Berechnbarkeit: Subset Sum...

Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex

@@ -3,6 +3,7 @@
 \usepackage[ngerman]{babel}
 \usepackage{amsmath}
 \usepackage{amssymb}
+\usepackage{marvosym} % Lightning
 \usepackage{multicol}
 \usepackage{framed}
 \usepackage{enumerate}
@@ -18,7 +19,7 @@
 	pdfborder={0 0 0}
 ]{hyperref}
 \usepackage{tabularx}
-%\usepackage{graphicx}
+\usepackage{graphicx}
 \usepackage[usenames,dvipsnames]{color}
 \usepackage{lastpage}
 \usepackage{fancyhdr}
@@ -1190,4 +1191,80 @@
 	Gefragt: \( \exists D \subseteq V, \left|D\right|\le k\)\\
 	so dass für jeden Knoten $v\in V$ gilt: $v$ oder ein Nachbar von $v$ ist in $D$.
 
+	\( VC \le^P DS:\)\\
+	Sei $G,k$ eine Eingabe für $VC$.\\
+	Ziel: konstruiere $G'$, so dass $(G,K)\in VC \Leftrightarrow (G',k) \in DS$
+
+	\includegraphics{bilder/ds-vc.eps}
+
+	Für jede Kante $(u,v)$ in $G$ führe neuen Knoten $K_{(u,v)}$ ein und Kanten $(u,K_{(u,v)})$ und $(v,K_{(u,v)})$ (ungerichtet).
+
+	\('\Rightarrow'\) Sei $C$ $VC$ in $G$, d.\,h. für jede Kante $(u,v) \in E$ ist $u\in C$ oder $v\in C$.
+
+	\(\Rightarrow\) der neue Knoten $K_{u,v}$ hat einen Nachbarn in $C$\\
+	\(\Rightarrow\) $C$ ist d.s. in $G'$.\\
+	\('\Leftarrow'\) Sei $D$ d.s. in $G'$\\
+	\includegraphics{bilder/ds-vc2.eps}
+	Enthält $D$ neue Knoten $K_{u,v}$ dann ersetze $K_{u,v}$ durch $u$ in $D$. Danach entsteht $D'$. Dann ist $D'$ (immer noch d.s.).\\
+	Dann gilt \( \left|D'\right| \le k\) und $D'$ ist V.C. in $G$:\\
+	wäre für Knate $(u,v)$ weder $u\in D'$ noch $v\in D'$, dann hätte $K_{u,v}$ keinen Nachbarn in $D'$.\\
+	Dann wäre $D'$ kein D.S. \Lightning
+
+	\underline{Subset Sum}
+	
+	Gegeben: \(a_1,a_2,…,a_n,b\).\\
+	Gefragt: \( \exists I \le \{ 1,2,…,n\} \quad \sum\limits_{i\in I} a_i - b \)
+
+	\bsp \(1,2,5,7,9,\underbrace{14}_{=b} \in \) Subset Sum\\
+	\( 4 \not\in \) Subset Sum
+
+	\( \in NP:\) rate $I$, teste Summe.\\
+	Subset Sum ist $NP$-Vollständig.\\
+	$3-SAT \le^P$ Subset Sum:
+
+	Sei \( F(x_1,…x_n) = C_1 \land … C_m \) .\\
+	\( b = \underbrace{33…3}_{m}{\color{Orange}|} \underbrace{11…1}_{n} \) dezimal.
+
+	\bsp \( F = (x_1 \lor \overline{x_2} \lor x_3) \land (\overline{x_1} \lor x_2 \lor x_4) \land (\overline{x_2} \lor \overline{x_4} \lor x_5) \)\\
+	Zahlen \( y_1, …, y_n \) für \( x_1,…,x_n \)\\
+	\( y_1 = \overbrace{100}^{m=3} {\color{Orange}|} \overbrace{1000}^{n=5} \)\\
+	$x_1$ kommt in $C_1$ vor\\
+	$X_1$ kommt nicht in $C_2,C_3$ vor.
+
+	\( y_2 = 010 {\color{Orange}|} 01000 \)\\
+	\( y_3 = 100 {\color{Orange}|} 00100 \)\\
+	\( y_4 = 010 {\color{Orange}|} 00010 \)\\
+	\( y_5 = 001 {\color{Orange}|} 00001 \)
+
+	Analog \( Z_1,…,Z_n \) für \( \overline{x_1},…,\overline{x_n}\)\\
+	\( Z_1 = 010 {\color{Orange}|} 10000 \)\\
+	\( Z_2 = 101 {\color{Orange}|} 01000 \)\\
+	\( Z_3 = 000 {\color{Orange}|} 00100 \)\\
+	\( Z_4 = 001 {\color{Orange}|} 00010 \)\\
+	\( Z_5 = 000 {\color{Orange}|} 00001 \)
+
+	Füllzahlen \( f_1,…f_m, g_1,…g_m \)\\
+	\( f_1 = g_1 = 100 {\color{Orange}|} 00000 \)\\
+	\( f_2 = g_2 = 010 {\color{Orange}|} 00000 \)\\
+	\( f_3 = g_3 = 001 {\color{Orange}|} 00000 \)
+
+	Es gibt nie Überträge, da im vorderen Teil kann man höchstens auf 3 Einsen kommen pro Klauselspalte mit $y-$ und $z-$Zahlen + max. 2 Einsen mit $f-$ und $g-$Zahlen.\\
+	\(\Rightarrow\) Summe pro Klauselspalte \(\le 5\)\\
+	Im hinteren Teil höchstens 2 Einsen pro Spalte.
+
+	\('\Rightarrow'\) Sei \( a = (a_1,…,a_n)\) erfüllende Belegung von $F$.\\
+	Wähle $y_i$, falls $a_i = 1$,\\
+	wähle $z_i$, falls $a_i = 0$\\
+	Summe ergibt \( s_1s_2…s_m 11…1 \), wobei \(s_i \in \{1,2,3\} \), da in jeder Klausel $\ge 1$ Literal den Wert 1 hat.\\
+	Dann wähle passende  Füllzahlen um auf $b$ zu kommen.\\
+	D.\,h. $f_i$ und $g_i$, falls $s_i=1$ und $f_i$, falls $s_i=2$.\\
+	\('\Leftarrow'\) Habe Subset Sum Problem eine Lösung.\\
+	Wegen 1-er Block in $b$ muss für jedes $i$ entweder $y_i$ oder $z_i$ in der Lösung sein.\\
+	\defin \(a_i=1\), falls $y_i$ in Lösung,\\
+	$a_i = 0$, falls $z_i$ in Lösung.
+
+	Dann ist $a=(a_1,…,a_n)$ erfüllende Belegung von F:\\
+	Da im vorderen Teil mit den Füll-Zahlen maximal 2 erreicht werden kann, muss die Summe der ausgewählten $y-$ und $z-$Zahlen $\ge 1$ je Klauselspalte sein.\\
+	\(\Rightarrow\) nach Konstruktion enthält also jede Klausel $\ge 1$ erfülltes Problem.
+
 \end{document}

Berechenbarkeits-KomplexTh/bilder/ds-vc.xml

@@ -0,0 +1,290 @@
+<?xml version="1.0"?>
+<!DOCTYPE ipe SYSTEM "ipe.dtd">
+<ipe version="70005" creator="Ipe 7.1.1">
+<info created="D:20111221095307" modified="D:20111221095336"/>
+<ipestyle name="basic">
+<symbol name="arrow/arc(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/farc(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="mark/circle(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</symbol>
+<symbol name="mark/disk(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+</path>
+</symbol>
+<symbol name="mark/fdisk(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+0.5 0 0 0.5 0 0 e
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</group>
+</symbol>
+<symbol name="mark/box(sx)" transformations="translations">
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</symbol>
+<symbol name="mark/square(sx)" transformations="translations">
+<path fill="sym-stroke">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+</path>
+</symbol>
+<symbol name="mark/fsquare(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+-0.5 -0.5 m
+0.5 -0.5 l
+0.5 0.5 l
+-0.5 0.5 l
+h
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="mark/cross(sx)" transformations="translations">
+<group>
+<path fill="sym-stroke">
+-0.43 -0.57 m
+0.57 0.43 l
+0.43 0.57 l
+-0.57 -0.43 l
+h
+</path>
+<path fill="sym-stroke">
+-0.43 0.57 m
+0.57 -0.43 l
+0.43 -0.57 l
+-0.57 0.43 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="arrow/fnormal(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/pointed(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/fpointed(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/linear(spx)">
+<path stroke="sym-stroke" pen="sym-pen">
+-1 0.333 m
+0 0 l
+-1 -0.333 l
+</path>
+</symbol>
+<symbol name="arrow/fdouble(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/double(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<pen name="heavier" value="0.8"/>
+<pen name="fat" value="1.2"/>
+<pen name="ultrafat" value="2"/>
+<symbolsize name="large" value="5"/>
+<symbolsize name="small" value="2"/>
+<symbolsize name="tiny" value="1.1"/>
+<arrowsize name="large" value="10"/>
+<arrowsize name="small" value="5"/>
+<arrowsize name="tiny" value="3"/>
+<color name="red" value="1 0 0"/>
+<color name="green" value="0 1 0"/>
+<color name="blue" value="0 0 1"/>
+<color name="yellow" value="1 1 0"/>
+<color name="orange" value="1 0.647 0"/>
+<color name="gold" value="1 0.843 0"/>
+<color name="purple" value="0.627 0.125 0.941"/>
+<color name="gray" value="0.745"/>
+<color name="brown" value="0.647 0.165 0.165"/>
+<color name="navy" value="0 0 0.502"/>
+<color name="pink" value="1 0.753 0.796"/>
+<color name="seagreen" value="0.18 0.545 0.341"/>
+<color name="turquoise" value="0.251 0.878 0.816"/>
+<color name="violet" value="0.933 0.51 0.933"/>
+<color name="darkblue" value="0 0 0.545"/>
+<color name="darkcyan" value="0 0.545 0.545"/>
+<color name="darkgray" value="0.663"/>
+<color name="darkgreen" value="0 0.392 0"/>
+<color name="darkmagenta" value="0.545 0 0.545"/>
+<color name="darkorange" value="1 0.549 0"/>
+<color name="darkred" value="0.545 0 0"/>
+<color name="lightblue" value="0.678 0.847 0.902"/>
+<color name="lightcyan" value="0.878 1 1"/>
+<color name="lightgray" value="0.827"/>
+<color name="lightgreen" value="0.565 0.933 0.565"/>
+<color name="lightyellow" value="1 1 0.878"/>
+<dashstyle name="dashed" value="[4] 0"/>
+<dashstyle name="dotted" value="[1 3] 0"/>
+<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
+<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
+<textsize name="large" value="\large"/>
+<textsize name="Large" value="\Large"/>
+<textsize name="LARGE" value="\LARGE"/>
+<textsize name="huge" value="\huge"/>
+<textsize name="Huge" value="\Huge"/>
+<textsize name="small" value="\small"/>
+<textsize name="footnote" value="\footnotesize"/>
+<textsize name="tiny" value="\tiny"/>
+<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
+<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
+<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
+<gridsize name="4 pts" value="4"/>
+<gridsize name="8 pts (~3 mm)" value="8"/>
+<gridsize name="16 pts (~6 mm)" value="16"/>
+<gridsize name="32 pts (~12 mm)" value="32"/>
+<gridsize name="10 pts (~3.5 mm)" value="10"/>
+<gridsize name="20 pts (~7 mm)" value="20"/>
+<gridsize name="14 pts (~5 mm)" value="14"/>
+<gridsize name="28 pts (~10 mm)" value="28"/>
+<gridsize name="56 pts (~20 mm)" value="56"/>
+<anglesize name="90 deg" value="90"/>
+<anglesize name="60 deg" value="60"/>
+<anglesize name="45 deg" value="45"/>
+<anglesize name="30 deg" value="30"/>
+<anglesize name="22.5 deg" value="22.5"/>
+<tiling name="falling" angle="-60" step="4" width="1"/>
+<tiling name="rising" angle="30" step="4" width="1"/>
+</ipestyle>
+<page>
+<layer name="alpha"/>
+<view layers="alpha" active="alpha"/>
+<use layer="alpha" name="mark/disk(sx)" pos="64 768" size="normal" stroke="red"/>
+<use name="mark/disk(sx)" pos="64 704" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="128 704" size="normal" stroke="red"/>
+<use name="mark/disk(sx)" pos="128 768" size="normal" stroke="black"/>
+<path stroke="black">
+64 768 m
+128 768 l
+128 704 l
+64 704 l
+64 768 l
+128 704 l
+</path>
+<path stroke="black">
+144 736 m
+176 736 l
+160 752 l
+</path>
+<path stroke="black">
+176 736 m
+160 720 l
+</path>
+<use name="mark/disk(sx)" pos="224 768" size="normal" stroke="red"/>
+<use name="mark/disk(sx)" pos="224 704" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="288 704" size="normal" stroke="red"/>
+<use name="mark/disk(sx)" pos="288 768" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="256 784" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="208 736" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="256 688" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="304 736" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="240 720" size="normal" stroke="black"/>
+<path stroke="black">
+224 768 m
+288 768 l
+288 704 l
+224 704 l
+224 768 l
+288 704 l
+</path>
+<path stroke="orange">
+224 768 m
+208 736 l
+224 704 l
+256 688 l
+288 704 l
+304 736 l
+288 768 l
+256 784 l
+224 768 l
+240 720 l
+288 704 l
+288 704 l
+</path>
+<text transformations="translations" pos="64 688" stroke="black" type="label" width="57.617" height="6.808" depth="0" valign="baseline">Vortex Cover</text>
+<text transformations="translations" pos="320 752" stroke="black" type="label" width="10.628" height="7.49" depth="0" valign="baseline">$G&apos;$</text>
+</page>
+</ipe>

Berechenbarkeits-KomplexTh/bilder/ds-vc2.xml

@@ -0,0 +1,251 @@
+<?xml version="1.0"?>
+<!DOCTYPE ipe SYSTEM "ipe.dtd">
+<ipe version="70005" creator="Ipe 7.1.1">
+<info created="D:20111221095933" modified="D:20111221095933"/>
+<ipestyle name="basic">
+<symbol name="arrow/arc(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/farc(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="mark/circle(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</symbol>
+<symbol name="mark/disk(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+</path>
+</symbol>
+<symbol name="mark/fdisk(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+0.5 0 0 0.5 0 0 e
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</group>
+</symbol>
+<symbol name="mark/box(sx)" transformations="translations">
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</symbol>
+<symbol name="mark/square(sx)" transformations="translations">
+<path fill="sym-stroke">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+</path>
+</symbol>
+<symbol name="mark/fsquare(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+-0.5 -0.5 m
+0.5 -0.5 l
+0.5 0.5 l
+-0.5 0.5 l
+h
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="mark/cross(sx)" transformations="translations">
+<group>
+<path fill="sym-stroke">
+-0.43 -0.57 m
+0.57 0.43 l
+0.43 0.57 l
+-0.57 -0.43 l
+h
+</path>
+<path fill="sym-stroke">
+-0.43 0.57 m
+0.57 -0.43 l
+0.43 -0.57 l
+-0.57 0.43 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="arrow/fnormal(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/pointed(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/fpointed(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/linear(spx)">
+<path stroke="sym-stroke" pen="sym-pen">
+-1 0.333 m
+0 0 l
+-1 -0.333 l
+</path>
+</symbol>
+<symbol name="arrow/fdouble(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/double(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<pen name="heavier" value="0.8"/>
+<pen name="fat" value="1.2"/>
+<pen name="ultrafat" value="2"/>
+<symbolsize name="large" value="5"/>
+<symbolsize name="small" value="2"/>
+<symbolsize name="tiny" value="1.1"/>
+<arrowsize name="large" value="10"/>
+<arrowsize name="small" value="5"/>
+<arrowsize name="tiny" value="3"/>
+<color name="red" value="1 0 0"/>
+<color name="green" value="0 1 0"/>
+<color name="blue" value="0 0 1"/>
+<color name="yellow" value="1 1 0"/>
+<color name="orange" value="1 0.647 0"/>
+<color name="gold" value="1 0.843 0"/>
+<color name="purple" value="0.627 0.125 0.941"/>
+<color name="gray" value="0.745"/>
+<color name="brown" value="0.647 0.165 0.165"/>
+<color name="navy" value="0 0 0.502"/>
+<color name="pink" value="1 0.753 0.796"/>
+<color name="seagreen" value="0.18 0.545 0.341"/>
+<color name="turquoise" value="0.251 0.878 0.816"/>
+<color name="violet" value="0.933 0.51 0.933"/>
+<color name="darkblue" value="0 0 0.545"/>
+<color name="darkcyan" value="0 0.545 0.545"/>
+<color name="darkgray" value="0.663"/>
+<color name="darkgreen" value="0 0.392 0"/>
+<color name="darkmagenta" value="0.545 0 0.545"/>
+<color name="darkorange" value="1 0.549 0"/>
+<color name="darkred" value="0.545 0 0"/>
+<color name="lightblue" value="0.678 0.847 0.902"/>
+<color name="lightcyan" value="0.878 1 1"/>
+<color name="lightgray" value="0.827"/>
+<color name="lightgreen" value="0.565 0.933 0.565"/>
+<color name="lightyellow" value="1 1 0.878"/>
+<dashstyle name="dashed" value="[4] 0"/>
+<dashstyle name="dotted" value="[1 3] 0"/>
+<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
+<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
+<textsize name="large" value="\large"/>
+<textsize name="Large" value="\Large"/>
+<textsize name="LARGE" value="\LARGE"/>
+<textsize name="huge" value="\huge"/>
+<textsize name="Huge" value="\Huge"/>
+<textsize name="small" value="\small"/>
+<textsize name="footnote" value="\footnotesize"/>
+<textsize name="tiny" value="\tiny"/>
+<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
+<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
+<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
+<gridsize name="4 pts" value="4"/>
+<gridsize name="8 pts (~3 mm)" value="8"/>
+<gridsize name="16 pts (~6 mm)" value="16"/>
+<gridsize name="32 pts (~12 mm)" value="32"/>
+<gridsize name="10 pts (~3.5 mm)" value="10"/>
+<gridsize name="20 pts (~7 mm)" value="20"/>
+<gridsize name="14 pts (~5 mm)" value="14"/>
+<gridsize name="28 pts (~10 mm)" value="28"/>
+<gridsize name="56 pts (~20 mm)" value="56"/>
+<anglesize name="90 deg" value="90"/>
+<anglesize name="60 deg" value="60"/>
+<anglesize name="45 deg" value="45"/>
+<anglesize name="30 deg" value="30"/>
+<anglesize name="22.5 deg" value="22.5"/>
+<tiling name="falling" angle="-60" step="4" width="1"/>
+<tiling name="rising" angle="30" step="4" width="1"/>
+</ipestyle>
+<page>
+<layer name="alpha"/>
+<view layers="alpha" active="alpha"/>
+<use layer="alpha" name="mark/disk(sx)" pos="64 768" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="128 768" size="normal" stroke="black"/>
+<use name="mark/disk(sx)" pos="96 784" size="normal" stroke="orange"/>
+<path stroke="black">
+64 768 m
+128 768 l
+</path>
+<path stroke="orange">
+64 768 m
+96 784 l
+128 768 l
+</path>
+<text transformations="translations" pos="48 752" stroke="black" type="label" valign="baseline">u</text>
+<text transformations="translations" pos="112 752" stroke="black" type="label" valign="baseline">v</text>
+<text transformations="translations" pos="96 800" stroke="black" type="label" valign="baseline">$K_{u,v}$</text>
+</page>
+</ipe>