Übungsblatt 4
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übung_4/agt26-blatt04.pdf
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übung_4/agt26-blatt04.pdf
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übung_4/agt_übung_3_korrpdf
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übung_4/agt_übung_4.pdf
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übung_4/agt_übung_4.tex
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übung_4/agt_übung_4.tex
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\documentclass{ngexrcs}
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\usepackage{hyperref}
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\setkeys{Gin}{pagebox=artbox, width=0.2\textwidth}
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\title{4. Übungsblatt}
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\subject{Algorithmische Graphentheorie}
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\author{Jasper Gude \and Pia Röttgers}
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\begin{document}
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\maketitle
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\points[\qquad]{20}
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\input{aufgabe_1.tex}
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\input{aufgabe_2.tex}
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\input{aufgabe_3.tex}
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\end{document}
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übung_4/aufgabe_1.tex
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übung_4/aufgabe_1.tex
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\section{Knotengrade}
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\begin{tasks}
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\item
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Einen solchen Graph gibt es. Siehe \autoref{fig:task1a}.
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\points{3}
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\begin{figure}
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\centering
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\includegraphics[page=1, width=0.8\textwidth]{figures.pdf}
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\caption{Ein Graph mit 11 Knoten und Knotengraden 1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 10.}
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\label{fig:task1a}
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\end{figure}
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\item
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Das Problem kann als Maximalflussproblem modelliert werden. Dafür haben wir
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eine Quelle $s$, eine Senke $t$ und Knoten $v_i$ mit $i \in \NN_{\leq n}$.
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Die Eingangskapazität eines Knoten $v_i$ entspricht dem Eingangsgrad $e_i$.
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Die Ausgangskapazität entspricht dem Ausgangsgrad $a_i$. Siehe \autoref{fig:task1b}.
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Wenn es einen vollständigen Fluss gibt, gibt es eine Lösung für das Problem,
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sonst nicht.
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\points{4}
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\begin{figure}
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\centering
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\includegraphics[page=2, width=0.5\textwidth]{figures.pdf}
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\caption{Das Problem als Maximalflussproblem.}
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\label{fig:task1b}
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\end{figure}
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\item
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Die Modellierung ist korrekt, da der Eingangs- und Ausgangsgrad jedes Knotens $v_i$
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von oben beschränkt wird von der Eingangskapazität $e_i$ bzw. Ausgangskapazität $a_i$.
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Zudem erfüllt die Flusserhaltung den Zweck, dass die Summe der Eingangsgrade gleich der Summe der Ausgangsgrade sind, was in einem Graphen erfüllt sein muss.
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\points{2}
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\end{tasks}
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übung_4/aufgabe_2.tex
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übung_4/aufgabe_2.tex
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\section{b-Flüsse}
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\begin{tasks}
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\item
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\item
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\end{tasks}
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übung_4/aufgabe_3.tex
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übung_4/aufgabe_3.tex
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\section{Minimale Schnitte}
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übung_4/aufgabe_4.tex
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übung_4/aufgabe_4.tex
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\section{Längste Wege}
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\begin{tasks}
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\item
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Da $s, t$ in $G'$ adjazent zu jedem Knoten in $G$ ist, können wir
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einen einfachen $s$-$t$-Weg der Länge $k+2$ erzeugen, indem wir
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einen einfachen Weg der Länge $k$ in $G$ nehmen, $s$ an das eine Ende und $t$ an das andere Ende hängen.
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Umgekehrt kann man aus einem einfachen $s$-$t$-Weg der Länge $k$
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in $G'$ einen einfachen Weg der Länge $k-2$ in $G$ konstruieren,
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indem wir $s$ und $t$ entfernen.
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\points{2}
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\item
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Ein Hamiltonweg ist ein Weg der alle Knoten in $G$ beinhaltet
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und somit Länge $n-1$ besitzt.
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Wie wir oben gezeigt haben, kann ein $s$-$t$-Weg der Länge $n+1$
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in $G'$ leicht in einen Weg der Länge $n-1$ in $G$ umgewandelt
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werden. Das heißt, dass wir einen Hamiltonweg in $G$ finden,
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wenn wir einen $s$-$t$-Weg finden.
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Umgekehrt können wir einen Hamiltonweg leicht in einen $s$-$t$-Weg
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umwandeln, also finden wir einen $s$-$t$-Weg wenn wir einen
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Hamiltonweg finden.
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Also finden wir einen Hamiltonweg genau dann, wenn wir einen
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$s$-$t$-Weg finden.
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\points{1}
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\item
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Da wir Hamiltonweg auf \algt{Längster $s$-$t$-Weg} reduziert
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haben, muss also \algt{Längster $s$-$t$-Weg} $\NPe$-schwer sein, denn
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wenn es in $\Pe$ liegen würde, könnten wir auch Hamiltonweg in
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polynomieller Zeit lösen. Da wir nicht von $\Pe = \NPe$ ausgehen,
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ist das nicht möglich.
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\points{2}
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\end{tasks}
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|
||||
<text matrix="1 0 0 1 128 -296" transformations="translations" pos="352 448" stroke="black" type="label" width="8.8704" height="11.3568" depth="0.36" halign="center" valign="top" size="script" style="math">8</text>
|
||||
<text matrix="1 0 0 1 32 -328" transformations="translations" pos="352 448" stroke="black" type="label" width="8.8704" height="10.9968" depth="0.168" halign="center" valign="top" size="script" style="math">7</text>
|
||||
<text matrix="1 0 0 1 224 -328" transformations="translations" pos="352 448" stroke="black" type="label" width="8.8704" height="11.3568" depth="0.36" halign="center" valign="top" size="script" style="math">6</text>
|
||||
<text matrix="1 0 0 1 32 -56" transformations="translations" pos="352 448" stroke="black" type="label" width="17.7408" height="11.3568" depth="0.36" halign="center" valign="bottom" size="script" style="math">10</text>
|
||||
</page>
|
||||
<page>
|
||||
<layer name="alpha"/>
|
||||
<view layers="alpha" active="alpha"/>
|
||||
<use layer="alpha" matrix="1 0 0 1 160 128" name="mark/disk(sx)" pos="256 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 32 96" name="mark/disk(sx)" pos="288 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 96 -256" name="mark/disk(sx)" pos="320 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 -96 32" name="mark/disk(sx)" pos="352 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 176 -208" name="mark/disk(sx)" pos="384 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 -192 -64" name="mark/disk(sx)" pos="416 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 160 -64" name="mark/disk(sx)" pos="448 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 -208 -208" name="mark/disk(sx)" pos="480 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 0 96" name="mark/disk(sx)" pos="512 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 32 32" name="mark/disk(sx)" pos="544 320" size="normal" stroke="black"/>
|
||||
<use matrix="1 0 0 1 16 -128" name="mark/disk(sx)" pos="400 384" size="normal" stroke="black"/>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
416 448 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
320 416 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
416 64 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
256 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
560 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
224 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
608 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
272 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
512 416 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
576 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
256 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
560 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
224 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
608 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
272 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
512 416 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
416 64 m
|
||||
576 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
560 112 m
|
||||
224 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
560 112 m
|
||||
608 256 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
560 112 m
|
||||
272 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
560 112 m
|
||||
512 416 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
560 112 m
|
||||
576 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
608 256 m
|
||||
272 112 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
608 256 m
|
||||
512 416 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
608 256 m
|
||||
576 352 l
|
||||
</path>
|
||||
<path stroke="black">
|
||||
512 416 m
|
||||
576 352 l
|
||||
</path>
|
||||
<text matrix="1 0 0 1 -64 -24" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">1</text>
|
||||
<text matrix="1 0 0 1 128 -24" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">1</text>
|
||||
<text matrix="1 0 0 1 -160 -184" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">2</text>
|
||||
<text matrix="1 0 0 1 -160 -312" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">3</text>
|
||||
<text matrix="1 0 0 1 -160 -56" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">4</text>
|
||||
<text matrix="1 0 0 1 128 -88" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">5</text>
|
||||
<text matrix="1 0 0 1 224 -56" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">5</text>
|
||||
<text matrix="1 0 0 1 128 -296" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">8</text>
|
||||
<text matrix="1 0 0 1 272 -184" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">7</text>
|
||||
<text matrix="1 0 0 1 288 -320" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">6</text>
|
||||
<text matrix="1 0 0 1 32 -56" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">10</text>
|
||||
<path stroke="black">
|
||||
416 256 m
|
||||
416 64 l
|
||||
</path>
|
||||
</page>
|
||||
<page>
|
||||
<layer name="alpha"/>
|
||||
<view layers="alpha" active="alpha"/>
|
||||
<path layer="alpha" stroke="black" arrow="ng-normal/normal">
|
||||
448 320 m
|
||||
576 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
448 256 m
|
||||
576 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
448 192 m
|
||||
576 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
192 256 m
|
||||
320 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 256 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 320 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 256 m
|
||||
448 256 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 192 m
|
||||
448 192 l
|
||||
</path>
|
||||
<path stroke="black" arrow="ng-normal/normal">
|
||||
320 320 m
|
||||
448 256 l
|
||||
</path>
|
||||
<use name="mark/disk(sx)" pos="192 256" size="normal" stroke="black"/>
|
||||
<use name="mark/disk(sx)" pos="320 320" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="320 256" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="320 192" size="normal" stroke="dark blue"/>
|
||||
<use name="mark/disk(sx)" pos="448 320" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="448 256" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="448 192" size="normal" stroke="dark orange"/>
|
||||
<use name="mark/disk(sx)" pos="576 256" size="normal" stroke="black"/>
|
||||
<text matrix="1 0 0 1 0 8" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_1</text>
|
||||
<text matrix="1 0 0 1 0 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_2</text>
|
||||
<text matrix="1 0 0 1 4 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="19.764" height="8.604" depth="4.824" halign="center" valign="top" size="footnote" style="math">v_n</text>
|
||||
<text matrix="1 0 0 1 128 8" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_1</text>
|
||||
<text matrix="1 0 0 1 -128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="7.7664" height="8.8368" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">s</text>
|
||||
<text matrix="1 0 0 1 128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_2</text>
|
||||
<text matrix="1 0 0 1 128 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="21.6624" height="8.8344" depth="4.824" halign="center" valign="top" size="footnote" style="math">e_m</text>
|
||||
<text matrix="1 0 0 1 256 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="8.34" height="10.9416" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">t</text>
|
||||
<text matrix="0.894704 0.446659 -0.446659 0.894704 112.622 -137.236" pos="320 320" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_1)</text>
|
||||
<text matrix="0.894427 -0.447214 0.447214 0.894427 -73.1492 134.135" pos="256 224" stroke="black" type="label" width="33.9288" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">B(v_n)</text>
|
||||
<text transformations="translations" pos="256 260" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_2)</text>
|
||||
<text matrix="0.894427 -0.447214 0.447214 0.894427 186.851 218.135" pos="256 224" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="top" size="tiny" style="math">r(e_1)</text>
|
||||
<text matrix="0.894704 0.446659 -0.446659 0.894704 368.622 -209.236" pos="320 320" stroke="black" type="label" width="31.6704" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">r(e_m)</text>
|
||||
<text matrix="1 0 0 1 256 0" transformations="translations" pos="256 260" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">r(e_2)</text>
|
||||
<text transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||
<text matrix="1 0 0 1 128 0" transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||
</page>
|
||||
</ipe>
|
||||
71
übung_4/ngexrcs.cls
Normal file
71
übung_4/ngexrcs.cls
Normal file
|
|
@ -0,0 +1,71 @@
|
|||
% vim: set filetype:tex
|
||||
|
||||
% Identification %
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesClass{ngexrcs}[2021/12/21 ADS Exercise class]
|
||||
|
||||
% Handle options %
|
||||
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}}
|
||||
\ProcessOptions\relax
|
||||
|
||||
% More declarations %
|
||||
\LoadClass{scrartcl}
|
||||
|
||||
\KOMAoptions{mpinclude=true}
|
||||
\recalctypearea
|
||||
|
||||
\RequirePackage{ngutils}
|
||||
|
||||
% define fonts
|
||||
\RequirePackage{fontspec}
|
||||
\RequirePackage[math-style=upright]{unicode-math}
|
||||
\setmainfont{TeX Gyre Pagella}
|
||||
\setsansfont{TeX Gyre Heros}
|
||||
\setmonofont{TeX Gyre Cursor}
|
||||
\setmathfont{Euler Math}
|
||||
|
||||
\setlength{\marginparwidth}{1.5\marginparwidth}
|
||||
\setlength{\fboxrule}{\heavyrulewidth}
|
||||
|
||||
% Use sansfont for all title elements
|
||||
\addtokomafont{titlehead}{\sffamily}
|
||||
\addtokomafont{subject}{\normalfont \sffamily \itshape}
|
||||
\addtokomafont{title}{\sffamily}
|
||||
\addtokomafont{subtitle}{\sffamily}
|
||||
\addtokomafont{author}{\sffamily}
|
||||
\addtokomafont{date}{\sffamily}
|
||||
\addtokomafont{publishers}{\sffamily}
|
||||
|
||||
% Use serif font for headings
|
||||
\addtokomafont{disposition}{\rmfamily}
|
||||
|
||||
% Let sections be formated as in: Aufgabe 1 -- Section title
|
||||
\renewcommand*{\sectionformat}{Aufgabe \thesection\autodot\enskip--\enskip}
|
||||
|
||||
% Let points of an exercise be printed as in: [__ / 2]
|
||||
\newkomafont{points}{\sffamily}
|
||||
\newcommand\points[2][\quad]{\marginline{\framebox{{\usekomafont{points}#1 \textbf{/} #2}}}}
|
||||
|
||||
\newcounter{task}
|
||||
\renewcommand{\thetask}{\@alph\c@task)}
|
||||
\newenvironment{tasks}
|
||||
{
|
||||
\begin{list}{\thetask}
|
||||
{
|
||||
\usecounter{task}
|
||||
\setlength{\leftmargin}{1.6em}
|
||||
}
|
||||
}{%
|
||||
\end{list}
|
||||
}
|
||||
|
||||
\DeclareNewTOC
|
||||
[
|
||||
type=algorithm,
|
||||
types=algorithms,
|
||||
float,
|
||||
floattype=4,
|
||||
name=Algorithmus,
|
||||
listname={Algorithmenverzeichnis}
|
||||
]
|
||||
{loa}
|
||||
124
übung_4/ngutils.sty
Normal file
124
übung_4/ngutils.sty
Normal file
|
|
@ -0,0 +1,124 @@
|
|||
\NeedsTeXFormat{LaTeX2e}
|
||||
\ProvidesPackage{ngutils}[2025/12/24 Never's LaTeX utils]
|
||||
|
||||
\RequirePackage[ngerman]{babel}
|
||||
\RequirePackage{graphicx}
|
||||
\RequirePackage{tabularx}
|
||||
\RequirePackage{booktabs}
|
||||
\RequirePackage{listings}
|
||||
|
||||
\lstnewenvironment{pseudocode}[1][] %defines the algorithm listing environment
|
||||
{
|
||||
\renewcommand{\lstlistingname}{Algorithmus}
|
||||
\lstset{ %this is the stype
|
||||
mathescape=true,
|
||||
columns=fullflexible,
|
||||
basicstyle=\normalfont,
|
||||
identifierstyle=\scshape,
|
||||
keywordstyle=\bfseries,
|
||||
keywords={, new, if, then, else, while, for, in, to, up, down, foreach, do, return},
|
||||
commentstyle=\itshape,
|
||||
comment=[l]//,
|
||||
delim=[is][\normalfont]{|}{|},
|
||||
tabsize=3,
|
||||
frame=tb,
|
||||
framerule=1pt,
|
||||
#1 % this is to add specific settings to an usage of this environment (for instnce, the caption and referable label)
|
||||
}
|
||||
}
|
||||
{}
|
||||
|
||||
\newcommand{\seilpmi}{\Longleftarrow}
|
||||
|
||||
\newcommand{\NN}{\mathbb{N}}
|
||||
\newcommand{\ZZ}{\mathbb{Z}}
|
||||
\newcommand{\QQ}{\mathbb{Q}}
|
||||
\newcommand{\RR}{\mathbb{R}}
|
||||
\newcommand{\CC}{\mathbb{C}}
|
||||
\newcommand{\PP}{\mathbb{P}}
|
||||
|
||||
\newcommand{\Pe}{\mathrm{P}}
|
||||
\newcommand{\NPe}{\mathrm{NP}}
|
||||
|
||||
\newcommand{\Oh}{\mathcal{O}}
|
||||
\newcommand{\oh}{\scriptstyle{\mathcal{O}}}
|
||||
|
||||
\newcommand{\Eh}{\mathop{\mathrm{E}}}
|
||||
\newcommand{\Var}{\mathop{\mathrm{Var}}}
|
||||
\newcommand{\pot}{\mathop{\mathcal{P}}}
|
||||
\newcommand{\argmax}{\mathop{\mathrm{arg\,max}}}
|
||||
\newcommand{\argmin}{\mathop{\mathrm{arg\,min}}}
|
||||
|
||||
\newcommand{\parens}[1]{\left(#1\right)}
|
||||
\newcommand{\brackets}[1]{\left[#1\right]}
|
||||
\newcommand{\braces}[1]{\left\{#1\right\}}
|
||||
\newcommand{\angled}[1]{\left\langle#1\right\rangle}
|
||||
|
||||
\newcommand{\tup}[1]{\parens{#1}}
|
||||
\newcommand{\ivl}[1]{\brackets{#1}}
|
||||
\newcommand{\set}[1]{\braces{#1}}
|
||||
\newcommand{\arr}[1]{\angled{#1}}
|
||||
\newcommand{\abs}[1]{\lvert#1\rvert}
|
||||
\newcommand{\norm}[1]{\lVert#1\rVert}
|
||||
\newcommand{\ceil}[1]{\lceil#1\rceil}
|
||||
\newcommand{\floor}[1]{\lfloor#1\rfloor}
|
||||
|
||||
\newcommand{\alg}[2]{{\normalfont\scshape#1}{\normalfont(#2)}}
|
||||
\newcommand{\algt}[1]{{\normalfont\scshape#1}}
|
||||
\newcommand{\com}[1]{{\normalfont\itshape/\!\!/ #1}}
|
||||
|
||||
\newenvironment{algorithmic}[1][1em]{
|
||||
\begin{minipage}[t]{\dimexpr\linewidth}
|
||||
\bfseries
|
||||
\begin{tabbing}
|
||||
\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\kill
|
||||
}{%
|
||||
\end{tabbing}
|
||||
\end{minipage}
|
||||
}
|
||||
|
||||
\newlength{\theoremskip}
|
||||
\setlength{\theoremskip}{6em}
|
||||
|
||||
\newcommand{\theoremfont}{\normalfont\bfseries}
|
||||
\newcommand{\definitionfont}{\normalfont\bfseries}
|
||||
|
||||
\newenvironment{theoremic}[2][\theoremfont]{
|
||||
{#1 #2}
|
||||
\hfill
|
||||
\begin{minipage}[t]{\the\dimexpr\linewidth-\theoremskip\relax}
|
||||
}{
|
||||
\end{minipage}
|
||||
}
|
||||
|
||||
\newenvironment{theorem}{
|
||||
\begin{theoremic}{Satz.}
|
||||
}{
|
||||
\end{theoremic}
|
||||
}
|
||||
|
||||
\newenvironment{lemma}{
|
||||
\begin{theoremic}{Lemma.}
|
||||
}{
|
||||
\end{theoremic}
|
||||
}
|
||||
|
||||
\newenvironment{proof}{
|
||||
\begin{theoremic}{Beweis.}
|
||||
}{
|
||||
\end{theoremic}
|
||||
}
|
||||
|
||||
\newenvironment{definition}{
|
||||
\begin{theoremic}[\definitionfont]{Definition.}
|
||||
}{
|
||||
\end{theoremic}
|
||||
}
|
||||
|
||||
\newenvironment{example}{
|
||||
\begin{theoremic}[\definitionfont]{Beispiel.}
|
||||
}{
|
||||
\end{theoremic}
|
||||
}
|
||||
|
||||
|
||||
Loading…
Add table
Add a link
Reference in a new issue