first commit

algo.org

@@ -0,0 +1,51 @@
+#+TITLE: algo
+
+* Orga
++ Klausur Fr 07.02.2020 14-16 Uhr
++ Nachklausur: Mo 06.04.2020 14-16 Uhr
+* 18.10.2019 2.VL
+
+** Merge Sort: \(T(n) = 2 T(\fraq{n}{2}) + n\)
+_Beh:_ T(n) = 0(n * log(n))
+_Bew:_ \( T(2) = 1 \le O(2n*log\,2) = o(1)\)
+_n/2 -> n:__ \(T(n) \le 2 * c * \fraq{n}{2} * log(\fraq{n}{2}) + n)\)
+           \(T(n) \le  c * n * log(\fraq{n}{2}) + n)\)
+           \(\le c * n * (log n - log 2) + n\
+         \(< c * n * log n - c * n + n \leq c * n * log(n)\)
+
+** Binäre Suche
+_Beh:_ T(n) = O(log n)
+_Bew:_ I.A:
+\(\begin{equation}
+T(2) = 2 = O(1) \\
+\fraq{n}{2} \rightarrow n \colon T(n) &\le c * log (\fraq{n}{2}) + 1\\
+&= c * (log n - log 2) + 1\\
+&= c * log n - c +1\\
+&\leq c *  log n
+\end{equation}\) für c \geq 1
+** Mastertheorem:
+log_{b}a = log_{2} 2 = _1_
+f(n) = n = n^{_1_} \Rightarrow T(n) = O(n^{1 * log_{2*}^{}n)
+2. T(n) = 9 * T (\fraq{n}{3} * + n^{2}
+   \rightarrow log_{b}a = log_{3}9 = 2
+   f(n) = n^{2} = n^{log_{3}9} = 2
+   f(n) = n² = n^{log_{3}9 \Rightarrow T(n) = O(n² * log_{3}n) = O(n² log(n)
+   f(n) = n = n^{2-1}
+   \Rightarrow T(n) = O(n²)
+
+** Bsp:
+
+a) \(T(n) = 4 * T (n/2) + n\\
+    log_{2} 4 = 2\,\,f(n) = n \le n^{2-\epsilon}
+    \rigtharrow T(n) = O(n^{2})\) für \epsilon = 1/2
+b) \( T(n) (\fraq{n}{2}) + n^{3/2} \le Tn^{2- \epsilion} \Rightarrow t(n) = O(n^{2}) \)
+c) \( T(n) (\fraq{n}{2}) + n^{2}  f(n) = n^{2} = n^{log_{b}a} \Rightarrow T(n) =
+O(n^{2} * log n\)
+d) \( T(n) (\fraq{n}{2}) + n^{3} f(n) = n^{3} > n^{2+ \epsilon }\Rightarrow T(n) = o (f(n))
+= O(n^{3})  \)
+e) \(T(n) = 8 * T(\frqa{n}{2} + n^{1} \rightarrow [1.] O(n^{3})\)
+f) \(T(n) = 8 * T(\frqa{n}{2} + n^{³} \rightarrow[2.] O(n^{3} * log n)\)
+
+T(n) = 2 * T(n/2) + n^{1 + \epsilon} , \epsilon > 0 \Rightarrow [3.] T(n) = O(n^{1+\epsilon})
+T(n) = 2 * T(n/2) + _n log n_ < n^{1}+\epsilon \forall \epsilon > 0 \Rightarrow M.T.
+nicht anwendbar