Chomksy, Schema F
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6
Makefile
6
Makefile
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MAIN = sheet
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FLAGS = -pdf
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FLAGS = -pdf -lualatex
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dev:
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latexmk $(FLAGS) -pvc $(MAIN)
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all:
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latexmk $(FLAGS) $(MAIN)
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dev:
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latexmk $(FLAGS) -pvc $(MAIN)
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clean:
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latexmk -C
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534
sheet.tex
534
sheet.tex
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@ -10,10 +10,14 @@
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]{geometry}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{makecell}
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\usepackage{multicol}
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\usepackage[noend]{algorithm2e}
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\usepackage[utf8]{inputenc}
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\usepackage{fancyhdr}
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\usepackage{tikz}
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\usetikzlibrary{arrows,automata,positioning, graphs, graphdrawing}
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\usegdlibrary {trees}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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@ -33,397 +37,217 @@
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\fancyhead[R]{Gero Beckmann - \url{https://github.com/Geronymos/}}
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\fancyfoot{}
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\fancyfoot[R]{\thepage}
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\section{Laufzeit}
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\newenvironment{definition}[1]{\noindent\textbf{#1:}}{}
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\section{Chomsky-Hierarchie}
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\hspace*{-.5cm}
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\begin{tabular}{ l l l l }
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Notations & Asymptotischer Vergleich & Formale Definition & Grenzen \\
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$f(n) \in \omega(g(n))$&
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$f(n)$ wächst schneller als $g(n)$ &
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$\forall c \exists n_0 \forall n > n_0 f(n) > c \cdot g(n)$ &
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$$$\lim\sup\limits_{n \rightarrow \infty}\frac{f}{g} = \infty$$$ \\
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\begin{tabular}{ l l l l l }
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Chomsky-Typ & Wortproblem & Definition & Bsp & Maschinenmodell \\
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$f(n) \in \Omega(g(n))$ &
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$f(n)$ wächst min. so schnell wie $g(n)$ &
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$\exists c \exists n_0 \forall n > n_0 c \cdot f(n) \leq g(n)$ &
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$$$0 < \liminf\limits_{n \rightarrow \infty}\frac{f}{g} \leq \infty$$$ \\
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Typ 0 &
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semi-entscheidbar &
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\makecell{$G = (\Sigma, V, S, R)$ \\ $R beliebig$ }&
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universelle Sprache &
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NTM/DTM akzeptiert L \\
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\( f(n) \in \Theta(g(n)) \) &
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$f(n)$ und $g(n)$ wachsen gleich schnell &
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$f(n) \in \mathcal{O}(g(n)) \wedge f(n) \in \Omega(g(n))$ &
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$$$0 < \lim\limits_{n \rightarrow \infty}\frac{f}{g} < \infty$$$ \\
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Typ 1 &
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NP-Schwer &
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\makecell{$u \rightarrow v, |u| \leq |v|$ \\ $u \in V^+, S \notin V$ \\ $S \rightarrow \epsilon$ } &
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$L = \{ a^ib^ic^i | i \leq 1 \}$ &
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\makecell{NTM mit Platzbedarf n \\ erkennt Wörter der Länge n in L \\ $\Rightarrow NTAPE(n)$ } \\
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\( f(n) \in \mathcal{O}(g(n)) \) &
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$f(n)$ wächst max. so schnell wie $g(n)$ &
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$\exists c \exists n_0 \forall n > n_0 f(n) \leq c \cdot g(n)$ &
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$$$0 \leq \limsup\limits_{n \rightarrow \infty}\frac{f}{g} < \infty$$$ \\
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\( f(n) \in o(g(n)) \) &
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$f(n)$ wächst langsamer als $g(n)$ &
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$\forall c \exists n_0 \forall n > n_0 c \cdot f(n) < g(n)$ &
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$$$\lim\limits_{n \rightarrow \infty} \frac{f}{g} = \infty$$$ \\
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Typ 2 &
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polynomiell &
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\makecell{$A \rightarrow v, A \in V$ \\ $v beliebig$} &
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$L = \{ a^ib^i | i \leq 1\}$ &
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CYK-Alg. erkennt L in polynom. Zeit, Chomsky-NF, NPDA \\
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Typ 3 &
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linear &
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\makecell{$A \rightarrow v, A \in V$ \\ $V \in \epsilon \cup \Sigma \cdot V$} &
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$L = \{ a^i | i \leq 1 \}$ &
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NEA/DEA erkennt L \\
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\end{tabular}
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\subsection{Vergleich}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
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$1$ & $\log^*n$ & $\log n$ & $\log^2n$ & $\sqrt[3]{n}$ &
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$\sqrt{n}$ & $n$ & $n^2$ & $n^3$ & $n^{\log n}$ &
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$2^{\sqrt{n}}$ & $2^n$ & $3^n$ & $4^n$ & $n!$ & $2^{n^2}$
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\subsection{Automaten}
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DEA $A = (Q, \Sigma, \delta: Q \times \Sigma \rightarrow Q, s \in Q, F \subseteq Q)$ \\
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NEA $A = (Q, \Sigma, \delta: Q \times (\Sigma \cup \{ \epsilon \} \rightarrow 2^Q, s \in Q, F \subseteq Q)$
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NPDA \\
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DPDA \\
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DTM \\
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NTM \\
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\subsection{Pumping-Lemma}
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\begin{multicols}{2}
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Erfüllt:
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\begin{itemize}
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\item["$\exists$"] Wähle $n = 2$
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\item["$\forall$"] Betrachte beliebiges $w \in L$ mit $|w| > 2$
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\item["$\exists$"] Wähle zerlegung $w = uvx$ mit $u = \epsilon, v = aa, x=a^{2(j-1)}$
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\item["$\forall$"] Für alle $i \in \mathbb{N}_0: uv^ix = a^{2i}a^2(j-1) = a^{2(i+j-1)} \in L$
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\end{itemize}
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Widerlegen:
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\begin{itemize}
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\item["$\exists$"] Wähle $n = 2$
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\item["$\forall$"] Betrachte beliebiges $w \in L$ mit $|w| > 2$
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\item["$\exists$"] Wähle zerlegung $w = uvx$ mit $u = \epsilon, v = aa, x=a^{2(j-1)}$
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\item["$\forall$"] Für alle $i \in \mathbb{N}_0: uv^ix = a^{2i}a^2(j-1) = a^{2(i+j-1)} \in L$
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\end{itemize}
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\end{multicols}
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\begin{multicols}{2}
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Potenzmengenkonstuktion NEA $\rightarrow$ DEA
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\begin{tabular}{c | c | c}
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Zustand & a & b \\
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\hline
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$\{\underline{s}\}$ & $\{s, q_1\}$ & $\{f\}$ \\
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$\{\underline{s}, q_1\}$ & $\{s, q_1\}$ & $\{f, q_2\}$ \\
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$\{f\}$ & $\{f\}$ & $\{f\}$ \\
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$\{f, q_2\}$ & $\{f\}$ & $\{f, q_1, q_2\}$ \\
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$\{f, \underline{s}\}$ & $\{f, s, q_1\}$ & $\{f\}$ \\
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$\{f, \underline{s}, q_1\}$ & $\{f, s, q_1\}$ & $\{f, q_2\}$ \\
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\end{tabular}
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\begin{multicols}{3}
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\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
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\subsubsection*{Transitivität}
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$f_1(n) \in \mathcal{O}(f_2(n)) \wedge f_2(n) \in\mathcal{O}(f_3(n))$ \\
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$\Rightarrow f_1(n) \in \mathcal{O}(f_3(n))$
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\subsubsection*{Summen}
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$f_1(n) \in \mathcal{O}f_3(n)) \wedge f_2(n) \in \mathcal{O}(f_3(n))$ \\
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$\Rightarrow f_1(n) + f_2(n) \in \mathcal{O}(f_3(n))$
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\subsubsection*{Produkte}
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$f_1(n) \in \mathcal{O}(g_1(n)) \wedge f_2(n) \in \mathcal{O}(g_2(n))$ \\
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$\Rightarrow f_1(n) \cdot f_2(n) \in \mathcal{O}(g_1(n) \cdot g_2(n))$
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\columnbreak
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\subsection{Master-Theorem}
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Sei $T(n) = a \cdot T(\frac{n}{b}) + f(n)$ mit $f(n) \in \Theta(n^c)$ und i
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$T(1) \in \Theta(1)$. Dann gilt
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$
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T(n) \in \begin{cases}
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\Theta(n^c) &\text{wenn } a < b^c, \\
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\Theta(n^c \log n) &\text{wenn } a = b^c, \\
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\Theta(n^{\log_b(a)}) &\text{wenn } a > b^c.
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\end{cases}
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$
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\subsubsection{Monome}
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\begin{itemize}
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\item $a \leq b \Rightarrow n^a \in \mathcal{O}(n^b)$
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\item $n^a \in \Theta(n^b) \Leftrightarrow a = b$
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\item $\sum_{v \in V}deg(v) = \Theta(m)$
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\item $\forall n \in \mathbb{N}: \sum^n_{k=0}k = \frac{n(n+1)}{2}$
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\item $
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\sum^b_{i=a}c^i \in \begin{cases}
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\Theta(c^a) &\text{wenn } c < 1, \\
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\Theta(c^b) &\text{wenn } c > 1, \\
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\Theta(b-a) &\text{wenn } c = 1.
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\end{cases}
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$
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\item $\log(ab) = \log(a) + \log(b)$
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\item $\log(\frac{a}{b}) = \log(a) - \log(b)$
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\item $a^{\log_a(b)} = b$
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\item $a^x = e^{ln(a) \cdot x}$
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\item $\log(a^b) = b \cdot \log(a)$
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\item $\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$
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\end{itemize}
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%\subsubsection{Konstante Faktoren}
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%
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%$a \cdot f(n) \in \Theta(f(n))$
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\node[state,initial,accepting] (S) {$S$};
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\node[state] (q_1) [right of=S] {$q_1$};
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\node[state] (q_2) [right of=q_1] {$q_2$};
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\node[state] (f) [below of=q_1] {$f$};
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\path[->]
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(S) edge [loop above] node {a} ()
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(S) edge node [below] {a} (q_1)
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(S) edge node [left] {b} (f)
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(q_1) edge [bend right] node [above] {a} (S)
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(q_1) edge node [below] {b} (q_2)
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(q_2) edge [bend right] node [above] {b} (q_1)
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(q_2) edge [loop right] node {b} ()
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(q_2) edge node {a} (f)
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(f) edge [loop left] node {a,b} ()
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;
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\end{tikzpicture}
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\end{multicols}
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\begin{minipage}{0.7\textwidth}
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\section{Sortieren}
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\begin{tabular}[t]{c || c | c | c | c}
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Algorithmus & best case & average & worst & Stabilität \\
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\hline
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Insertion-Sort &
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$\mathcal{O}(n)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & stabil\\
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Bubble-Sort &
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$\mathcal{O}(n)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & stabil\\
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Merge-Sort &
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$\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & stabil\\
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Quick-Sort &
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$\mathcal{O}(n \log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & i.A. nicht stabil\\
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Heap-Sort &
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$\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & nicht stabil\\
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\hline
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Bucket-Sort &
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$\Theta(n+m)$ & $\Theta(n+m)$ & $\Theta(n+m)$ &
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stabil $e \in [0, m)$\\
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Radix-Sort &
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$\Theta(c \cdot n)$ & $\Theta(c\cdot n)$ & $\Theta(c\cdot n)$ &
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stabil $e \in [0, n^c)$\\
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\end{tabular}
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\textwidth}
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\subsection{Heaps}
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\begin{tabular}[t]{c || c}
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Bin.-Heap & Laufzeit \\
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\hline
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push(x) & $\mathcal{O}(\log n)$ \\
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popMin() & $\mathcal{O}(\log n)$ \\
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decPrio(x, x') & $\mathcal{O}(\log n)$ \\
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build([$\mathbb{N}$; n]) & $\mathcal{O}(n)$
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\end{tabular}
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\begin{itemize}
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\item linkes Kind: $2v + 1$
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\item rechts Kind: $2v + 2$
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\item Elternknoten: $ \lfloor \frac{v - 1}{2} \rfloor $
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\end{itemize}
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\end{minipage}
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\begin{multicols}{2}
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Entfernen von $\epsilon$-Übergängen
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\section{Datenstrukturen}
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\begin{tabular}{c | c | c}
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Zustand & a & b \\
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\hline
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$S$ & $q_1$ & $S, q_1, q_2, q_3$ \\
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$q_1$ & $q_2, q_3$ & $q_3$ \\
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$q_2$ & $q_1$ & $S, q_2, q_3$ \\
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$q_3$ & $q_1$ & $S, q_2, q_3$ \\
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\end{tabular}
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\subsection{Listen}
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\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
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\begin{tabular}{c || c | c | c || c}
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Operation & DLL & SLL & Array & Erklärung(*) \\
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\hline
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first & 1 & 1 & 1 & \\
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last & 1 & 1 & 1 & \\
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insert & 1 & 1* & n & nur insertAfter \\
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remove & 1 & 1* & n & nur removeAfter \\
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pushBack & 1 & 1 & 1* & amortisiert \\
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pushFront & 1 & 1 & n & \\
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popBack & 1 & n & 1* & amortisiert \\
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popFront & 1 & 1 & n & \\
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concat & 1 & 1 & n & \\
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splice & 1 & 1 & n \\
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findNext & n & n & n
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\node[state,initial] (S) {$S$};
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\node[state,accepting] (q_1) [right of=S] {$q_1$};
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\node[state,accepting] (q_2) [below of=q_1] {$q_2$};
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\node[state] (q_3) [below of=S] {$q_3$};
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\end{tabular}
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\subsection{Hash-Tabelle}
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$\mathcal{H}$ heißt \textbf{universell}, wenn für ein zufälliges gewähltes
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$h \in \mathcal{H}$ gilt: $U \rightarrow \{0, ..., m-1\}$ \\
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$\forall k, l \in U, k \neq l: Pr[h(k) = h(l)] = \frac{1}{m}$ \\
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$h_{a,b}(k) = ((a\cdot k + b) \mod p) \mod m$
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\subsection{Graphen}
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\begin{tabular}{c || c}
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Algorithmus & Laufzeit \\
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\hline
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BFS/DFS & $\Theta(n+m)$\\
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topoSort & $\Theta(n)$\\
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Kruskal & $\Theta(m \log n)$\\
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Prim & $\Theta((n+m)\log n)$ \\
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Dijksta & $\Theta((n + m) \log n)$\\
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Bellmann-Ford & $\Theta(nm)$\\
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Floyd-Warshall & $\Theta(n^3)$ \\
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\end{tabular}
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\path[->]
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(S) edge node {b} (q_1)
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(S) edge node [above left] {$\epsilon$} (q_2)
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(q_1) edge node {a} (q_2)
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(q_1) edge [bend left] node [above right] {b} (q_3)
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(q_2) edge node {\epsilon} (q_3)
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(q_3) edge node [below left] {a} (q_1)
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(q_3) edge node {b} (S)
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(q_3) edge [loop left] node {b} ()
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;
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\end{tikzpicture}
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\end{multicols}
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\newpage
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Minimierung von Automaten
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\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
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\begin{multicols}{2}
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\node[state,initial] (S) {$S$};
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\node[state] (p) [right of=S] {$p$};
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\node[state] (q) [right of=p] {$q$};
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\node[state] (t) [below of=p] {$t$};
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\node[state,accepting] (r) [below of=q] {$r$};
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\node[state] (v) [below of=t] {$v$};
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\node[state] (u) [below of=r] {$u$};
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\subsubsection{DFS}
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\path[->]
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(S) edge [loop above] node {0} ()
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(S) edge node {1} (p)
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(p) edge [loop above] node {1} ()
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(p) edge node {0} (q)
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(q) edge [bend left] node {0} (S)
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(q) edge node {1} (r)
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(t) edge node [right] {0} (S)
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(t) edge [bend right] node {1} (r)
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(r) edge [bend right] node [above] {0} (t)
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(r) edge node {1} (u)
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(v) edge node {0} (S)
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(v) edge node[left] {1} (r)
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(u) edge node {0} (v)
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(u) edge [loop right] node {1} ()
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;
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\begin{tabular}{c || c | c}
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Kante & DFS & FIN \\
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\hline
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Vorkante & klein $\rightarrow$ groß & groß $\rightarrow$ klein \\
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Rückkante & groß $\rightarrow$ klein & klein $\rightarrow$ groß \\
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Querkante & groß $\rightarrow$ klein & groß $\rightarrow$ klein \\
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Baumkante & klein $\rightarrow$ groß & groß $\rightarrow$ klein \\
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\end{tabular}
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\subsection{Bäume}
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\subsubsection{Heap}
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Priorität eines Knotens $\geq (\leq)$ Priorität der Kinder.
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\textbf{BubbleUp}, \textbf{SinkDown}. \textbf{Build} mit \textbf{sinkDown}
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beginnend mit letztem Knoten der vorletzten Ebene weiter nach oben.
|
||||
\textbf{decPrio} entweder updaten, Eigenschaft wiederherstellen; löschen,
|
||||
mit neuer Prio einfügen oder Lazy Evaluation.
|
||||
\end{tikzpicture}
|
||||
|
||||
\subsubsection{(ab)-Baum}
|
||||
Balanciert. \textbf{find}, \textbf{insert}, \textbf{remove} in
|
||||
$\Theta(log n)$. Zu wenig Kinder: \textbf{rebalance} / \textbf{fuse}.
|
||||
Zu viele Kinder: \textbf{split}.
|
||||
\begin{tikzpicture}[initial text=,shorten >=1pt,node distance=2cm,on grid,auto]
|
||||
|
||||
Linker Teilbaum $\leq$ Schlüssel k $<$ rechter Teilbaum
|
||||
\node[state,initial] (S) {$[S]$};
|
||||
\node[state] (p) [right of=S] {$[p]$};
|
||||
\node[state] (q) [right of=p] {$[q]$};
|
||||
\node[state,accepting] (r) [right of=q] {$[r]$};
|
||||
|
||||
Unendlich-Trick, für Invarianten.
|
||||
\path[->]
|
||||
(S) edge [loop above] node {0} ()
|
||||
(S) edge node {1} (p)
|
||||
(p) edge [loop above] node {1} (p)
|
||||
(p) edge node {0} (q)
|
||||
(q) edge[bend left] node {0} (S)
|
||||
(q) edge node {1} (r)
|
||||
(r) edge[bend right] node [above] {1} (p)
|
||||
;
|
||||
|
||||
\subsection{Union-Find}
|
||||
Rang: höhe des Baums, damit ist die Höhe h mind. $2^h$ Knoten, h $\in
|
||||
\mathcal{O}(\log n)$.
|
||||
Union hängt niedrigen Baum an höherrängigen Baum. Pfadkompression hängt alle
|
||||
Knoten bei einem \textbf{find} an die Wurzel.
|
||||
\end{tikzpicture}
|
||||
|
||||
|
||||
\columnbreak
|
||||
\section{Amortisierte Analyse}
|
||||
|
||||
\subsection{Aggregation}
|
||||
Summiere die Kosten für alle Operationen. Teile Gesamtkkosten durch Anzahl
|
||||
Operationen.
|
||||
\begin{tikzpicture} [binary tree layout]
|
||||
\node[align=center] (1) {s,p,q,r,t,a,v \\ $\epsilon$ trennt}
|
||||
child {
|
||||
node {r}
|
||||
}
|
||||
child { node[align=center] {s,p,q,t,u,v \\ 1 trennt}
|
||||
child { node[align=center] {s,p,u \\ 0 trennt}
|
||||
child { node {s} }
|
||||
child { node {p,u} }
|
||||
}
|
||||
child{
|
||||
node {q,t,v}
|
||||
}
|
||||
};
|
||||
\end{tikzpicture}
|
||||
|
||||
\subsection{Charging}
|
||||
Verteile Kosten-Tokens von teuren zu günstigen Operationen (Charging). Zeige:
|
||||
jede Operation hat am Ende nur wenige Tokens.
|
||||
\subsection{Nerode-Relation}
|
||||
|
||||
\subsection{Konto}
|
||||
Günstige Operationen bezahlen mehr als sie tatsächlich kosten (ins Konto
|
||||
einzahlen). Teure Operationen bezahlen tatsächliche Kosten zum Teil mit
|
||||
Guthaben aus dem Konto. \textbf{Beachte: Konto darf nie negativ sein!}
|
||||
\subsection{Chomsky-NF}
|
||||
|
||||
\subsection{Potential (Umgekehrte Kontomethode)}
|
||||
Definiere Kontostand abhängig vom Zustand der Datenstruktur
|
||||
(Potentialfunktion)
|
||||
\section{NP-Vollständigkeit}
|
||||
|
||||
amortisierten Kosten = tatsächliche Kosten
|
||||
$+ \Phi(S_\text{nach}) -\Phi(S_\text{vor})$
|
||||
\section{Kellerautomaten}
|
||||
|
||||
\end{multicols}
|
||||
\subsection{$4COLOR \in NP$}
|
||||
|
||||
\subsection{$3COLOR \propto 4COLOR$}
|
||||
|
||||
\subsubsection{Transformation}
|
||||
\subsubsection{Äquivalenz/Korrektheit}
|
||||
|
||||
\section{Approximationsalgorithmen}
|
||||
|
||||
\section{Huffman-Kodierung}
|
||||
|
||||
\section{Pseudocode}
|
||||
\scriptsize
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
DFS(Graph G, Node v) \\
|
||||
mark v \\
|
||||
dfs[v] := dfsCounter++ \\
|
||||
low[v] := dfs[v] \\
|
||||
\For{u $\in$ N(v)}{
|
||||
\eIf{not marked u}{
|
||||
dist[u] := dist[v] + 1 \\
|
||||
par[u] := v \\
|
||||
DFS(G, u) \\
|
||||
low[v] := min(low[v], low[u]) \\
|
||||
}{low[v] := min(low[v], dfs[u])}
|
||||
}
|
||||
fin[v] := fin++ \\
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
topoSort(Graph G) \\
|
||||
fin := [$\infty$; n] \\
|
||||
curr := 0 \\
|
||||
\For{Node v in V}{
|
||||
\If{v is colored}{DFS(G,v)}
|
||||
}
|
||||
return V sorted by decreasing fin \\
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Kruskal(Graph G) \\
|
||||
U := Union-Find(G.v) \\
|
||||
PriorityQueue Q := empty \\
|
||||
\For{Edge e in E}{Q.push(e, len(e))}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
e := Q.popMin() \\
|
||||
\If{U.find(v) $\neq$ U.find(u)}{
|
||||
L.add(e) \\
|
||||
U.union(v, u) \\
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Prim(Graph G) \\
|
||||
Priority Queue Q := empty \\
|
||||
p := [0; n] \\
|
||||
\For{Node v in V}{
|
||||
Q.push(v, $\infty$) \\
|
||||
}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.popMin() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{v $\in$ Q $\wedge$ (len(u, v) $<$ Q.prio(v))}{
|
||||
p[v] = u \\
|
||||
Q.decPrio(v, len(u, v) \\
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
BFS(Graph G, Start s, Goal z) \\
|
||||
Queue Q := empty queue \\
|
||||
Q.push(s) \\
|
||||
s.layer = 0 \\
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.pop() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{v.layer = $-\infty$}{
|
||||
Q.push(v) \\
|
||||
v.layer = u.layer + 1
|
||||
}
|
||||
\If{v = z}{
|
||||
return z.layer
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Dijkstra(Graph G, Node s) \\
|
||||
d := [$\infty$; n] \\
|
||||
d[s] := 0 \\
|
||||
PriorityQueue Q := empty priority queue \\
|
||||
\For{Node v in V}{
|
||||
Q.push(v, d[v])
|
||||
}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.popMin() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
d[v] := d[u] + len(u, v) \\
|
||||
Q.decPrio(v, d[v]) \\
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
BellManFord(Graph G, Node s) \\
|
||||
d := [$\infty$, n] \\
|
||||
d[s] := 0 \\
|
||||
\For{n-1 iterations}{
|
||||
\For{(u, v) $\in$ E}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
d[v] := d[u] + len(u, v)
|
||||
}
|
||||
}
|
||||
}
|
||||
\For{(u, v) $\in$ E}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
return negative cycle
|
||||
}
|
||||
}
|
||||
return d
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
FloydWarshall(Graph G) \\
|
||||
D := [$\infty$, n $\times$ n] \\
|
||||
\For{(u, v) $\in$ E}{D[u][v] := len(u, v)}
|
||||
\For{v $\in$ V}{D[v][v] := 0}
|
||||
\For{i $\in 1,...,n$}{
|
||||
\For{(u,v) $\in V \times V$}{
|
||||
D[u][v] := min(D[u][v], D[u][$v_i$] + D[$v_i$][v]) \\
|
||||
}
|
||||
}
|
||||
return D
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\end{document}
|
||||
|
|
|
|||
Loading…
Reference in a new issue