From 3c933395062b74ca41408e82c94331ea3241f205 Mon Sep 17 00:00:00 2001 From: Never Gude Date: Thu, 23 Apr 2026 13:22:15 +0200 Subject: [PATCH] =?UTF-8?q?=C3=BCbungsblatt=202?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- übung_1/ngexrcs.cls => ngexrcs.cls | 1 - übung_1/ngutils.sty => ngutils.sty | 0 übung_1/agt_übung_1.fdb_latexmk | 18 +- übung_1/agt_übung_1.fls | 14 +- übung_1/agt_übung_1.log | 70 +- übung_1/agt_übung_1.pdf | Bin 49645 -> 49648 bytes übung_1/agt_übung_1.synctex.gz | Bin 22594 -> 0 bytes übung_1/agt_übung_1.tex | 3 +- übung_2/agt26-blatt02.pdf | Bin 0 -> 117586 bytes übung_2/agt_übung_1.aux | 23 + übung_2/agt_übung_1.fdb_latexmk | 184 ++++ übung_2/agt_übung_1.fls | 186 ++++ übung_2/agt_übung_1.log | 1436 ++++++++++++++++++++++++++++ übung_2/agt_übung_1.pdf | Bin 0 -> 49648 bytes übung_2/agt_übung_2.tex | 19 + übung_2/aufgabe_1.tex | 47 + übung_2/aufgabe_2.tex | 78 ++ übung_2/aufgabe_3.tex | 26 + übung_2/figures.pdf | Bin 0 -> 10235 bytes 19 files changed, 2055 insertions(+), 50 deletions(-) rename übung_1/ngexrcs.cls => ngexrcs.cls (98%) rename übung_1/ngutils.sty => ngutils.sty (100%) delete mode 100644 übung_1/agt_übung_1.synctex.gz create mode 100644 übung_2/agt26-blatt02.pdf create mode 100644 übung_2/agt_übung_1.aux create mode 100644 übung_2/agt_übung_1.fdb_latexmk create mode 100644 übung_2/agt_übung_1.fls create mode 100644 übung_2/agt_übung_1.log create mode 100644 übung_2/agt_übung_1.pdf create mode 100644 übung_2/agt_übung_2.tex create mode 100644 übung_2/aufgabe_1.tex create mode 100644 übung_2/aufgabe_2.tex create mode 100644 übung_2/aufgabe_3.tex create mode 100644 übung_2/figures.pdf diff --git a/übung_1/ngexrcs.cls b/ngexrcs.cls similarity index 98% rename from übung_1/ngexrcs.cls rename to ngexrcs.cls index 1663cd6..f7863a5 100644 --- a/übung_1/ngexrcs.cls +++ b/ngexrcs.cls @@ -14,7 +14,6 @@ \KOMAoptions{mpinclude=true} \recalctypearea -\RequirePackage{ngutils} \RequirePackage{enumitem} diff --git a/übung_1/ngutils.sty b/ngutils.sty similarity index 100% rename from übung_1/ngutils.sty rename to ngutils.sty diff --git a/übung_1/agt_übung_1.fdb_latexmk b/übung_1/agt_übung_1.fdb_latexmk index 4ce6b79..34047a0 100644 --- a/übung_1/agt_übung_1.fdb_latexmk +++ b/übung_1/agt_übung_1.fdb_latexmk @@ -1,5 +1,7 @@ # Fdb version 4 -["lualatex"] 1776697900.74979 "/home/never/Documents/InNa/26_SoSe/AGT/agt_exercise/übung_1/agt_übung_1.tex" "agt_übung_1.pdf" "agt_übung_1" 1776697902.55915 0 +["lualatex"] 1776943071.00209 "agt_übung_1.tex" "agt_übung_1.pdf" "agt_übung_1" 1776943072.74298 0 + "../ngexrcs.cls" 1776942999.96868 2044 cc75cac272b2a86e50ef3bd66a963cf3 "" + "../ngutils.sty" 1776942782.83951 4493 7f39562722b52c0e9a76c17c4438174e "" "/home/never/.texlive2025/texmf-var/luatex-cache/generic/fonts/otl/euler-math.luc" 1770519352.28514 417132 89f085d929a7f52a226e499fed2b660f "" "/home/never/.texlive2025/texmf-var/luatex-cache/generic/fonts/otl/lmroman10-bold.luc" 1770519351.28214 128405 86240d2e88c12243734a13e4bf6704ec "" "/home/never/.texlive2025/texmf-var/luatex-cache/generic/fonts/otl/lmroman10-regular.luc" 1770519350.86413 127322 64667bb671c8def3fa5d85c341ebd817 "" @@ -17,9 +19,6 @@ "/home/never/.texlive2025/texmf-var/luatex-cache/generic/fonts/otl/texgyrepagella-regular.luc" 1770519351.40914 310182 64e3ff98a9959bea5d737ee8b9a06ff2 "" "/home/never/.texlive2025/texmf-var/luatex-cache/generic/names/luaotfload-lookup-cache.luc" 1776608606.25376 23014 4a1f49530b644afddd7c9abad44fb6f5 "" "/home/never/.texlive2025/texmf-var/luatex-cache/generic/names/luaotfload-names.luc.gz" 1770519350.80013 381420 8c34679872cd58fe28bd24cc6f0940ba "" - "/home/never/Documents/InNa/26_SoSe/AGT/agt_exercise/übung_1/agt_übung_1.tex" 1776697897.66689 304 d56b2c71c7672c3ab6ee2cdcfe258fc5 "" - "/home/never/texmf/tex/latex/ngexrcs/ngexrcs.cls" 1776614186.59271 2041 f7d79ac75cecdfbe676fa2a93e60930b "" - "/home/never/texmf/tex/latex/ngutils/ngutils.sty" 1776618253.70411 4493 7f39562722b52c0e9a76c17c4438174e "" "/usr/local/texlive/2025/texmf-dist/fonts/opentype/public/euler-math/Euler-Math.otf" 1761682777 433104 26cb5ed06a913ce89a33ff41234f5c7a "" "/usr/local/texlive/2025/texmf-dist/fonts/opentype/public/tex-gyre/texgyreheros-bold.otf" 1257299502 135204 f5f20609880fff666e17267854df804d "" "/usr/local/texlive/2025/texmf-dist/fonts/opentype/public/tex-gyre/texgyreheros-italic.otf" 1257299502 139208 418e1daec3a0c0b2d8ebb349c646ec22 "" @@ -171,14 +170,13 @@ "/usr/local/texlive/2025/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1770517628 5541377 824b1e5beafd2ebda40585373b35bb6b "" "/usr/local/texlive/2025/texmf-var/tex/generic/config/language.dat" 1770517624.99251 6800 99336720d16c8191172704bea65615e7 "" "/usr/local/texlive/2025/texmf-var/web2c/luahbtex/lualatex.fmt" 1770517696 5655839 756767735d369e65a483b47207bff649 "" - "agt_übung_1.aux" 1776697902.28891 2429 81986a9a774647930a5ac9e94cda7c51 "lualatex" - "agt_übung_1.tex" 1776697897.66689 304 d56b2c71c7672c3ab6ee2cdcfe258fc5 "" - "aufgabe_1.tex" 1776697508.46334 1313 9a143a43ac6e7341ab427860198035fe "" - "aufgabe_2.tex" 1776697727.20732 2574 4462577f3400843a00f378bf393c326c "" + "agt_übung_1.aux" 1776943072.50539 2429 81986a9a774647930a5ac9e94cda7c51 "lualatex" + "agt_übung_1.tex" 1776943024.09058 383 7ebaaa039d7287d23b7b487b932e7358 "" + "aufgabe_1.tex" 1776703998.38047 1292 93f78c81bfd1373cd3c7f231730eab51 "" + "aufgabe_2.tex" 1776699060.19025 2580 a0ca1bea6a4b1f6c64b382a891bc1e2a "" "aufgabe_3.tex" 1776697749.8824 1092 7e430e37e65a2141c25db6271e3e6e91 "" "aufgabe_4.tex" 1776697772.78547 718 3d1c5acc0d22695136e721c12d5042b8 "" - "dfs.eps" 1776689817.37668 5896 bfdfa03e5701d0b1d7555aa426023dcf "" - "msb.eps" 1776424417.18606 5913 81dafb89427e3a448260b7481993e119 "" + "figures.pdf" 1776704841.30975 10235 8eb27faa5a8fa3298021aeccb552f048 "" (generated) "agt_übung_1.aux" "agt_übung_1.log" diff --git a/übung_1/agt_übung_1.fls b/übung_1/agt_übung_1.fls index e64a2d8..f6fcf83 100644 --- a/übung_1/agt_übung_1.fls +++ b/übung_1/agt_übung_1.fls @@ -1,6 +1,6 @@ PWD /home/never/Documents/InNa/26_SoSe/AGT/agt_exercise/übung_1 INPUT /usr/local/texlive/2025/texmf-var/web2c/luahbtex/lualatex.fmt -INPUT /home/never/Documents/InNa/26_SoSe/AGT/agt_exercise/übung_1/agt_übung_1.tex +INPUT ./agt_übung_1.tex OUTPUT agt_übung_1.log INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/tex-ini-files/lualatexquotejobname.lua INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/base/ltluatex.lua @@ -61,7 +61,7 @@ INPUT /usr/local/texlive/2025/texmf-dist/tex/luatex/luaotfload/luaotfload-case.l INPUT /usr/local/texlive/2025/texmf-dist/tex/luatex/luaotfload/luaotfload-bcp47.lua INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/l3kernel/expl3.lua INPUT /usr/local/texlive/2025/texmf-dist/tex/luatex/lua-uni-algos/lua-uni-stage-tables.lua -INPUT /home/never/texmf/tex/latex/ngexrcs/ngexrcs.cls +INPUT ../ngexrcs.cls INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/scrartcl.cls INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/scrkbase.sty INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/scrbase.sty @@ -74,7 +74,7 @@ INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/scrsize11pt.clo INPUT /home/never/.texlive2025/texmf-var/luatex-cache/generic/names/luaotfload-names.luc.gz INPUT /home/never/.texlive2025/texmf-var/luatex-cache/generic/fonts/otl/lmroman10-regular.luc INPUT /usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/typearea.sty -INPUT /home/never/texmf/tex/latex/ngutils/ngutils.sty +INPUT ../ngutils.sty INPUT /usr/local/texlive/2025/texmf-dist/tex/generic/babel/babel.sty INPUT /usr/local/texlive/2025/texmf-dist/tex/generic/babel/luababel.def INPUT /usr/local/texlive/2025/texmf-var/tex/generic/config/language.dat @@ -175,10 +175,10 @@ INPUT /usr/local/texlive/2025/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm INPUT /usr/local/texlive/2025/texmf-dist/fonts/tfm/public/cm/cmmi10.tfm INPUT /usr/local/texlive/2025/texmf-dist/fonts/tfm/public/cm/cmmi7.tfm INPUT ./aufgabe_1.tex -INPUT ./msb-eps-converted-to.pdf -INPUT ./msb-eps-converted-to.pdf -INPUT ./dfs-eps-converted-to.pdf -INPUT ./dfs-eps-converted-to.pdf +INPUT ./figures.pdf +INPUT ./figures.pdf +INPUT ./figures.pdf +INPUT ./figures.pdf INPUT ./aufgabe_2.tex INPUT /usr/local/texlive/2025/texmf-var/fonts/map/pdftex/updmap/pdftex.map INPUT ./aufgabe_3.tex diff --git a/übung_1/agt_übung_1.log b/übung_1/agt_übung_1.log index b5a8513..711a16a 100644 --- a/übung_1/agt_übung_1.log +++ b/übung_1/agt_übung_1.log @@ -1,4 +1,4 @@ -This is LuaHBTeX, Version 1.22.0 (TeX Live 2025) (format=lualatex 2026.2.8) 20 APR 2026 18:53 +This is LuaHBTeX, Version 1.22.0 (TeX Live 2025) (format=lualatex 2026.2.8) 23 APR 2026 13:17 restricted system commands enabled. **agt_übung_1.tex (./agt_übung_1.tex @@ -39,7 +39,11 @@ Inserting `luaotfload.aux.fixup_fontdata' in `luaotfload.patch_font_unsafe'. Inserting `luaotfload.aux.set_capheight' in `luaotfload.patch_font'. Inserting `luaotfload.aux.set_xheight' in `luaotfload.patch_font'. Inserting `luaotfload.rewrite_fontname' in `luaotfload.patch_font'. -Inserting `tracingstacklevels' in `input_level_string'. (./ngexrcs.cls +Inserting `tracingstacklevels' in `input_level_string'. (../ngexrcs.cls + +LaTeX Warning: You have requested document class `../ngexrcs', + but the document class provides `ngexrcs'. + Document Class: ngexrcs 2021/12/21 ADS Exercise class (/usr/local/texlive/2025/texmf-dist/tex/latex/koma-script/scrartcl.cls Document Class: scrartcl 2026/02/02 v3.49.2 KOMA-Script document class (article @@ -188,9 +192,13 @@ Package typearea Info: These are the values describing the layout: (typearea) \footskip = 47.6pt (typearea) \baselineskip = 13.6pt (typearea) on input line 15. -(./ngutils.sty +(../ngutils.sty + +LaTeX Warning: You have requested package `../ngutils', + but the package provides `ngutils'. + Package: ngutils 2025/12/24 Never's LaTeX utils - (/usr/local/texlive/2025/texmf-dist/tex/generic/babel/babel.sty +(/usr/local/texlive/2025/texmf-dist/tex/generic/babel/babel.sty Package: babel 2026/01/26 v26.2 The multilingual framework for LuaLaTeX, pdfLaT eX and XeLaTeX @@ -1298,7 +1306,7 @@ Class scrartcl Info: loading recommended package `bookmark'. (scrartcl) `bookmarkpackage=false' before \begin{document} and (scrartcl) you can avoid this message adding: (scrartcl) \usepackage{bookmark} -(scrartcl) before \begin{document} on input line 9. +(scrartcl) before \begin{document} on input line 10. (/usr/local/texlive/2025/texmf-dist/tex/latex/bookmark/bookmark.sty Package: bookmark 2023-12-10 v1.31 PDF bookmarks (HO) @@ -1313,26 +1321,26 @@ Package babel Info: 'ngerman' activates 'german' shorthands. ) \openout1 = agt_übung_1.aux -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. -LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 9. -LaTeX Font Info: ... okay on input line 9. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for TU/lmr/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. +LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 10. +LaTeX Font Info: ... okay on input line 10. (/usr/local/texlive/2025/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] @@ -1357,9 +1365,9 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) \c@lstlisting=\count345 -Package tocbasic Info: setting babel extension for `lol' on input line 9. +Package tocbasic Info: setting babel extension for `lol' on input line 10. \scr@dte@lstlisting@maxnumwidth=\skip73 -Package hyperref Info: Link coloring OFF on input line 9. +Package hyperref Info: Link coloring OFF on input line 10. \g__cctab_next_cctab=\catcodetable17 (./aufgabe_1.tex @@ -1401,15 +1409,15 @@ L3 programming layer <2026-01-19> ) Here is how much of LuaTeX's memory you used: - 22934 strings out of 476076 + 22938 strings out of 476076 100000,1373583 words of node,token memory allocated 3192 words of node memory still in use: 61 hlist, 40 vlist, 38 rule, 9 local_par, 78 glue, 31 kern, 9 penalty, 20 gly ph, 213 attribute, 62 glue_spec, 107 attribute_list, 1 write nodes avail lists: 1:1,2:649,3:303,4:4,5:433,6:52,7:4721,8:16,9:473,10:4,11:478 - 45027 multiletter control sequences out of 65536+600000 + 45029 multiletter control sequences out of 65536+600000 115 fonts using 27962135 bytes - 122i,8n,114p,11236b,574s stack positions out of 10000i,1000n,20000p,200000b,200000s + 122i,8n,114p,11238b,574s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on agt_übung_1.pdf (4 pages, 49645 bytes). +Output written on agt_übung_1.pdf (4 pages, 49648 bytes). 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b/übung_2/agt_übung_2.tex @@ -0,0 +1,19 @@ +\documentclass{../ngexrcs} +\usepackage{../ngutils} +\usepackage{hyperref} +\setkeys{Gin}{pagebox=artbox, width=0.2\textwidth} + +\title{1. Übungsblatt} +\subject{Algorithmische Graphentheorie} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[\qquad]{20} + +\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 new file mode 100644 index 0000000..b214c4b --- /dev/null +++ b/übung_2/aufgabe_1.tex @@ -0,0 +1,47 @@ +\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} + diff --git a/übung_2/aufgabe_2.tex b/übung_2/aufgabe_2.tex new file mode 100644 index 0000000..266a448 --- /dev/null +++ b/übung_2/aufgabe_2.tex @@ -0,0 +1,78 @@ +\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} + diff --git a/übung_2/aufgabe_3.tex b/übung_2/aufgabe_3.tex new file mode 100644 index 0000000..ef594c1 --- /dev/null +++ b/übung_2/aufgabe_3.tex @@ -0,0 +1,26 @@ +\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} + diff --git a/übung_2/figures.pdf b/übung_2/figures.pdf new file mode 100644 index 0000000000000000000000000000000000000000..181f17abcbc24349f80fa63437e7d72b762f5a2a GIT binary patch literal 10235 zcmd^lc|4R~|93@bvSv>-${uEAW;Do>BE(pdErY>WW;8Rhr4m^x5+#&fQbJiH4M~e# zN?EciyC@=+e%Da=es$mX^ZmVE&;9)Ic)eV6u5-@koX>e*pZ&TdjkI+XV2Vm06c`F7 z+q;6XScr+2J6{K9<entkz9bKO=cob>WiEy=>GW-sT6AUBMA5T=hE930Ht6Dlf?Stg z_kA_&jnFHO7W2L}_E7^P(KGW~?Wf25y^$b`GV6UJclhnPM}m_xo^2jYzdqQFNR2t4 z->bN9|JPB3^cAnJI088XW=PUKe2zawL4}`Cdd2*VgPVpi{j^%ZP1XZsu8kLOvbh+_ z)|OOcZ+0?Q!N0X|4fZPt$BH~$6lg@B{ZTnr%&RyE^PTOg$dkS0tZ6)HQziNK$Js;O zv-VDw6Ikr|bT{clA%%u>uaf(}*KIy1Y?wJTC#GH2E;n-PsKdD8ojkD)jU{!`?evZ|#l7o~qdd;rt*i z!F3Ly7H6Xv>d1s`>!gIiv$Bh}s~V6iJ2!5BZ6KWGNAcrWC{lr)w|=sXmGwD|@*Eb> zP#2h$Xd9ni2ewIqh=>k1IfX7zk%^F|Aw$A9YP-AQfg|5H?@oyR2_ldjm$eZLv;f5h zp;s0B-zhR2x>k=J?3(R_HW#v9e2r?^xZ!}^2G{iqV^$YtSPH;C05(ILKy{!HJ!xcy zy#VeF@g7SCMaRVG@E${gx0Znko003MM_U@K)9!p3HW684J{Ft4j10sQ>Q)Pfc{mkOL&m*hldn05ebA|MvNU>OZSC(%hS`H;K6_q5QjAYz*mp)?}P|}My{2j2%L>u6_hWaDer+l_e`p? 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