung_1/ads_übung_1.tex | 86 ----- übung_1/{ads_übung_1.aux => agt_übung_1.aux} | 14 +- ...ng_1.fdb_latexmk => agt_übung_1.fdb_latexmk} | 25 +- übung_1/{ads_übung_1.fls => agt_übung_1.fls} | 16 +- übung_1/{ads_übung_1.log => agt_übung_1.log} | 347 ++++++++++-------- übung_1/agt_übung_1.pdf | Bin 0 -> 45629 bytes übung_1/agt_übung_1.synctex.gz | Bin 0 -> 18514 bytes übung_1/agt_übung_1.tex | 157 ++++++++ 10 files changed, 385 insertions(+), 260 deletions(-) delete mode 100644 übung_1/ads_übung_1.pdf delete mode 100644 übung_1/ads_übung_1.synctex.gz delete mode 100644 übung_1/ads_übung_1.tex rename übung_1/{ads_übung_1.aux => agt_übung_1.aux} (58%) rename übung_1/{ads_übung_1.fdb_latexmk => agt_übung_1.fdb_latexmk} (93%) 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+(/usr/local/texlive/2025/texmf-dist/tex/latex/listings/lstmisc.sty +File: lstmisc.sty 2025/11/14 1.11b (Carsten Heinz) +\c@lstnumber=\count294 +\lst@skipnumbers=\count295 +\lst@framebox=\box58 +) +(/usr/local/texlive/2025/texmf-dist/tex/latex/listings/listings.cfg +File: listings.cfg 2025/11/14 1.11b listings configuration +)) +Package: listings 2025/11/14 1.11b (Carsten Heinz) + +==> First Aid for listings.sty no longer applied! + Expected: + 2024/09/23 1.10c (Carsten Heinz) + but found: + 2025/11/14 1.11b (Carsten Heinz) + so I'm assuming it got fixed. +\c@algorithm=\count296 +\theoremskip=\skip67 +) (/usr/local/texlive/2025/texmf-dist/tex/latex/enumitem/enumitem.sty Package: enumitem 2025/02/06 v3.11 Customized lists \labelindent=\skip68 -\enit@outerparindent=\dimen171 -\enit@toks=\toks20 -\enit@inbox=\box57 -\enit@count@id=\count288 -\enitdp@description=\count289 +\enit@outerparindent=\dimen175 +\enit@toks=\toks21 +\enit@inbox=\box59 +\enit@count@id=\count297 +\enitdp@description=\count298 ) (/usr/local/texlive/2025/texmf-dist/tex/latex/fontspec/fontspec.sty (/usr/local/texlive/2025/texmf-dist/tex/latex/l3packages/xparse/xparse.sty @@ -408,7 +442,7 @@ Package: expl3 2026-01-19 L3 programming layer (loader) (/usr/local/texlive/2025/texmf-dist/tex/latex/l3backend/l3backend-luatex.def File: l3backend-luatex.def 2025-10-09 L3 backend support: PDF output (LuaTeX) -\l__color_backend_stack_int=\count290 +\l__color_backend_stack_int=\count299 Inserting `l3color' in `luaotfload.parse_color'.)) Package: xparse 2025-10-09 L3 Experimental document command parser ) @@ -416,20 +450,20 @@ Package: fontspec 2025/09/29 v2.9g Font selection for XeLaTeX and LuaLaTeX Lua module: fontspec 2025/09/29 v2.9g Font selection for XeLaTeX and LuaLaTeX (/usr/local/texlive/2025/texmf-dist/tex/latex/fontspec/fontspec-luatex.sty Package: fontspec-luatex 2025/09/29 v2.9g Font selection for XeLaTeX and LuaLaT eX -\l__fontspec_script_int=\count291 -\l__fontspec_language_int=\count292 -\l__fontspec_strnum_int=\count293 -\l__fontspec_tmp_int=\count294 -\l__fontspec_tmpa_int=\count295 -\l__fontspec_tmpb_int=\count296 -\l__fontspec_tmpc_int=\count297 -\l__fontspec_em_int=\count298 -\l__fontspec_emdef_int=\count299 -\l__fontspec_strong_int=\count300 -\l__fontspec_strongdef_int=\count301 -\l__fontspec_tmpa_dim=\dimen172 -\l__fontspec_tmpb_dim=\dimen173 -\l__fontspec_tmpc_dim=\dimen174 +\l__fontspec_script_int=\count300 +\l__fontspec_language_int=\count301 +\l__fontspec_strnum_int=\count302 +\l__fontspec_tmp_int=\count303 +\l__fontspec_tmpa_int=\count304 +\l__fontspec_tmpb_int=\count305 +\l__fontspec_tmpc_int=\count306 +\l__fontspec_em_int=\count307 +\l__fontspec_emdef_int=\count308 +\l__fontspec_strong_int=\count309 +\l__fontspec_strongdef_int=\count310 +\l__fontspec_tmpa_dim=\dimen176 +\l__fontspec_tmpb_dim=\dimen177 +\l__fontspec_tmpc_dim=\dimen178 (/usr/local/texlive/2025/texmf-dist/tex/latex/base/fontenc.sty Package: fontenc 2025/07/18 v2.1d Standard LaTeX package @@ -716,61 +750,61 @@ Package: amstext 2024/11/17 v2.01 AMS text (/usr/local/texlive/2025/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 generic functions -\@emptytoks=\toks21 -\ex@=\dimen175 +\@emptytoks=\toks22 +\ex@=\dimen179 )) (/usr/local/texlive/2025/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d Bold Symbols -\pmbraise@=\dimen176 +\pmbraise@=\dimen180 ) (/usr/local/texlive/2025/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 2022/04/08 v2.04 operator names ) -\inf@bad=\count302 +\inf@bad=\count311 LaTeX Info: Redefining \frac on input line 235. -\uproot@=\count303 -\leftroot@=\count304 +\uproot@=\count312 +\leftroot@=\count313 LaTeX Info: Redefining \overline on input line 398. LaTeX Info: Redefining \colon on input line 409. -\classnum@=\count305 -\DOTSCASE@=\count306 +\classnum@=\count314 +\DOTSCASE@=\count315 LaTeX Info: Redefining \ldots on input line 495. LaTeX Info: Redefining \dots on input line 498. LaTeX Info: Redefining \cdots on input line 619. -\Mathstrutbox@=\box58 -\strutbox@=\box59 +\Mathstrutbox@=\box60 +\strutbox@=\box61 LaTeX Info: Redefining \big on input line 721. LaTeX Info: Redefining \Big on input line 722. LaTeX Info: Redefining \bigg on input line 723. LaTeX Info: Redefining \Bigg on input line 724. -\big@size=\dimen177 +\big@size=\dimen181 LaTeX Font Info: Redeclaring font encoding OML on input line 742. LaTeX Font Info: Redeclaring font encoding OMS on input line 743. -\macc@depth=\count307 +\macc@depth=\count316 LaTeX Info: Redefining \bmod on input line 904. LaTeX Info: Redefining \pmod on input line 909. LaTeX Info: Redefining \smash on input line 939. LaTeX Info: Redefining \relbar on input line 969. LaTeX Info: Redefining \Relbar on input line 970. -\c@MaxMatrixCols=\count308 +\c@MaxMatrixCols=\count317 \dotsspace@=\muskip17 -\c@parentequation=\count309 -\dspbrk@lvl=\count310 -\tag@help=\toks22 -\row@=\count311 -\column@=\count312 -\maxfields@=\count313 -\andhelp@=\toks23 -\eqnshift@=\dimen178 -\alignsep@=\dimen179 -\tagshift@=\dimen180 -\tagwidth@=\dimen181 -\totwidth@=\dimen182 -\lineht@=\dimen183 -\@envbody=\toks24 +\c@parentequation=\count318 +\dspbrk@lvl=\count319 +\tag@help=\toks23 +\row@=\count320 +\column@=\count321 +\maxfields@=\count322 +\andhelp@=\toks24 +\eqnshift@=\dimen182 +\alignsep@=\dimen183 +\tagshift@=\dimen184 +\tagwidth@=\dimen185 +\totwidth@=\dimen186 +\lineht@=\dimen187 +\@envbody=\toks25 \multlinegap=\skip70 \multlinetaggap=\skip71 -\mathdisplay@stack=\toks25 +\mathdisplay@stack=\toks26 LaTeX Info: Redefining \[ on input line 2950. LaTeX Info: Redefining \] on input line 2951. ) @@ -780,13 +814,13 @@ Package: lualatex-math 2022/01/01 v1.12 Patches for mathematics typesetting wit h LuaLaTeX (/usr/local/texlive/2025/texmf-dist/tex/latex/etoolbox/etoolbox.sty Package: etoolbox 2025/10/02 v2.5m e-TeX tools for LaTeX (JAW) -\etb@tempcnta=\count314 +\etb@tempcnta=\count323 ) Lua module: lualatex-math 2013/08/03 v1.3 Patches for mathematics typesetting wi th LuaLaTeX) -\g__um_fam_int=\count315 -\g__um_fonts_used_int=\count316 -\l__um_primecount_int=\count317 +\g__um_fam_int=\count324 +\g__um_fonts_used_int=\count325 +\l__um_primecount_int=\count326 \g__um_primekern_muskip=\muskip18 (/usr/local/texlive/2025/texmf-dist/tex/latex/unicode-math/unicode-math-table.t @@ -858,13 +892,13 @@ Package fontspec Info: LaTeX Font Info: Overwriting math alphabet `\mathrm' in version `normal' (Font) OT1/cmr/m/n --> TU/TeXGyrePagella(1)/m/n on input line -29. +28. LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' (Font) OT1/cmr/m/it --> TU/TeXGyrePagella(1)/m/it on input lin -e 29. +e 28. LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' (Font) OT1/cmr/bx/n --> TU/TeXGyrePagella(1)/b/n on input line - 29. + 28. Package fontspec Info: (fontspec) Font family 'TeXGyreHeros(0)' created for font 'TeX @@ -927,11 +961,11 @@ Package fontspec Info: (fontspec) Heros/BI:mode=base;script=latn;language=dflt;+smcp;" LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `normal' -(Font) OT1/cmss/m/n --> TU/TeXGyreHeros(1)/m/n on input line 3 -0. +(Font) OT1/cmss/m/n --> TU/TeXGyreHeros(1)/m/n on input line 2 +9. LaTeX Font Info: Overwriting math alphabet `\mathsf' in version `bold' (Font) OT1/cmss/bx/n --> TU/TeXGyreHeros(1)/b/n on input line -30. +29. Package fontspec Info: (fontspec) Font family 'TeXGyreCursor(0)' created for font 'TeX @@ -1012,10 +1046,10 @@ Package fontspec Info: LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `normal' (Font) OT1/cmtt/m/n --> TU/TeXGyreCursor(1)/m/n on input line -31. +30. LaTeX Font Info: Overwriting math alphabet `\mathtt' in version `bold' (Font) OT1/cmtt/m/n --> TU/TeXGyreCursor(1)/b/n on input line -31. +30. Package fontspec Info: (fontspec) Could not resolve font "Euler Math/B" (it probably @@ -1062,13 +1096,13 @@ etex;+ssty=0;"<-6.57>"name:Euler etex;+ssty=1;" LaTeX Font Info: Encoding `OT1' has changed to `TU' for symbol font -(Font) `operators' in the math version `normal' on input line 32. +(Font) `operators' in the math version `normal' on input line 31. LaTeX Font Info: Overwriting symbol font `operators' in version `normal' -(Font) OT1/cmr/m/n --> TU/EulerMath(1)/m/n on input line 32. +(Font) OT1/cmr/m/n --> TU/EulerMath(1)/m/n on input line 31. LaTeX Font Info: Encoding `OT1' has changed to `TU' for symbol font -(Font) `operators' in the math version `bold' on input line 32. +(Font) `operators' in the math version `bold' on input line 31. LaTeX Font Info: Overwriting symbol font `operators' in version `bold' -(Font) OT1/cmr/bx/n --> TU/EulerMath(1)/b/n on input line 32. +(Font) OT1/cmr/bx/n --> TU/EulerMath(1)/b/n on input line 31. Package fontspec Info: (fontspec) Could not resolve font "Euler Math/B" (it probably @@ -1096,13 +1130,13 @@ etex;mathfontdimen=tex2;+ssty=0;"<-6.57>"name:Euler etex;mathfontdimen=tex2;+ssty=1;" LaTeX Font Info: Encoding `OMS' has changed to `TU' for symbol font -(Font) `symbols' in the math version `normal' on input line 32. +(Font) `symbols' in the math version `normal' on input line 31. LaTeX Font Info: Overwriting symbol font `symbols' in version `normal' -(Font) OMS/cmsy/m/n --> TU/EulerMath(2)/m/n on input line 32. +(Font) OMS/cmsy/m/n --> TU/EulerMath(2)/m/n on input line 31. LaTeX Font Info: Encoding `OMS' has changed to `TU' for symbol font -(Font) `symbols' in the math version `bold' on input line 32. +(Font) `symbols' in the math version `bold' on input line 31. LaTeX Font Info: Overwriting symbol font `symbols' in version `bold' -(Font) OMS/cmsy/b/n --> TU/EulerMath(2)/b/n on input line 32. +(Font) OMS/cmsy/b/n --> TU/EulerMath(2)/b/n on input line 31. Package fontspec Info: (fontspec) Could not resolve font "Euler Math/B" (it probably @@ -1131,14 +1165,14 @@ etex;mathfontdimen=tex3;+ssty=1;" LaTeX Font Info: Encoding `OMX' has changed to `TU' for symbol font (Font) `largesymbols' in the math version `normal' on input line 3 -2. +1. LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal' -(Font) OMX/cmex/m/n --> TU/EulerMath(3)/m/n on input line 32. +(Font) OMX/cmex/m/n --> TU/EulerMath(3)/m/n on input line 31. 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Package hyperref Info: Backreferencing OFF on input line 4221. Package hyperref Info: Implicit mode ON; LaTeX internals redefined. Package hyperref Info: Bookmarks ON on input line 4468. -\c@Hy@tempcnt=\count322 +\c@Hy@tempcnt=\count331 (/usr/local/texlive/2025/texmf-dist/tex/latex/url/url.sty \Urlmuskip=\muskip19 Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. ) LaTeX Info: Redefining \url on input line 4807. -\XeTeXLinkMargin=\dimen185 +\XeTeXLinkMargin=\dimen189 (/usr/local/texlive/2025/texmf-dist/tex/generic/bitset/bitset.sty Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO) @@ -1228,9 +1262,9 @@ Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO) Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO ) )) -\Fld@menulength=\count323 -\Field@Width=\dimen186 -\Fld@charsize=\dimen187 +\Fld@menulength=\count332 +\Field@Width=\dimen190 +\Fld@charsize=\dimen191 Package hyperref Info: Hyper figures OFF on input line 6084. 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Package epstopdf Info: Output file is already uptodate. File: msb-eps-converted-to.pdf Graphic file (type pdf) -Package luatex.def Info: msb-eps-converted-to.pdf used on input line 33. +Package luatex.def Info: msb-eps-converted-to.pdf used on input line 35. (luatex.def) Requested size: 71.69984pt x 74.25963pt. 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+\documentclass{ngexrcs} +\usepackage{hyperref} +\title{1. Übungsblatt} +\subject{Algorithmische Graphentheorie} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[\qquad]{20} +\section{Spannbäume \& Breitensuche} +Sei $G = \tup{V, E}$ ein zusammenhängender Graph mit Kantengewichten $w: E \to \NN$ +und $s \in V \deg$ 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[width=0.2\textwidth]{msb.eps} + \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} +Sei $w(e) = 1$ für +alle $e \in E$ so gilt das Gegenbeispiel von oben. Also ist die Aussage +falsch. +\points{2} + +\end{tasks} + +\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} + +\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 herasgenommen. + Somit läuft der Algorithmus in $\Oh(\abs{V})$. + +\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. +\end{tasks} + +\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} + + +\section{Graphmodellierung} + +\end{document}