430 lines
13 KiB
TeX
430 lines
13 KiB
TeX
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\documentclass[11pt, a4paper, twoside]{article}
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\usepackage[
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a4paper,
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headsep=5mm,
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footskip=0mm,
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top=12mm,
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left=10mm,
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right=10mm,
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bottom=10mm
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]{geometry}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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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{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=blue,
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filecolor=magenta,
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urlcolor=cyan,
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pdftitle={Overleaf Example},
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pdfpagemode=FullScreen,
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}
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\setlength{\algomargin}{0pt}
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\begin{document}
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\pagestyle{fancy}
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\fancyhead{}
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\fancyhead[L]{Theoretische Grundlagen der Informatik}
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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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\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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$f(n) \in \Omega(g(n))$ &
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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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\( f(n) \in \Theta(g(n)) \) &
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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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\( f(n) \in \mathcal{O}(g(n)) \) &
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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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\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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\end{tabular}
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\begin{multicols}{3}
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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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\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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\section{Datenstrukturen}
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\subsection{Listen}
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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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\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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\end{multicols}
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\newpage
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\begin{multicols}{2}
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\subsubsection{DFS}
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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.
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\textbf{decPrio} entweder updaten, Eigenschaft wiederherstellen; löschen,
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mit neuer Prio einfügen oder Lazy Evaluation.
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\subsubsection{(ab)-Baum}
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Balanciert. \textbf{find}, \textbf{insert}, \textbf{remove} in
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$\Theta(log n)$. Zu wenig Kinder: \textbf{rebalance} / \textbf{fuse}.
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Zu viele Kinder: \textbf{split}.
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Linker Teilbaum $\leq$ Schlüssel k $<$ rechter Teilbaum
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Unendlich-Trick, für Invarianten.
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\subsection{Union-Find}
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Rang: höhe des Baums, damit ist die Höhe h mind. $2^h$ Knoten, h $\in
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\mathcal{O}(\log n)$.
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Union hängt niedrigen Baum an höherrängigen Baum. Pfadkompression hängt alle
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Knoten bei einem \textbf{find} an die Wurzel.
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\columnbreak
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\section{Amortisierte Analyse}
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\subsection{Aggregation}
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Summiere die Kosten für alle Operationen. Teile Gesamtkkosten durch Anzahl
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Operationen.
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\subsection{Charging}
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Verteile Kosten-Tokens von teuren zu günstigen Operationen (Charging). Zeige:
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jede Operation hat am Ende nur wenige Tokens.
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\subsection{Konto}
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Günstige Operationen bezahlen mehr als sie tatsächlich kosten (ins Konto
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einzahlen). Teure Operationen bezahlen tatsächliche Kosten zum Teil mit
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Guthaben aus dem Konto. \textbf{Beachte: Konto darf nie negativ sein!}
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\subsection{Potential (Umgekehrte Kontomethode)}
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Definiere Kontostand abhängig vom Zustand der Datenstruktur
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(Potentialfunktion)
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amortisierten Kosten = tatsächliche Kosten
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$+ \Phi(S_\text{nach}) -\Phi(S_\text{vor})$
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\end{multicols}
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\section{Pseudocode}
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\scriptsize
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\begin{minipage}{.25\linewidth}
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\begin{algorithm}[H]
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DFS(Graph G, Node v) \\
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mark v \\
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dfs[v] := dfsCounter++ \\
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low[v] := dfs[v] \\
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\For{u $\in$ N(v)}{
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\eIf{not marked u}{
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dist[u] := dist[v] + 1 \\
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par[u] := v \\
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DFS(G, u) \\
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low[v] := min(low[v], low[u]) \\
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}{low[v] := min(low[v], dfs[u])}
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}
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fin[v] := fin++ \\
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\end{algorithm}
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\end{minipage}
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\begin{minipage}{.25\linewidth}
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\begin{algorithm}[H]
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topoSort(Graph G) \\
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fin := [$\infty$; n] \\
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curr := 0 \\
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\For{Node v in V}{
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\If{v is colored}{DFS(G,v)}
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}
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return V sorted by decreasing fin \\
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\end{algorithm}
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\end{minipage}
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\begin{minipage}{.25\linewidth}
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\begin{algorithm}[H]
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Kruskal(Graph G) \\
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U := Union-Find(G.v) \\
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PriorityQueue Q := empty \\
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\For{Edge e in E}{Q.push(e, len(e))}
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\While{Q $\neq \emptyset$}{
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e := Q.popMin() \\
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\If{U.find(v) $\neq$ U.find(u)}{
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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}
|