From 47a5b2aa8cc60683a3c7cf0410632a7ba1ac9bd8 Mon Sep 17 00:00:00 2001 From: Never Gude Date: Sat, 13 Jun 2026 18:27:25 +0200 Subject: [PATCH] blatt 8 --- übung_6/agt_übung_6.pdf | Bin 70261 -> 70353 bytes übung_8/agt26-blatt08.pdf | Bin 0 -> 96296 bytes übung_8/agt_übung_8.pdf | Bin 0 -> 36383 bytes übung_8/agt_übung_8.tex | 18 +++++ übung_8/aufgabe_1.tex | 23 ++++++ übung_8/aufgabe_2.tex | 25 ++++++ übung_8/aufgabe_3.tex | 0 übung_8/aufgabe_4.tex | 38 +++++++++ übung_8/figures.pdf | Bin 0 -> 8376 bytes übung_8/ngexrcs.cls | 71 +++++++++++++++++ übung_8/ngutils.sty | 159 ++++++++++++++++++++++++++++++++++++++ übung_8/vw.jpg | Bin 0 -> 81265 bytes 12 files changed, 334 insertions(+) create mode 100644 übung_8/agt26-blatt08.pdf create mode 100644 übung_8/agt_übung_8.pdf create mode 100644 übung_8/agt_übung_8.tex create mode 100644 übung_8/aufgabe_1.tex create mode 100644 übung_8/aufgabe_2.tex create mode 100644 übung_8/aufgabe_3.tex create mode 100644 übung_8/aufgabe_4.tex create mode 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Übungsblatt} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[2em]{20} + +\input{aufgabe_1.tex} +\input{aufgabe_2.tex} +\input{aufgabe_3.tex} + +\end{document} diff --git a/übung_8/aufgabe_1.tex b/übung_8/aufgabe_1.tex new file mode 100644 index 0000000..0bfc858 --- /dev/null +++ b/übung_8/aufgabe_1.tex @@ -0,0 +1,23 @@ +\section{Kleinste Schnitte} +\begin{tasks} + \item + Siehe \autoref{fig:1a}. + \points{2} + + \begin{figure} + \centering + \includegraphics[page=1, width=0.5\textwidth]{figures.pdf} + \caption{Gegenbeispiel; Minimaler Schnitt $\tup{S, T}$ und nicht minimaler Schnitt $\tup{\set{v}, V \setminus \set{v}}$ in rot.} + \label{fig:1a} + + \end{figure} + + \item + Die Wahrscheinlichkeit, dass \alg*{Contract} in keiner Iteration + eine Kante aus $C = \set{uv \in E \mid u \in S, v \in T}$ mit minimalem Schnitt $\tup{S, T}$ kontrahiert, also immer die falschen Knoten auswählt, + ist laut Vorlesung $\frac{2}{n(n-1)}$. Das heißt die Zahl der + richtigen Knoten wächst quadratisch, da die Wahrscheinlichkeit quadratisch + abnimmt. + \points{3} + +\end{tasks} diff --git a/übung_8/aufgabe_2.tex b/übung_8/aufgabe_2.tex new file mode 100644 index 0000000..84af509 --- /dev/null +++ b/übung_8/aufgabe_2.tex @@ -0,0 +1,25 @@ +\section{Implementierung von \textsc{Contract}} +\begin{enumerate} + \item $G$ nach $H$ kopieren. \hfill $\Oh(1)$ + \item Wenn $\abs{V_H} \leq 2$, dann ist die Zerlegung $\tup{S, T}$ von $G$, + die den beiden letzen Knoten in $H$ entspricht, das Ergebnis. + \label{step2} + \item Wähle eine Zufallszahl $z$ im Intervall $\interval{1; \abs{V_H}}$. \hfill $\Oh(1)$ + \item Nimm den Knoten $a = V_H[z]$ und wähle eine Zufallszahl $z'$ im Intervall $\interval{1, \abs{Adj[a]}}$. + + Nimm den Knoten $b = Adj_{z'}[a]$. \hfill $\Oh(1)$ + \item Bestimme für jeden zu $a$ oder $b$ adjazenten Knoten $c_i$ die Anzahl der Kanten + zwischen $c_i$ und $a$ oder $b$. \hfill $\Oh(V_H) = \Oh(n)$. + \item Kontrahiere die Kante $ab$. Lösche dazu die Knoten $a, b$ sowie alle zu $a$ oder $b$ inzidenten Kanten. Da Mehrfachkanten als Zahl implementiert sind, sind + nur maximal 2 Einträge pro $c_i$ zu löschen. \hfill $\Oh(n)$ + + Füge einen neuen Knoten $d$ ein. Füge für jeden Knoten $c_i$ die vorher bestimmte Anzahl + an Kanten zwischen $c_i$ und $a$ oder $b$ als Kanten zwischen $c_i$ und $d$ ein. + \hfill $\Oh(n)$ + \item Gehe zurück zu \autoref{step2} +\end{enumerate} + +Da der Knoten $a$ durch eine gleichverteilte Zufallszahl $z$ ausgewählt wird und der +Knoten $b$ ebenfalls gleichverteilt ausgewählt wird, ist die Kante $ab$ ebenfalls +gleichverteilt ausgewählt. +\points{6} diff --git a/übung_8/aufgabe_3.tex b/übung_8/aufgabe_3.tex new file mode 100644 index 0000000..e69de29 diff --git a/übung_8/aufgabe_4.tex b/übung_8/aufgabe_4.tex new file mode 100644 index 0000000..4535a11 --- /dev/null +++ b/übung_8/aufgabe_4.tex @@ -0,0 +1,38 @@ +\section{Längste Wege} +\begin{tasks} + \item + Da $s, t$ in $G'$ adjazent zu jedem Knoten in $G$ ist, können wir + einen einfachen $s$-$t$-Weg der Länge $k+2$ erzeugen, indem wir + einen einfachen Weg der Länge $k$ in $G$ nehmen, $s$ an das eine Ende und $t$ an das andere Ende hängen. + + Umgekehrt kann man aus einem einfachen $s$-$t$-Weg der Länge $k$ + in $G'$ einen einfachen Weg der Länge $k-2$ in $G$ konstruieren, + indem wir $s$ und $t$ entfernen. + \points{2} + + \item + Ein Hamiltonweg ist ein Weg der alle Knoten in $G$ beinhaltet + und somit Länge $n-1$ besitzt. + + Wie wir oben gezeigt haben, kann ein $s$-$t$-Weg der Länge $n+1$ + in $G'$ leicht in einen Weg der Länge $n-1$ in $G$ umgewandelt + werden. Das heißt, dass wir einen Hamiltonweg in $G$ finden, + wenn wir einen $s$-$t$-Weg finden. + + Umgekehrt können wir einen Hamiltonweg leicht in einen $s$-$t$-Weg + umwandeln, also finden wir einen $s$-$t$-Weg wenn wir einen + Hamiltonweg finden. + + Also finden wir einen Hamiltonweg genau dann, wenn wir einen + $s$-$t$-Weg finden. + \points{1} + + \item + Da wir Hamiltonweg auf \algt{Längster $s$-$t$-Weg} reduziert + haben, muss also \algt{Längster $s$-$t$-Weg} $\NPe$-schwer sein, denn + wenn es in $\Pe$ liegen würde, könnten wir auch Hamiltonweg in + polynomieller Zeit lösen. Da wir nicht von $\Pe = \NPe$ ausgehen, + ist das nicht möglich. + \points{2} + +\end{tasks} diff --git a/übung_8/figures.pdf b/übung_8/figures.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e67be4300fb97484d932d89a7ab42ff3aa4d29b2 GIT binary patch literal 8376 zcmc(E2{@E*_b?^;O_tIqRC-!18D^g$*~v~b$dVW{gUK?(jHN|UmdaL0iKr+dWUZ7X zQk1QbvM+^D_NAi!&x}HE{onVy-s}3lZ?22yxzByh{hV{|bIyIvxp$dpYQqq67zhr4 z19bZ%fRd6t(bJxFltBgLHFPz#>254jZ~)f;Cw3xqZ-+9`#%TsmkU`w+v;JU@;>&I&b>LeZc;~e`8Ykcy1x@Smq_s`x%UDIhkH$& zsT4bnY~xkZQ{7F3M?iEaDtVBY)i(gNUGp+cjj6c`|sJgQi_mZ8=b+G9713c)`t*@P`wEP zAq1+2B#5+WoY8=`jZf%7Pava+6X%H{EKia1M4|{%PY-f;a!OyXVQM_&^;CnO-uWpe zaOC`yS_`XvZeE?`t8~p~{_%BK+qRPG0#t_X4clhq+q(R@fyCCnBdPV z;~|@g<57ftM8bp~QeQ?)RP@2rgQ-}5(n_9c{_4OE;sG49rbLd_l0aj&yeoxGav-o4EeZe@%!KJz)*t@n=GWqeMCMOJWI z>b|CVo15SEwgzi;#^PsNHb&1=l*?xudL&WxpH*xNor{X~J#&pl5?-N-;(BQ_^r{kB zv#JuV^-^2b4{Ytef3dlf)a!;N4{MK4r~(yySRqlGDest+mYJeotfT+UdP_ 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