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;$%9be;1e1~uF3x2>)s;SQAk7}c`f1||uk1e&B##W^)#oAQlo5oJ1vcY6~ zmFk}+TU2WQn(UxbmucPAm}448m8K?>om5&pOdg=p8g8;Px}rOJqBr{DAPm64Ds4&D zx3rgA?I^O^8EEy@H>+<$&B;M3UH?oTqVl!>zM;Lj*yLc9AKfNLsQm1oDooY>m&wtp zCzk#jMXI`GnH<-f>rIYPjczn~m}-oR$?>Ww&An4;dalTS6eP7wmuYJO91J)zGB!C1 JB_%~qMhX!L@}B?z diff --git a/DiMa/dima.tex b/DiMa/dima.tex index ef23a14..c5212a7 100644 --- a/DiMa/dima.tex +++ b/DiMa/dima.tex @@ -147,6 +147,30 @@ Jeder ist dazu aufgerufen, sich an der Entwicklung zu beteiligen! Also ist $f(n) = \Theta$, wenn $\exists c_1, c_2$, so dass $c_1 \cdot |g(n)| \le |f(n)| \le |c_2 \cdot g(n)|$. + \subsection{Master-Theorem} + Das \idx{Master-Theorem} lässt sich auf Rekursionen der folgenden Struktur + anwenden: + \begin{alignat*}{2} + T(n) &= a \cdot T\left(\ceil{\rfrac{n}{b}}\right) + c(n) \; &&, n > n_0\\ + T(n) &= \Theta(1) &&, n \le n_0 + \end{alignat*} + $c$ muss dabei monoton wachsend sein. Nun gilt: + \begin{enumerate} + \item Ist $c(n) = \O\left(n^{\log_b a - \epsilon}\right)$ für ein + $\epsilon $ mit $ 0 < \epsilon < \log_b a$, so ist + \[T(n) = \O\left(n^{\log_b a}\right)\] + \item Ist $c(n) = \Theta(n^{\log_b a})$, so ist + \[T(n) = \Theta\left(n^{\log_b a} \cdot \log_b n\right)\] + \item Erfüllt $c$ die Bedingung $\exists \gamma \in (0, 1)$, so dass + $ac(\ceil{\frac{a}{b}}) \le \gamma c(n)$ für ein hinreichend großes + $n$ und ist $c(n) = \Omega\left(n^{\log_b a + \epsilon}\right)$, so + ist + \[T(n) = \Theta\left(c\left(n\right)\right)\] + \end{enumerate} + Zahlreiche Übungen lassen sich im Internet + finden\footnote{\url{http://www.csd.uwo.ca/~moreno/CS433-CS9624/Resources/master.pdf}}. + + \section{Graphentheorie} \subsection{Ungerichtete Graphen} \subsubsection{Kreis}\label{graph:kreis} diff --git a/env/commands.tex b/env/commands.tex index 00e4216..9a61d36 100644 --- a/env/commands.tex +++ b/env/commands.tex @@ -6,6 +6,8 @@ \newcommand{\K}{\mathbb{K}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} +\renewcommand*{\O}{\mathcal{O}} +\renewcommand*{\epsilon}{\varepsilon} \newcommand{\Rq}{\R^2} \newcommand{\Qq}{\Q^2}