#+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