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\documentclass[11pt]{scrartcl}
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\usepackage[utf8]{inputenc}
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\usepackage[ngerman]{babel}
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\usepackage{amsmath}
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\usepackage{enumerate}
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\usepackage{latexsym}
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%\usepackage{amssymb}
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\usepackage{multicol}
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%\usepackage{booktabs}
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%\usepackage{pstricks}
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%\usepackage{pst-node}
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\usepackage[paper=a4paper,left=30mm,right=20mm,top=20mm,bottom =25mm]{geometry}
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\usepackage[
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pdftitle={Uebungsblatt 4: Graphentheorie - AlgoDS2},
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pdfsubject={Algorithmen und Datenstrukturen 2; Uebungsblatt 4: Graphentheorie},
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pdfauthor={Thomas Battermann},
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pdfkeywords={Uebungsblatt 4: Graphentheorie - AlgoDS2},
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pdfborder={0 0 0}
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]{hyperref}
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\usepackage{listings}
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basicstyle=\ttfamily,
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}
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\pagestyle{fancy} %eigener Seitenstil
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\fancyhf{} %alle Kopf- und Fußzeilenfelder bereinigen
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\fancyhead[L]{Übungsblatt 4\\Graphentheorie} %Kopfzeile links
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\fancyhead[C]{} %zentrierte Kopfzeile
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\fancyhead[R]{\textsc{Algorithmen \& Datenstrukturen 2}\\12. Januar 2011} %Kopfzeile rechts
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\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie
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\newcommand{\spa}{\hspace*{4mm}}
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\newcommand{\defin}{\textcolor{darkgreen}{\textbf{Def.: }}}
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\newcommand{\rrfloor}{\right\rfloor}
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\newcommand{\llfloor}{\left\lfloor}
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\title{Übung Graphentheorie 1 -- AlgoDS2}
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\author{Thomas Battermann}
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\date{12. Januar 2011}
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\begin{document}
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\pagestyle{fancy}
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\section*{Aufgabe 1}
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\addcontentsline{toc}{section}{Aufgabe 1}
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\subsection*{a)}
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\addcontentsline{toc}{subsection}{a)}
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\includegraphics{bilder/1a.eps}
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\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
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a&q&s&t&w&v&x&y&z&r&u\\\hline
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$d(a)$ & 0&1&1&1&2&2&2&3&$\infty$&$\infty$\\\hline
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$\pi(a)$ & -&q&q&q&s&t&t&x&-&-
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\end{tabular}
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\subsection*{b)}
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\addcontentsline{toc}{subsection}{b)}
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Vorgängergraph:\\
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\includegraphics{bilder/1b.eps}
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\subsection*{c)}
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\addcontentsline{toc}{subsection}{c)}
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\includegraphics{bilder/1c.eps}
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\section*{Aufgabe 2}
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\addcontentsline{toc}{section}{Aufgabe 2}
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\subsection*{a)}
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\addcontentsline{toc}{subsection}{a)}
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Kante \((u,v) \in E \)
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Angenommen $(u,v)$ ist eine TreeEdge (\textcolor{Red}{ForwardEdge})\\
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\begin{itemize}
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\item[$\Rightarrow$] In der Tiefensuche wird $v$ von $u$ aus entdeckt.
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\item[$\Rightarrow$] $v$ ist ein Nachkomme von $u$ im Tiefensuchebaum
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\item[$\Rightarrow$] Verschachtelungssatz:\\
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\( d[u] < d[v] < f[v] < f[u] \)
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\end{itemize}
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\('\Rightarrow'\) Angenommen \( d[u] < d[v] < f[v] < f[u] \)
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\begin{itemize}
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\item[$\Rightarrow$] Verschachtelungssatz:\\
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$v$ ist ein Nachkomme von $u$ um selben Tiefensuchebaum.\\
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\begin{itemize}
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\item[Fall 1:]
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$v$ wurde von $u$ aus entdeckt $\Rightarrow (u,v)$ TreeEdge
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\item[Fall 2:]
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$v$ wurde nicht von $u$ aus entdeckt $\Rightarrow (u,v)$ ForwardEdge
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\end{itemize}
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\end{itemize}
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\subsection*{b)}
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\addcontentsline{toc}{subsection}{b)}
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$'\Rightarrow'$ Angenommen $(u,v)$ ist eine BackEdge\\
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d.\,h. $u$ ist ein Nachfolger von $v$ im selben Tiefesuchebaum\\
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$\Leftrightarrow$ Verschachtelungssatz \( d[v] < d[u] < f[u] y f[v] \)
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\includegraphics{bilder/2b.eps}
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\subsection*{c)}
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\addcontentsline{toc}{subsection}{c)}
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\includegraphics{bilder/2c.eps}
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$'\Rightarrow': (u,v)$ CrossEdge
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Also sind $u$ und $v$ in verschiedenen Tiefensuchebäumen.
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\(\Rightarrow\) $v$ muss vor $u$ in der Tiefensuche entdeckt worden sein.\\
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\(\Rightarrow d[v] < f[v] < d[u] < f[u] \)\\
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\(\Leftarrow\) Angenommen: \( d[v] < f[v] < d[u] < f[u] \)\\
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\(\Rightarrow\) Verschachtelungssatz: $u$ und $v$ sind in zwei verschiedenen TS-Bäumen\\
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\(\Rightarrow (u,v) \) ist CrossEdge
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\section*{Aufgabe 3}
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\addcontentsline{toc}{section}{Aufgabe 3}
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Laufzeit: \( O(\|\|V\|\|^2) \)\\
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sowohl Breitensuche und Tiefensuche
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$\Rightarrow$ Es müssen alle Spalten/Zeilen durchgegangen werden.
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\section*{Aufgabe 4}
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\addcontentsline{toc}{section}{Aufgabe 4}
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Annahme: \includegraphics{bilder/4.eps}
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Behauptung: Jede Tiefensuche liefert $d[v] < f[u]$ als Ergebnis.
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Betrachte eine beliebige Tiefensuche zu dem Zeitpunkt, an dem $v$ entdeckt wird.
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\underline{Fall 1:} $u$ wurde vor $v$ entdeckt\\
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Weißer Pfadsatz: $v$ ist ein Nachfolger von u \(\to d[v] < f[u]\)
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\underline{Fall 2:} $u$ wurde noch nicht entdeckt:\\
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\begin{enumerate}[a)]
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\item \(v\overset{P}{\leadsto}u \Rightarrow u\) Nachkomme von $v$\\
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Verschachtelungssatz: \( d[v] < d[u] < f[u] < f[v] \)
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\item $u$ ist von $v$ aus nicht erreichbar.\\
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Also wird $v$ komplett abgearbeitet, bevor $u$ entdeckt wird:\\
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\( d[v] < f[v] < d[u] < f[u] \)
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\end{enumerate}
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\section*{Aufgabe 5}
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\addcontentsline{toc}{section}{Aufgabe 5}
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Graph $G=(V,E)$ (gerichtet)\\
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$E^T = \{ (v,u) \mid (u,v) \in E \}$
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\begin{lstlisting}[frame=single]
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MatrixTranspose(a):
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n = size(a)
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for i:= 1 to n do
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for j:= 1 to n do
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B[i][j] = A[j][i]
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return B
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\end{lstlisting}
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Laufzeit: \( \Theta(n^2)\)
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\begin{lstlisting}[frame=single]
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ListTranspose(G=(V,E)):
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Erzeuge einen leeren Graphen G2 mit der Knotenmenge V
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for jeden Knoten u in V do
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for jeden Knoten v in Adj[u] do
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Fuege Kante (v,u) zu G2 hinzu
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return G2
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\end{lstlisting}
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Laufzeit: \( O(\|V\| + \|E\| ) \)
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\section*{Aufgabe 6}
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\addcontentsline{toc}{section}{Aufgabe 6}
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\includegraphics{bilder/6.eps}
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\underline{Ansatz:} Führe Tiefensuche durch und speichere die benutzten Kanten in einer Liste.
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\begin{lstlisting}[frame=single]
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Search(G,s):
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for each v in V do
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color[v] := WHITE
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Initialisiere die Liste p
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Visit(G,s,p)
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\end{lstlisting}
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\begin{lstlisting}[frame=single]
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Visit(G,v,p):
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color[v] := GRAY
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Haenge v an das Ende von p
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for each u in Neighbour(v) do
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if color[u] = WHITE then
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Visit(G,u,p)
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Haenge v an das Ende von P
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else if color[u] = GRAY then
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Haenge (u,v) an das Ende von p
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color[v] := BLACK
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\end{lstlisting}
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\end{document}
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<path fill="sym-stroke" fillrule="eofill">
|
||||
-0.6 -0.6 m
|
||||
0.6 -0.6 l
|
||||
0.6 0.6 l
|
||||
-0.6 0.6 l
|
||||
h
|
||||
-0.4 -0.4 m
|
||||
0.4 -0.4 l
|
||||
0.4 0.4 l
|
||||
-0.4 0.4 l
|
||||
h
|
||||
</path>
|
||||
</group>
|
||||
</symbol>
|
||||
<symbol name="mark/cross(sx)" transformations="translations">
|
||||
<group>
|
||||
<path fill="sym-stroke">
|
||||
-0.43 -0.57 m
|
||||
0.57 0.43 l
|
||||
0.43 0.57 l
|
||||
-0.57 -0.43 l
|
||||
h
|
||||
</path>
|
||||
<path fill="sym-stroke">
|
||||
-0.43 0.57 m
|
||||
0.57 -0.43 l
|
||||
0.43 -0.57 l
|
||||
-0.57 0.43 l
|
||||
h
|
||||
</path>
|
||||
</group>
|
||||
</symbol>
|
||||
<symbol name="arrow/fnormal(spx)">
|
||||
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||
0 0 m
|
||||
-1 0.333 l
|
||||
-1 -0.333 l
|
||||
h
|
||||
</path>
|
||||
</symbol>
|
||||
<symbol name="arrow/pointed(spx)">
|
||||
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
|
||||
0 0 m
|
||||
-1 0.333 l
|
||||
-0.8 0 l
|
||||
-1 -0.333 l
|
||||
h
|
||||
</path>
|
||||
</symbol>
|
||||
<symbol name="arrow/fpointed(spx)">
|
||||
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||
0 0 m
|
||||
-1 0.333 l
|
||||
-0.8 0 l
|
||||
-1 -0.333 l
|
||||
h
|
||||
</path>
|
||||
</symbol>
|
||||
<symbol name="arrow/linear(spx)">
|
||||
<path stroke="sym-stroke" pen="sym-pen">
|
||||
-1 0.333 m
|
||||
0 0 l
|
||||
-1 -0.333 l
|
||||
</path>
|
||||
</symbol>
|
||||
<symbol name="arrow/fdouble(spx)">
|
||||
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||
0 0 m
|
||||
-1 0.333 l
|
||||
-1 -0.333 l
|
||||
h
|
||||
-1 0 m
|
||||
-2 0.333 l
|
||||
-2 -0.333 l
|
||||
h
|
||||
</path>
|
||||
</symbol>
|
||||
<symbol name="arrow/double(spx)">
|
||||
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
|
||||
0 0 m
|
||||
-1 0.333 l
|
||||
-1 -0.333 l
|
||||
h
|
||||
-1 0 m
|
||||
-2 0.333 l
|
||||
-2 -0.333 l
|
||||
h
|
||||
</path>
|
||||
</symbol>
|
||||
<pen name="heavier" value="0.8"/>
|
||||
<pen name="fat" value="1.2"/>
|
||||
<pen name="ultrafat" value="2"/>
|
||||
<symbolsize name="large" value="5"/>
|
||||
<symbolsize name="small" value="2"/>
|
||||
<symbolsize name="tiny" value="1.1"/>
|
||||
<arrowsize name="large" value="10"/>
|
||||
<arrowsize name="small" value="5"/>
|
||||
<arrowsize name="tiny" value="3"/>
|
||||
<color name="red" value="1 0 0"/>
|
||||
<color name="green" value="0 1 0"/>
|
||||
<color name="blue" value="0 0 1"/>
|
||||
<color name="yellow" value="1 1 0"/>
|
||||
<color name="orange" value="1 0.647 0"/>
|
||||
<color name="gold" value="1 0.843 0"/>
|
||||
<color name="purple" value="0.627 0.125 0.941"/>
|
||||
<color name="gray" value="0.745"/>
|
||||
<color name="brown" value="0.647 0.165 0.165"/>
|
||||
<color name="navy" value="0 0 0.502"/>
|
||||
<color name="pink" value="1 0.753 0.796"/>
|
||||
<color name="seagreen" value="0.18 0.545 0.341"/>
|
||||
<color name="turquoise" value="0.251 0.878 0.816"/>
|
||||
<color name="violet" value="0.933 0.51 0.933"/>
|
||||
<color name="darkblue" value="0 0 0.545"/>
|
||||
<color name="darkcyan" value="0 0.545 0.545"/>
|
||||
<color name="darkgray" value="0.663"/>
|
||||
<color name="darkgreen" value="0 0.392 0"/>
|
||||
<color name="darkmagenta" value="0.545 0 0.545"/>
|
||||
<color name="darkorange" value="1 0.549 0"/>
|
||||
<color name="darkred" value="0.545 0 0"/>
|
||||
<color name="lightblue" value="0.678 0.847 0.902"/>
|
||||
<color name="lightcyan" value="0.878 1 1"/>
|
||||
<color name="lightgray" value="0.827"/>
|
||||
<color name="lightgreen" value="0.565 0.933 0.565"/>
|
||||
<color name="lightyellow" value="1 1 0.878"/>
|
||||
<dashstyle name="dashed" value="[4] 0"/>
|
||||
<dashstyle name="dotted" value="[1 3] 0"/>
|
||||
<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
|
||||
<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
|
||||
<textsize name="large" value="\large"/>
|
||||
<textsize name="Large" value="\Large"/>
|
||||
<textsize name="LARGE" value="\LARGE"/>
|
||||
<textsize name="huge" value="\huge"/>
|
||||
<textsize name="Huge" value="\Huge"/>
|
||||
<textsize name="small" value="\small"/>
|
||||
<textsize name="footnote" value="\footnotesize"/>
|
||||
<textsize name="tiny" value="\tiny"/>
|
||||
<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
|
||||
<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
|
||||
<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
|
||||
<gridsize name="4 pts" value="4"/>
|
||||
<gridsize name="8 pts (~3 mm)" value="8"/>
|
||||
<gridsize name="16 pts (~6 mm)" value="16"/>
|
||||
<gridsize name="32 pts (~12 mm)" value="32"/>
|
||||
<gridsize name="10 pts (~3.5 mm)" value="10"/>
|
||||
<gridsize name="20 pts (~7 mm)" value="20"/>
|
||||
<gridsize name="14 pts (~5 mm)" value="14"/>
|
||||
<gridsize name="28 pts (~10 mm)" value="28"/>
|
||||
<gridsize name="56 pts (~20 mm)" value="56"/>
|
||||
<anglesize name="90 deg" value="90"/>
|
||||
<anglesize name="60 deg" value="60"/>
|
||||
<anglesize name="45 deg" value="45"/>
|
||||
<anglesize name="30 deg" value="30"/>
|
||||
<anglesize name="22.5 deg" value="22.5"/>
|
||||
<tiling name="falling" angle="-60" step="4" width="1"/>
|
||||
<tiling name="rising" angle="30" step="4" width="1"/>
|
||||
</ipestyle>
|
||||
<page>
|
||||
<layer name="alpha"/>
|
||||
<view layers="alpha" active="alpha"/>
|
||||
<path layer="alpha" stroke="black">
|
||||
8 0 0 8 64 768 e
|
||||
</path>
|
||||
<path stroke="black">
|
||||
8 0 0 8 128 768 e
|
||||
</path>
|
||||
<text transformations="translations" pos="64 768" stroke="black" type="label" width="5.703" height="4.289" depth="0" halign="center" valign="center">$u$</text>
|
||||
<text transformations="translations" pos="128 768" stroke="black" type="label" width="5.187" height="4.289" depth="0" halign="center" valign="center">$v$</text>
|
||||
<path stroke="black">
|
||||
72 768 m
|
||||
120 768 l
|
||||
</path>
|
||||
<path stroke="blue" rarrow="normal/normal">
|
||||
122.343 773.657 m
|
||||
96 784
|
||||
69.6569 773.657 s
|
||||
</path>
|
||||
<path stroke="orange" rarrow="normal/normal">
|
||||
69.6569 762.343 m
|
||||
96 752
|
||||
122.343 762.343 s
|
||||
</path>
|
||||
<text matrix="1 0 0 1 32 0" transformations="translations" pos="64 792" stroke="blue" type="label" width="74.886" height="6.918" depth="0" halign="center" valign="center">rekursiver Aufruf</text>
|
||||
<text matrix="1 0 0 1 32 8" transformations="translations" pos="64 736" stroke="orange" type="label" width="100.484" height="6.918" depth="0" halign="center" valign="center">Ende rekursiver Aufruf</text>
|
||||
</page>
|
||||
</ipe>
|
||||
Loading…
Reference in new issue