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| - | ====== Hausaufgaben ====== | ||
| - | ===== Tag 1 ===== | ||
| - | ==== Mathematische Grundlagen, Asymptotik, Landau-Notation ==== | ||
| - | |||
| - | - Welche Rechenregeln für Logarithmen gibt es? | ||
| - | - Vereinfache die folgenden Terme: | ||
| - | * $\log_2(8)$ | ||
| - | * $\log_3(1)$ | ||
| - | * $\log_2(n^4)$ | ||
| - | * $\log_{10}\left(\frac{1}{n^2}\right)$ | ||
| - | * $2^{\log_4(n)}$ | ||
| - | * $\log_n(n)$ | ||
| - | * $\log_a(b) \cdot \log_b(n)$ | ||
| - | * $\log_2(4n^3)$ | ||
| - | * $\log_2(\sqrt{n})$ | ||
| - | * $\log_2\left( \sum_{i=0}^n i \right)$ | ||
| - | * $\log_3\left(\prod_{i=1}^n 3^i \right)$ | ||
| - | * $\sum_{i=1}^n n \cdot i$ | ||
| - | * $\sum_{k=0}^n 2^{-k}$ | ||
| - | * $\sum_{k=1}^{\frac{n}{2}} 4^{-k}$ | ||
| - | * $\sum_{i=1}^n \frac{2}{i} < | ||
| - | - Wahr oder falsch? Begründe deine Antwort. | ||
| - | * $1 = \Theta(3)$ | ||
| - | * $2 = \Omega(1)$ | ||
| - | * $3 = \omega(2)$ | ||
| - | * $n = \omega(1)$ | ||
| - | * $f = \mathcal{O}(g)$ und $g = \mathcal{O}(f) \iff f = \Theta(g)$ | ||
| - | * $\sum_{k=0}^n 2^{-k} = \mathcal{O}(1)$ | ||
| - | * $\log_2(n) = \Theta(\log_8(n))$ | ||
| - | * $2^{\log_2(2n)} = \mathcal{O}(n^2)$ | ||
| - | * $f_1 = \Omega(g_1)$ und $f_2 = \Omega(g_2) \implies f_1 + f_2 = \Omega(g_1 + g_2)$ | ||
| - | * $f = o(g) \iff f = \mathcal{O}(g)$ | ||
| - | * $f(n) = g(\frac{n}{2}) \implies f = \Omega(g)$ | ||
| - | * $f = o(g) \implies 2^f = o(2^g)$ | ||
| - | |||
| - | ==== Pseudocode und Laufzeitanalyse ==== | ||
| - | Welchen Wert hat k in Abhängigkeit von n nach der Ausführung des Programms? | ||
| - | - < | ||
| - | k = 1 | ||
| - | for i=1,2,...,n | ||
| - | k = k + 1 | ||
| - | </ | ||
| - | - < | ||
| - | k = 1 | ||
| - | for i=1,2,...,n | ||
| - | k = k + 1 | ||
| - | for j=1, | ||
| - | k = k + 2 | ||
| - | </ | ||
| - | - < | ||
| - | k = 1 | ||
| - | for i=1, | ||
| - | k = k + 2 | ||
| - | </ | ||
| - | - < | ||
| - | k = 1 | ||
| - | for i=n, | ||
| - | k = k + 1 | ||
| - | </ | ||
| - | - < | ||
| - | k = 1 | ||
| - | for i=1,2,...,n | ||
| - | k = k + i | ||
| - | </ | ||
| - | - < | ||
| - | k = 1 | ||
| - | while i < n | ||
| - | i = i + i | ||
| - | k = k + 1 | ||
| - | </ | ||
| - | - < | ||
| - | i = 0 | ||
| - | k = 0 | ||
| - | while i < n | ||
| - | i = i + k | ||
| - | k = k + 1 | ||
| - | </ | ||
| - | |||
| - | Umgekehrt: Schreibe Pseudocode mit den folgenden Laufzeiten: | ||
| - | - $\Theta(1)$ | ||
| - | - $\Theta(n)$ | ||
| - | - $\Theta(n^3)$ | ||
| - | - $\Theta(2^n)$ | ||
| - | - $\Theta(\sqrt{n})$ | ||
| - | - $\Theta(\log{n})$ | ||
| - | - $\Theta(\log{n^3})$ | ||
| - | - $\Theta(n!)$ | ||
| - | |||
| - | ==== Rekursionsgleichungen aufstellen und lösen ==== | ||
| - | Aufstellen: | ||
| - | - < | ||
| - | def f(n): | ||
| - | if n == 1 | ||
| - | return 1 | ||
| - | else | ||
| - | return n + f(n-1) | ||
| - | </ | ||
| - | - < | ||
| - | def g(n): | ||
| - | if n > 1: | ||
| - | return g(n/2) | ||
| - | </ | ||
| - | |||
| - | Lösen: | ||
| - | - $T(1) = 0$ und $T(n) = 2 T(\frac{n}{2}) + \frac{n}{2}$ für $n > 1$ | ||
| - | - $T(2) = 1$ und $T(n) = T(\frac{n}{2}) + n - 1$ | ||