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 jpfender [HA]: C&J Übung 3: Aufgabe 1a 187d33c jpfender [HA]: C&J Übung 3: Aufgabe 2: Partial, only those solvable with MT 06a31d5
 @@ -59,8 +59,64 @@ \maketitle \section{Schnelle Matrizenmultiplikation} +\begin{enumerate} +\item Da bei der Multiplikation zweier $n \times n$~-Matrizen jedes + Element der ersten mit $n$ Elementen der zweiten Matrix + multipliziert werden muss, werden insgesamt $n^2 \cdot n = n^3$ + Multiplikationen sowie $n \cdot n \cdot n-1 = (n-1)n^2$ + Additionen durchgeführt. + +\item +\end{enumerate} \section{Rekursionsgleichungen} +\begin{enumerate} +\item \begin{align} + T(n) = T(9n/10)+n \\ + a = 1, b = 10, f(n) = n \\ + \log _b a = log _{10} 1 = 0 \\ + \Rightarrow f(n) = n = \Theta(n^{\log _b a}) \\ + \Rightarrow T(n) = \Theta(\log n) \tag{Fall 2 des Master-Theorems} + \end{align} +\item \begin{align} + T(n) = T(n-a) + T(a) + n, a \geq 1 \\ + \end{align} + MT nicht anwendbar. +\item \begin{align} + T(n) = T(\sqrt{n}) + 1 \\ + \end{align} +\item \begin{align} + T(n) = 2T(n/4) + \sqrt{n} \\ + a = 2, b = 4, f(n) = \sqrt{n} \\ + \log _b a = \log _4 2 = \frac{1}{2} \\ + \Rightarrow \exists \varepsilon > 0, f(n) = \sqrt{n} = O(n^{\log + _b a+\varepsilon}), \exists c < 1, 2f(n/4) \leq cf(n) \\ + \Rightarrow T(n) = \Theta(f(n)) \tag{Fall 3 des Master-Theorems} + \end{align} +\item \begin{align} + T(n) = 7T(n/3) + n^2 \\ + a = 7, b = 3, f(n) = n^2 \\ + \log _b a = \log _3 7 \approx 1.77 \\ + \Rightarrow \exists \varepsilon > 0, f(n) = n^2 = O(n^{\log _b a-\varepsilon}) \\ + \Rightarrow T(n) = \Theta(n^{log _3 7}) \tag{Fall 1 des Master-Theorems} + \end{align} +\item \begin{align} + T(n) = 2T(n/2) + n \log n \\ + a = 2, b = 2, f(n) = n \log n \\ + \log _b a = \log _2 2 = 1 + \end{align} + MT nicht anwendbar, da $\nexists \varepsilon > 0, f(n) = n \log + n = \Omega(n^{\log _b a+\varepsilon}), \nexists \varepsilon > 0, + f(n) = n \log n = O(n^{\log _b a-\varepsilon}), f(n) \neq + \Theta(n^{\log _b a})$ +\item \begin{align} + T(n) = T(n-1) + \frac{1}{n} \\ + a = 1, b = 1, f(n) = \frac{1}{n} \\ + \log _b a = \log _1 1 = 1 \\ + \Rightarrow f(n) = \frac{1}{n} = \Theta(n^{\log _b a}) \\ + \Rightarrow T(n) = \Theta(n \log n) \tag{Fall 2 des Master-Theorems} + \end{align} +\end{enumerate} \section{Implementierung}