From 09337ff9bc0f50ab6b048348adccf8887cbeb965 Mon Sep 17 00:00:00 2001
From: Thomas Ba <git@thomasba.de>
Date: Wed, 11 Jan 2012 11:46:35 +0100
Subject: [PATCH] Update

---
 .../Berechenbarkeits-KomplexTh.tex            |  70 ++++
 Berechenbarkeits-KomplexTh/Makefile           |   2 +-
 .../bilder/approx_bin.xml                     | 286 +++++++++++++++++
 .../bilder/approx_bin_1.xml                   | 270 ++++++++++++++++
 .../bilder/approx_bin_2.xml                   | 297 +++++++++++++++++
 .../bilder/approx_bin_3.xml                   | 298 ++++++++++++++++++
 Berechenbarkeits-KomplexTh/bilder/p-np.xml    | 268 ++++++++++++++++
 7 files changed, 1490 insertions(+), 1 deletion(-)
 create mode 100644 Berechenbarkeits-KomplexTh/bilder/approx_bin.xml
 create mode 100644 Berechenbarkeits-KomplexTh/bilder/approx_bin_1.xml
 create mode 100644 Berechenbarkeits-KomplexTh/bilder/approx_bin_2.xml
 create mode 100644 Berechenbarkeits-KomplexTh/bilder/approx_bin_3.xml
 create mode 100644 Berechenbarkeits-KomplexTh/bilder/p-np.xml

diff --git a/Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex b/Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex
index aafbfcd..5797351 100644
--- a/Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex
+++ b/Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex
@@ -1327,4 +1327,74 @@
 	Konstruiere Eingabe für Bin Packing:\\
 		\( a_1,…,a_n,  B= \frac A2, k=2 \) mit \( A = \sum\limits_{i=1}^n a_i \)
 
+	\includegraphics{bilder/p-np.eps}
+
+	\section{Approximationsalgorithmen}
+
+	\subsection{Bin Packing}
+
+	geg. $a_1,a_2,…,a_n$
+
+	\includegraphics{bilder/approx_bin.eps}
+
+
+	\underline{1. Strategie}
+
+	\begin{itemize}
+		\item sortiere Eingabe aufsteigend zu \( a_1 \le a_2 \le .. \le a_n \)
+		\item fülle bin $i$ so weit wie möglich, für $i=1,2,...$ mit Gegenständen in sortierter Reihenfolge\\
+			\includegraphics{bilder/approx_bin_1.eps}
+	\end{itemize}
+
+	\underline{2. Strategie}
+
+	\begin{itemize}
+		\item sortiere Eingabe absteigend zu \( a_1\ge a_2\ge ... \ge a_n \)
+		\item setze $a_i$ in den ersten bin, in den es noch hinein passt, für $i=1,…,n$.
+	\end{itemize}
+
+	\includegraphics{bilder/approx_bin_2.eps}
+
+
+	\underline{Strategie 3: First Fit (FF)}
+
+	setze $a_i$ in den ersten bin, in den es noch hinein passt, für $i=1,2,…,n$
+
+	Betrachtung zu FF:\\
+	Sei $k^*$ die Anzahl der bins bei einer optimalen Lösung und $k$ die Anzahl der bins, die FF auf $a_1,...,a_n$ benützt.
+
+	Es gilt: $ \sum\limits_{i=1}^n a_i \le k^* $
+
+	Für FF gilt: alle bins, bis auf evtl. den letzten, haben Füllhöhe \(>\frac12\)
+
+	\includegraphics{bilder/approx_bin_3.eps}
+
+	\( \Rightarrow 2 \sum\limits_{i=1}^n a_i > k \)\\
+	= \# bins, wenn alle genau $\frac12$ voll sind.
+
+	Folglich gilt:\\
+	\( k < 2 \sum\limits_{i=1}^n a_i \le 2k^* \)\\
+	\( \Rightarrow k < 2 k^* \)
+
+	Man kann sogar zeigen: $l\le \frac{17}{10} k^* $\\
+	Für FFD gilt sogar: $ k \le \frac{11}{9} k^* $
+
+	\subsection{Multi Processor Scheduling (MPS)}
+
+	geg. $m$ Prozessoren, $n$ Jobs\\
+	mit Laufzeiten $t_1,t_2,…,t_n$ und deadline D.
+
+	gefr. kann man die Jobs so auf die Prozessoren verteilen, dass alle Jobs in Zeit D fertig sind.
+
+	MPS ist NP-vollständig:\\
+	BinPacking $\le^P$ MPS\\
+	\( a_1,…,a_n,k,B \to t_i = a_i, i=1,…,n, m = k, D=B \)
+
+	\underline{Greedy-Strategie:}
+
+	\begin{itemize}
+		\item Ordne Job $i$ dem Prozessor zu, der aktuell die kleinste Ladung hat.
+	\end{itemize}
+
+	Sei $T$ die maximale Laufzeit, 
 \end{document}
diff --git a/Berechenbarkeits-KomplexTh/Makefile b/Berechenbarkeits-KomplexTh/Makefile
index 0429315..0474773 100644
--- a/Berechenbarkeits-KomplexTh/Makefile
+++ b/Berechenbarkeits-KomplexTh/Makefile
@@ -1,3 +1,3 @@
 Berechenbarkeits-KomplexTh.pdf: Berechenbarkeits-KomplexTh.tex bilder/*.xml
-	for FILE in $(find bilder/ -iname '*.xml') ; do ipetoipe -eps -runlatex $FILE ${FILE%.xml}.eps ; done
+	for FILE in $(find bilder/ -iname '*.xml') ; do ipetoipe -eps -runlatex ${FILE} ${FILE%.xml}.eps ; done
 	pdflatex Berechenbarkeits-KomplexTh.tex
diff --git a/Berechenbarkeits-KomplexTh/bilder/approx_bin.xml b/Berechenbarkeits-KomplexTh/bilder/approx_bin.xml
new file mode 100644
index 0000000..901b074
--- /dev/null
+++ b/Berechenbarkeits-KomplexTh/bilder/approx_bin.xml
@@ -0,0 +1,286 @@
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diff --git a/Berechenbarkeits-KomplexTh/bilder/approx_bin_1.xml b/Berechenbarkeits-KomplexTh/bilder/approx_bin_1.xml
new file mode 100644
index 0000000..40fbae1
--- /dev/null
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new file mode 100644
index 0000000..5be5a2e
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diff --git a/Berechenbarkeits-KomplexTh/bilder/approx_bin_3.xml b/Berechenbarkeits-KomplexTh/bilder/approx_bin_3.xml
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index 0000000..7d451fb
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new file mode 100644
index 0000000..dfebc5a
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