From 2af4ef67e2d89b09cf7dca1bd07307aa3565888f Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 13 Nov 2019 21:34:24 +0100 Subject: [PATCH] add ipi3 --- ws2019/ipi/uebungen/binomial.cpp | 18 +++ ws2019/ipi/uebungen/binomial_fast.cpp | 25 ++++ ws2019/ipi/uebungen/ipi3.pdf | Bin 0 -> 113204 bytes ws2019/ipi/uebungen/ipi3.tex | 177 ++++++++++++++++++++++++++ ws2019/ipi/uebungen/uebung3.cpp | 14 ++ 5 files changed, 234 insertions(+) create mode 100644 ws2019/ipi/uebungen/binomial.cpp create mode 100644 ws2019/ipi/uebungen/binomial_fast.cpp create mode 100644 ws2019/ipi/uebungen/ipi3.pdf create mode 100644 ws2019/ipi/uebungen/ipi3.tex create mode 100644 ws2019/ipi/uebungen/uebung3.cpp diff --git a/ws2019/ipi/uebungen/binomial.cpp b/ws2019/ipi/uebungen/binomial.cpp new file mode 100644 index 0000000..337dcda --- /dev/null +++ b/ws2019/ipi/uebungen/binomial.cpp @@ -0,0 +1,18 @@ +#include "cpp_headers/fcpp.hh" + +// Berechnet den Binomialkoeffizienten (n ueber k) rekursiv +int binomial(int n, int k) { + return cond(k > n, + // Falls k > n: n ueber k = 0 + 0, + cond(k == 0 || n == k, + // Falls k = 0 oder n = k: n ueber k = 1 + 1, + // sonst rekursiv berechnen + binomial(n-1, k-1) + binomial(n-1, k))); +} + +int main(int argc, char **argv) { + return print(binomial(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +} diff --git a/ws2019/ipi/uebungen/binomial_fast.cpp b/ws2019/ipi/uebungen/binomial_fast.cpp new file mode 100644 index 0000000..08a5ee5 --- /dev/null +++ b/ws2019/ipi/uebungen/binomial_fast.cpp @@ -0,0 +1,25 @@ +#include "cpp_headers/fcpp.hh" + +// Helper Funktion fuer Fakultaet +// berechnet Fakultaet linear iterativ +int fakIter(int produkt, int zaehler, int ende) { + return cond(zaehler>ende, + produkt, + fakIter(produkt*zaehler,zaehler+1,ende)); +} + +// Berechnet die Fakulaet einer Zahl +int fakultaet(int n) { + return fakIter(1,1,n); +} + +// Berechnet den Binomial Koeffizienten (n ueber k) durch seine explizite +// Darstellung +int binomial_fast(int n, int k) { + return fakultaet(n)/(fakultaet(k) * fakultaet(n-k)); +} + +int main(int argc, char **argv) { + return print(binomial_fast(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +} diff --git a/ws2019/ipi/uebungen/ipi3.pdf b/ws2019/ipi/uebungen/ipi3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..29202baac2840a9991b1e8d4bf04c0b1ad79402a GIT binary patch literal 113204 zcmbrlWmFv6)~1U~VTHQ~3s!}@ySuvw5AIF^1b24`9!NrPcXti$5FofqvyXgzZ*TgZ zoN?<1qsHLRT5nBxrYtHYF$qR66FY!ve7a{Iz)A)pb1<<6@bLkd<;?9ZU98AhI9RyI z{_z1YOT4pnF?S+kmasK;F&8s8bucpr2nYb2U7XB~?EoH&N4mNWn;bZPV_(J#g2TD_ znpr>5jYILmE;6~*Vde^1-WkzEn@+!9oL@k_`*p1pZ?2M0JD@?$+TWB1g!Fa$D$-&o(qc9H(GTvX5Z^2aeSk%ju|~~o?%MLVf8GL$--W&eVjcB8rJ+X- zqCMi=jZ;<&@4dTt>C?l`!QrJYHA~^=)WIrySyUa_&iei;`>)ptUvlhIzTMs?ej9*# zCJ=yhdK#K>$4;7a5cYagNF_q>P*}Z>%HlVm=Qe7j;onTywuOnqp!bK(z#z_Rb!ycbsV|=Vy3Jx8w`kCL@)~7> zf}t+Yvb_5^S!^#FYw^JWSUPN>3h_O9QPj36(=Pd^YeN*Qn2$b`HY26nXS2&n4WrB7 zn%RWX;`NDXJ6LUbs(#&tWWyzcap`RBKxjU3XiZ*qb7_j}1|i}egxx(C4; zKJOib>8E_h?*>my5J3vogm0-MT72tnF5fw;-l)q%p@%_uP2w; zQx-E40rr^bhbZ84DJ8BWpyvbC-(f6+J_8DB_y;h&QwvPg`a6NYGN`3c)&+)5I{
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zi34907JbD#Y)Uj(M+?i#J}Qa|?fT8M5bxXPr67apC_Js^<T=IN61tG|Yq>tn&GlB?PA6jBq4$~kLoVeG%0G!p539eE-;jwhhUg^a9 zDDQ04A#9+<5+2jKE$7fFDRjX1G_-cks!HL}T0ounn<_$PvP1u=svX2=$f@OKegVA} zq(DW`FH#a5v6TeSdB5~|b;ilZIvn0gNS5z_Ft;(77&28PnoZ`;p-pXC3{ao-zkgUA z3G26G;{n>}WLeJm~w qN>5cP)clh9fN(!3=LqnIRpY#fx>uYWUk*r@%{}6k^S`m literal 0 HcmV?d00001 diff --git a/ws2019/ipi/uebungen/ipi3.tex b/ws2019/ipi/uebungen/ipi3.tex new file mode 100644 index 0000000..5baf18b --- /dev/null +++ b/ws2019/ipi/uebungen/ipi3.tex @@ -0,0 +1,177 @@ +\documentclass{../../../lecture} + +\usepackage{enumerate} + +\begin{document} + +\begin{aufgabe} Algorithmische Komplexität +\begin{enumerate}[a)] + \item Laufzeiten der verschiedenen algorithmischen Komplexitäten $f(n)$ für $2n$ in Abhängigkeit von + der Ausgangslaufzeit für $n$. + + \begin{tabular}{|l|l|l|l|} + \hline + $f(n)$ & 4 Sekunden & 10 Sekunden & 100 Sekunden \\ \hline + $\text{ld}(2n)$ & 5s & 11s & 101s \\ \hline + $2n$ & 8s & 20s & 200s \\ \hline + $2n \text{ ld}(2n)$ & 13,49s & 29,13s & 244,64s\\ \hline + $(2n)^{3}$ & 32s & 80s & 800s \\ \hline + $2^{2n}$ & 16s & 100s & 10000s\\ \hline + \end{tabular} + + \item + \begin{enumerate}[1.] + \item $1$ + \item $\log \log n$ + \item $\log n$ + \item $n^{\epsilon}$ + \item $n^{c}$ + \item $n^{\log n}$ + \item $c^{n}$ + \item $n^{n}$ + \item $c^{\left( c^{n} \right) }$ + \end{enumerate} + + \item Beweisen Sie folgende Behauptungen + \begin{enumerate} + \item $x^{a} = O(x^{b}) \iff a -b \le 0$ + \begin{proof} + Seien $a, b, x \in \R$ mit $x > 1 $ + + Zu zeigen: $\exists c \in \R$: $x^{a} \le c x^{b} \iff a \le b$ + + Wegen $x > 1$ ist $\ln(x) > 0$ und $\ln(x)$ streng monoton steigend, wähle $c \ge 1$, dann + folgt: + \begin{align*} + &a \le b \\ + \iff &\ln(x) \cdot a \le \ln(x) \cdot b \le \ln(c) + \ln(x) \cdot b \\ + \iff &\ln(x^{a}) \le \ln(c\cdot b^{x})\\ + \iff & x^{a} \le c x^{b} + .\end{align*} + \end{proof} + \item $\log_a(x) = \Theta(\log_b(x))$ $\forall a, b \in \R^{+}$ + \begin{proof} + Seien $a, b, x \in \R^{+}$. + + Zu zeigen: $\exists c \in \R$: $\log_a(x) = c \cdot \log_b(x)$. + \begin{align*} + &\log_a(x) = \log_a(x) \\ + \implies & \log_a(x) = \log_a(b) \cdot \frac{\log_a(x)}{\log_a(b)} \\ + \implies & \log_a(x) = \log_a(b) \cdot \log_b(x) + .\end{align*} + Mit $c := \log_a(b)$ folgt damit: + \[ + \log_a(x) = c \cdot \log_b(x) + .\] + \end{proof} + \item $a^{x} = O(b^{x}) \iff 0 \le a \le b$ + \begin{proof} + Seien $a, b, x \in \R$ mit $x \ge 0$ + + Zu zeigen: $\exists c \in \R$: $a^{x} \le c b^{x} \iff 0 \le a \le b$ + \begin{align*} + &0 \le a \le b \\ + \iff &a^{x} \le b^{x} \le b^{x+1} = b \cdot b^{x} + .\end{align*} + Mit $c := b$ folgt damit: + \[ + a^{x} \le c \cdot b^{x} \iff 0 \le a \le b + .\] + \end{proof} + \end{enumerate} +\end{enumerate} + +\end{aufgabe} + +\begin{aufgabe} Klassischer Euklidischer Algorithmus + + Sei $a, b \in \N_0, a+b > 0$ gegeben. + \[ + \text{ggT}(a, b) = \begin{cases} + a & b =0 \\ + \text{ggT}(b,a) & a < b \\ + \text{ggT}(a - b, b) & a \ge b \\ + \end{cases} + .\] + + \begin{enumerate} + \item Für $a \ge b > 0$ gilt: + \[ + \text{ggT}(a,b) = \text{ggT}(a -b, b) + .\] + \begin{proof} + Wegen $b \neq 0$ und $a \ge b$ folgt nach Definition: + \[ + \text{ggT}(a, b) = \text{ggT}(a -b, b) + .\] + \end{proof} + \item Der klassische Euklidische Algorithmus terminiert. + \begin{proof} + Seien $(a_n)_{n\in\N} \in \N_0$ und $(b_n)_{n\in\N} \in \N_0$ Folgen mit $a_1 = a$ und + $b_1 = b$. + + Sei $n \in \N$ beliebig. + + \begin{itemize} + \item Falls $b_n = 0$ terminiert der Algorithmus direkt. + \item + Falls $a_n < b_n$, folgt nach Definition: + \[ + a_{n+1} = b_n \text{ und } b_{n+1} = a_n < b_n + .\] Damit folgt: + \[ + b_{n+1} < b_n + .\] + \item Falls $a_n \ge b_n$ folgt nach Definition: + \[ + a_{n+1} = a_n - b_n \text{ und } b_{n+1} = b_n + .\] Wegen $a_{n+1} < a_n $ folgt, dass $\exists k \in \N$: $a_{n+k} < b_n$. Dann + tritt wieder der zweite Fall ein, d.h. + \[ + b_{n+k+1} < b_n + .\] + \end{itemize} + + Damit folgt, dass $(b_n)_{n\in\N}$ für fast alle $n \in \N$ streng monoton fällt. + Da $(a_n - b_n) \in \N$, folgt: + \[ + \exists k \in \N\text{: } b_k = 0 + .\] + Damit terminiert der \textit{klassische Euklidische Algorithmus} immer. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Binomialkoeffizient + \begin{enumerate}[a)] + \item Programm siehe \textit{binomial.cc} + + Für $n = 35$ und $k = 18$ benötigt das Programm mehr als 20 Sekunden für die Berechnung. + + Für $n = 34$ und $k = 18$ liefert das Programm $-2091005866$. Das liegt an der + begrenzten Größe des Datentyps \textbf{int}. Bei Überschreitung der maximalen Größe + beginnt der Wert erneut bei dem Minimalwert des Datentyps \textbf{int}. Deshalb können + wir dann negative Ergebnisse erhalten. + \item Der Rechenaufwand für die rekursive Berechnung des Binomialkoeffizienten ist: + \begin{align*} + A_{n, 0} = A_{n, n} = A_{n, k>n} = 1 \\ + A_{n,k} = A_{n-1, k-1} + A_{n-1, k} + .\end{align*} + \item Programm siehe \textit{binomial\_fast.cc} + + Die schnellere Variante hat die Komplexität $O(n)$, da die Komplexität der Fakultätsfunktion + $O(n)$ ist und diese einfach dreimal ausgeführt wird. + + Programm (c) ist deutlich schneller als Programm (a) für größere Zahlen $n$ und $k$. Aufgrund + der Verwendung der Fakultät, wird allerdings schneller die maximale Größe eines \textbf{int}'s + erreicht. Dadurch erhalten wir bereits für recht kleine Werte für $n$ und $k$ falsche Ergebnisse. + \item Ein effizienter Algorithmus berechnet jede Zeile linear iterativ, aus der vorhergehenden + Zeile, damit ist die Komplexität $O(n)$. + + Der Unterschied entsteht daraus, dass bei (a) einzelne Binomialkoeffizienten mehrfach + ausgerechnet werden müssen und bei (d) jeder Binomialkoeffizient genau einmal ausgerechnet wird. + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2019/ipi/uebungen/uebung3.cpp b/ws2019/ipi/uebungen/uebung3.cpp new file mode 100644 index 0000000..d1e6e05 --- /dev/null +++ b/ws2019/ipi/uebungen/uebung3.cpp @@ -0,0 +1,14 @@ +#include "cpp_headers/fcpp.hh" + +int binomial(int n, int k) { + return cond(k > n, + 0, + cond(k == 0 || n == k, + 1, + binomial(n-1, k-1) + binomial(n-1, k))); +} + +int main(int argc, char **argv) { + return print(binomial(readarg_int(argc, argv, 1), + readarg_int(argc, argv, 2))); +}