From 6475d1b263840fa9b34eddd27ae6ee5e22fa8448 Mon Sep 17 00:00:00 2001 From: Never Gude Date: Sat, 30 May 2026 20:05:41 +0200 Subject: [PATCH] =?UTF-8?q?=C3=BCbung=206?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...gt_übung_3_korrpdf => agt_übung_3_korr.pdf} | Bin übung_5/agt_übung_5.pdf | Bin 43248 -> 45544 bytes übung_5/agt_übung_5_korr.pdf | Bin 0 -> 1005268 bytes übung_5/aufgabe_3.tex | 22 ++++ übung_6/agt26-blatt06.pdf | Bin 0 -> 116641 bytes übung_6/agt_übung_6.tex | 17 +++ übung_6/aufgabe_1.tex | 82 ++++++++++++ übung_6/aufgabe_2.tex | 24 ++++ übung_6/aufgabe_3.tex | 24 ++++ übung_6/aufgabe_4.tex | 38 ++++++ übung_6/figures.pdf | Bin 0 -> 38992 bytes übung_6/ngexrcs.cls | 71 ++++++++++ übung_6/ngutils.sty | 124 ++++++++++++++++++ 13 files changed, 402 insertions(+) rename übung_3/{agt_übung_3_korrpdf => agt_übung_3_korr.pdf} (100%) create mode 100644 übung_5/agt_übung_5_korr.pdf create mode 100644 übung_6/agt26-blatt06.pdf 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enthalten. Wenn wir eine Kante zum Matching + hinzufügen, verringert sich die Anzahl der offenen Knoten um $2$. + Insbesondere verringert sich auch der Grad von $\frac{k}{2} - 1$ + Knoten um $1$ in $A$ und $B$. Da jeder Knoten mindestens Grad $\frac{k}{2}$ hat und jedes mal + ein offenes Knotenpaar entfernt wird. + $k$ mal eine Kante zum Matching hinzufügen. Also wird jeder Knoten gematcht. +\end{itemize} \points{5} diff --git a/übung_6/agt26-blatt06.pdf b/übung_6/agt26-blatt06.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0d027a76be4d692d6fff9b7e4975cdf99b6635d5 GIT binary patch literal 116641 zcma(2Ly#^^u!f6%ZQC|h+qR9>wr$(CZQHhO+xBX^`~3ID*@F{%vTKxC!>WkP_su7g zOhH79mXVGfnrwb~WF4A=kb%(7&=Q)52bx~S#MaE&oREoylbP`Uf1v5bEUcYP90}>g ztPPw^L`;nAj7^~V_@JGf9Zd{upxrlbG_T^e+L3;H{X*vDnL1DJ$^LeV2jD)O?~sXZ z?o0vi^dL0KZcR=)QcY~=|9;HIC#xk^^cdshj}+v|L<}9+m{BsV%sCZCYQI@>)-AWM8r9?m^O8vHw(P&Vk0f zzNHU$b<^5@4_~)U+_;)iwe2g^Z11OKfQEJb|ijQy;s&96ns@nZcq;lAuQmIPf>+Txlvpv57I4mvLW{^49}A^c0C5dUXktQn;LB$4%d5_>%N+jcVEiL7Ks*@ zhGAu-N;HWgkOT)gnc%A6Tjcv1yp1@M9`5wYuU+x`PXC+}i3uf%OB}uVZmFB&xqqTZ zO1a$TW_(lFV{qIssfxq*vQqA;A1Yaun8MeC;T?I?pyDV<)1aLJ>@Q8QY^+3Lp>2*J+Ba(H>Lh99;E{6U9N*`l77 zyJw`SQRk|!5ZOI***W$BY0RmkpQ0A7wxeDurt&)14~raw0ZE6Mvx>B|VjGd|5{+^Q z)UKjeSUZp^Ep52roNcE?q z7bv};RmTo&n^^UumffXyo0A@B4}B&yT@PUm&7@b^V$JcKC2M1U|{#;5chRlOQg-ob1f 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Übungsblatt} +\author{Jasper Gude \and Pia Röttgers} + +\begin{document} +\maketitle +\points[2em]{25} + +\input{aufgabe_1.tex} +\input{aufgabe_2.tex} +\input{aufgabe_3.tex} + +\end{document} diff --git a/übung_6/aufgabe_1.tex b/übung_6/aufgabe_1.tex new file mode 100644 index 0000000..665e6d9 --- /dev/null +++ b/übung_6/aufgabe_1.tex @@ -0,0 +1,82 @@ +\newcommand{\OPT}{\mathrm{OPT}} + +\section{Paarungen (Matchings) in Bäumen} +\begin{tasks} +\item + \label{1a} + Ein Baum ist ein kreisfreier, zusammenhängender Graph. Ein Baum besitzt ein + perfektes Matching genau dann, wenn die Anzahl der Knoten gerade ist und + jedes Blatt keine Geschwisterknoten hat. + + Die Anzahl der Knoten muss gerade sein, da in einem perfekten Match\-ing jeder + Knoten mit einem anderen gematcht wird. + + Jedes Blatt muss Einzelblatt sein, da der Elternknoten nur mit einem Blatt + gematcht werden kann. + + Also ist ein perfektes Matching im Baum eindeutig, da jedes Blatt mit seinem + Elternknoten gematcht werden muss. Diese Kante $uv$ ist sicher im Matching enthalten und alle zu $u$ oder $v$ inzidenten Kanten können aus dem Baum gelöscht werden. So entsteht ein neues Blatt, für das + die selbe Regel gilt. + \points{3} +\item + Der \autoref{alg:treematching} berechnet ein größtes Matching. + + Der Algorithmus ist korrekt, da der Baum $T$ entweder ein perfektes Matching + enthält und somit ein perfektes Matching berechnet (Siehe \autoref{1a}), + oder I dont fucking know. + + Die Laufzeit für \algt{Initialize} liegt in $\Oh(V)$, da sich der Baum in linearer Zeit augmentieren lässt und es $\abs{V} - 1$ viele Kanten gibt. + + Die Laufzeit für \algt{MatchaTeeRec} liegt in $\Oh(V)$. Der Algorithmus wird genau + einmal für jeden Knoten aufgerufen. Wenn eine Kante zum + Match\-ing hinzugefügt wird, werden die Knoten und Kanten aus dem Baum gelöscht + die nicht mehr zum Matching hinzugefügt werden können und somit + nicht mehr bearbeitet werden. + + Also liegt auch \algt{MatchaTee} in $\Oh(V)$. + \begin{algorithm} + \centering + \caption{Größtes Matching in Bäumen} + \label{alg:treematching} + \begin{algorithmic} + \alg{MatchaTee}{$T = \tup{V, E}$} \+ \\ + \com{Den ungerichten Baum zu einem gerichteten Baum mit $parent$ und} \\ + \com{$children$ pro Knoten und einer Wurzel $root$ augmentieren.} \\ + \alg{Initialize}{$T$} \\ + return \alg{MatchaTeeRec}{$T$, $T.root$} \- \\ + \\ + \alg{MatchaTeeRec}{$T$, $r$} \+ \\ + \com{Rekursiv für die Kinder aufrufen} \\ + $m \gets \emptyset$ \\ + foreach $v \in r.children$ do \+ \\ + $m \gets m \cup \text{\alg{MatchaTeeRec}{T, v}}$ \- \\ + \com{Wenn Knoten Blatt ist, dann die Kante zum Elternknoten hinzufügen.} \\ + if $r \neq nil$ and $deg(r) = 1$ then \+ \\ + \com{Die Knoten und inzidente Kanten aus dem Baum entfernen.} \\ + $T \gets T \setminus \set{r.parent, r.parent.children}$ \\ + $m \gets m \cup \set{\set{r, r.parent}}$ \- \\ + return $m$ + \end{algorithmic} + \end{algorithm} + \points{5} +\item + Wenn eine Kante $rv$ zu einem Matching hinzugefügt wird, kann keine zu $v$ + inzidente Kante hinzugefügt werden. Das müssen wir auch bei unserem dynamischen + Programm beachten. + \[ + \OPT(r) = \max_{v \in r.children} + \begin{cases} + rv + \sum_{i \in v.children} \OPT(i) + \sum_{j \in r.children \setminus v} \OPT(j) \\ + \sum_{k \in r.children} \OPT(k) + \end{cases} + \] + + Der Algorithmus hält sich an die Eigenschaften eines Matchings, also ist + er korrekt. + + Der Algorithmus probiert alle Kindknoten von $r$ aus. Geht man mit $r$ im Baum von + unten nach oben sind das also höchstens so viele Iterationen wie, Knoten im Baum + existieren (ohne die Blätter). Also läuft das Dynamische Programm in $\Oh(V)$. + + \points{5} +\end{tasks} diff --git a/übung_6/aufgabe_2.tex b/übung_6/aufgabe_2.tex new file mode 100644 index 0000000..872c2fd --- /dev/null +++ b/übung_6/aufgabe_2.tex @@ -0,0 +1,24 @@ +\section{Hamiltonkreise} +\begin{tasks} +\item + Ein Kreis hat im allgemeinen mindestens die Länge 3. Da der Turniergraph mindestens + 3 Knoten enthält und stark zusammenhängend ist, liegt jeder Knoten auf einem Kreis + mit Länge $\geq 3$. + \points{1} +\item + Gegeben ist ein Kreis $K$ mit Länge $\geq 3$. + Um $K$ zu verkleinern wählen wir uns drei aufeinanderfolgende Knoten $a, b, c$ aus. + Nach Definition gibt es für jedes Knotenpaar eine gerichtete Kante. Wir ersetzen + jetzt einen Weg der Länge 2 durch eine Kante die nach Annahme existieren muss. Das + machen wir solange, bis der Kreis die Länge 3 hat. + + \points{2} +\item + \points{1} +\item + \points{1} +\item + \points{1} +\item + \points{1} +\end{tasks} diff --git a/übung_6/aufgabe_3.tex b/übung_6/aufgabe_3.tex new file mode 100644 index 0000000..33b6aed --- /dev/null +++ b/übung_6/aufgabe_3.tex @@ -0,0 +1,24 @@ +% vim: ft=tex +\section{Perfekte Matchings in bipartiten Graphen} +\begin{quote} + Ein bipartiter Graph $G = \tup{A \cupdot B, E}$ mit $\abs{A} = \abs{B} = k$ + und jeder Knoten hat mindestens Grad $\frac{k}{2}$ $\iff$ $G$ enthält + perfektes Matching. +\end{quote} +\begin{itemize} + \item[$\seilpmi$] Wenn $G$ ein perfektes Matching enthält, dann muss + $\abs{A} = \abs{B}$, sonst würden Knoten übrig bleiben. + + Außerdem muss jeder Knoten mindestens Grad $1$ haben, da sonst Knoten + nicht gematcht werden könnten. + \item[$\implies$] Man kann jeden Knoten in $A$ einem + Knoten in $B$ zuordnen, da $\abs{A} = \abs{B}$. + + Alle Knoten sind im Matching enthalten. Wenn wir eine Kante zum Matching + hinzufügen, verringert sich die Anzahl der offenen Knoten um $2$. + Insbesondere verringert sich auch der Grad von $\frac{k}{2} - 1$ + Knoten um $1$ in $A$ und $B$. Da jeder Knoten mindestens Grad $\frac{k}{2}$ hat und jedes mal + ein offenes Knotenpaar entfernt wird. + $k$ mal eine Kante zum Matching hinzufügen. Also wird jeder Knoten gematcht. +\end{itemize} +\points{5} diff --git a/übung_6/aufgabe_4.tex b/übung_6/aufgabe_4.tex new file mode 100644 index 0000000..4535a11 --- /dev/null +++ b/übung_6/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_6/figures.pdf b/übung_6/figures.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dac87d043102e87e2c05ce715ab1337fae0e46ad GIT binary patch literal 38992 zcmeFa1zc6l)<2GPgMfhKp+)K(&f#zfkyg698O z<^Dc6{N?6O$L2P3c(4eZ(wKa9t2eCP*nMD3*-S=6s_Ba&jJKsg9B+`xq#bX#wYJi< zGz4-F!stI)Oko8NYtyk?fxo)gK;WvkG0HjW_wXzxWb%ERgTe>7JIrd<7+WvXs{Fe| z**T!8%U_n=nG=*3%9=3*Uzm)q&-d+R?Q8_7Ab-xp{3}T<$LfK zXNAgfKRV5n^)&drlf6ooykD#KJ$kNZ*;x-GAJfTk;`g6V&Z2h}%%@T#DHG|dUvm@6 zu}aklv|b!8lpF%o)CoqTBFI=8 zYbep92%O#E9Tj8$`F=cUrFf!VWNn-jHk=L-t`zSqB#ujWR0Cf0Tv0WWkmYV>m=NWm z90}%3HHe~^#fBO^*U~S^ikObTvVVE%nN8wdt<5CY`kPq!BHlTTXqu`opBRxk7I7qYcbPs`W^GlsGZA_;D9s)|4Q0*0jw8ykh!H<+E}OI z^XDuBU-62qkelN3RT%Qmh{Rm`LK^VE;)@*2E~}XUIE_E6{00B~os z40`*W+6I@rifu*~Yc}Ll=vu-!3PkFw(?z2i-U#VAo9jYsarz}o@iFK{+gV=s-S-?F z3Fcbu0>vhyYo=w%WD7(so*=xqZp3ZJ36Ttr3)nVadohA%CdU*NJ78}&hBvp!tnKR1 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