From c9789d9251bd52dc18da6dfcf25770f022ec8e17 Mon Sep 17 00:00:00 2001 From: Lukas Rost Date: Sun, 28 Apr 2019 19:42:30 +0200 Subject: [PATCH] Alles fertig! Endrunde, ich komme! --- .../Aufgabe1-Dokumentation.log | 415 ++++++----- .../Aufgabe1-Dokumentation.pdf | Bin 406786 -> 412219 bytes .../Aufgabe1-Dokumentation.synctex.gz | Bin 180852 -> 201821 bytes .../Aufgabe1-Dokumentation.tex | 33 +- .../svg-inkscape/lisarennt-ext_svg-tex.pdf | Bin 1164 -> 1164 bytes .../svg-inkscape/lisarennt-ha_svg-tex.pdf | Bin 1184 -> 1184 bytes .../lisarennt1-visgraph_svg-tex.pdf | Bin 1187 -> 1187 bytes .../svg-inkscape/lisarennt1_svg-tex.pdf | Bin 1154 -> 1154 bytes .../lisarennt2-vel-visgraph_svg-tex.pdf | Bin 1465 -> 1465 bytes .../svg-inkscape/lisarennt2-vel_svg-tex.pdf | Bin 1252 -> 1252 bytes .../lisarennt2-visgraph_svg-tex.pdf | Bin 1465 -> 1465 bytes .../svg-inkscape/lisarennt2_svg-tex.pdf | Bin 1252 -> 1252 bytes .../lisarennt3-visgraph_svg-tex.pdf | Bin 2499 -> 2499 bytes .../svg-inkscape/lisarennt3_svg-tex.pdf | Bin 1467 -> 1467 bytes .../lisarennt4-visgraph_svg-tex.pdf | Bin 2932 -> 2932 bytes .../svg-inkscape/lisarennt4_svg-tex.pdf | Bin 1537 -> 1537 bytes .../lisarennt5-visgraph_svg-tex.pdf | Bin 2855 -> 2855 bytes .../svg-inkscape/lisarennt5_svg-tex.pdf | Bin 1492 -> 1492 bytes .../svg-inkscape/reg1000-visgraph_svg-tex.pdf | Bin 25022 -> 25022 bytes .../svg-inkscape/reg1000_svg-tex.pdf | Bin 6918 -> 6918 bytes .../Aufgabe2-Dokumentation.aux | 83 +-- .../Aufgabe2-Dokumentation.log | 657 +++++++++--------- .../Aufgabe2-Dokumentation.pdf | Bin 205214 -> 226758 bytes .../Aufgabe2-Dokumentation.synctex.gz | Bin 146726 -> 181292 bytes .../Aufgabe2-Dokumentation.tex | 252 +++++-- .../Aufgabe2-Dokumentation.toc | 34 +- ...2284FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex} | 16 +- .../svg-inkscape/dreiecke1_svg-tex.pdf | Bin 1080 -> 1080 bytes .../svg-inkscape/dreiecke2_svg-tex.pdf | Bin 1096 -> 1096 bytes .../svg-inkscape/dreiecke3_svg-tex.pdf | Bin 1257 -> 1257 bytes .../svg-inkscape/dreiecke4_svg-tex.pdf | Bin 1499 -> 1499 bytes .../svg-inkscape/dreiecke5_svg-tex.pdf | Bin 1830 -> 1830 bytes Aufgabe2-Implementierung/triangles.cpp | 16 +- 33 files changed, 831 insertions(+), 675 deletions(-) rename Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/{7B2F7C6D41DCE6772411798425F1F75884FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex => B09D3170810788694402390E3C916A2284FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex} (99%) diff --git a/Aufgabe1-Dokumentation/Aufgabe1-Dokumentation.log b/Aufgabe1-Dokumentation/Aufgabe1-Dokumentation.log index 7e668c2..5f87e78 100644 --- a/Aufgabe1-Dokumentation/Aufgabe1-Dokumentation.log +++ b/Aufgabe1-Dokumentation/Aufgabe1-Dokumentation.log @@ -1,4 +1,4 @@ -This is XeTeX, Version 3.14159265-2.6-0.99999 (TeX Live 2018/Arch Linux) (preloaded format=xelatex 2019.4.9) 25 APR 2019 13:12 +This is XeTeX, Version 3.14159265-2.6-0.99999 (TeX Live 2018/Arch Linux) (preloaded format=xelatex 2019.4.9) 28 APR 2019 18:00 entering extended mode \write18 enabled. %&-line parsing enabled. @@ -497,33 +497,26 @@ Package: amsthm 2017/10/31 v2.20.4 \thm@headsep=\skip67 \dth@everypar=\toks27 ) +(/usr/share/texmf-dist/tex/latex/koma-script/scrlayer-scrpage.sty +Package: scrlayer-scrpage 2019/02/01 v3.26b KOMA-Script package (end user inter +face for scrlayer) -Class scrartcl Warning: Usage of package `fancyhdr' together -(scrartcl) with a KOMA-Script class is not recommended. -(scrartcl) I'd suggest to use -(scrartcl) package `scrlayer' or `scrlayer-scrpage', because -(scrartcl) they support KOMA-Script classes. -(scrartcl) With `fancyhdr' several features of class `scrartcl' -(scrartcl) like options `headsepline', `footsepline' or command -(scrartcl) `\MakeMarkcase' and the commands `\setkomafont' and -(scrartcl) `\addtokomafont' for the page style elements need -(scrartcl) explicite user intervention to work. -(scrartcl) Nevertheless, using requested -(scrartcl) package `fancyhdr' on input line 20. - -(/usr/share/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty -Package: fancyhdr 2019/01/31 v3.10 Extensive control of page headers and footer -s -\f@nch@headwidth=\skip68 -\f@nch@O@elh=\skip69 -\f@nch@O@erh=\skip70 -\f@nch@O@olh=\skip71 -\f@nch@O@orh=\skip72 -\f@nch@O@elf=\skip73 -\f@nch@O@erf=\skip74 -\f@nch@O@olf=\skip75 -\f@nch@O@orf=\skip76 +(/usr/share/texmf-dist/tex/latex/koma-script/scrlayer.sty +Package: scrlayer 2019/02/01 v3.26b KOMA-Script package (defining layers and pa +ge styles) +Package scrlayer Info: Using already defined \footheight +(scrlayer) hoping, that this is a length and +(scrlayer) not only a macro on input line 109. +Package scrlayer Info: patching LaTeX kernel macro \pagestyle on input line 208 +3. ) +Package scrlayer-scrpage Info: Makeing stand-alone element `pagehead' from +(scrlayer-scrpage) alias to `pageheadfoot' on input line 1241. +Package scrlayer-scrpage Info: deactivating warning for font element +(scrlayer-scrpage) `pagefoot' on input line 1268. +) +Package scrlayer-scrpage Info: auto-selection of `pagestyleset=KOMA-Script'. + (/usr/share/texmf-dist/tex/latex/lastpage/lastpage.sty Package: lastpage 2015/03/29 v1.2m Refers to last page's name (HMM; 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Package hyperref Warning: Token not allowed in a PDF string (Unicode): -(hyperref) removing `math shift' on input line 322. +(hyperref) removing `math shift' on input line 323. -Underfull \hbox (badness 10000) in paragraph at lines 323--326 +Underfull \hbox (badness 10000) in paragraph at lines 324--327 [] -Underfull \hbox (badness 10000) in paragraph at lines 323--326 +Underfull \hbox (badness 10000) in paragraph at lines 324--327 [] -Underfull \hbox (badness 3568) in paragraph at lines 327--328 +Underfull \hbox (badness 3568) in paragraph at lines 328--329 \TU/lmr/m/n/12 2016 (Präsentation), []$[][][][] [] [] [] [][][][][][][][][] [] [][][][][][][][][][][][][][] [] [][] [] [][][][][][][] [] [][][] [] [][][][] [] [] -Underfull \hbox (badness 10000) in paragraph at lines 329--330 +Underfull \hbox (badness 10000) in paragraph at lines 330--331 []\TU/lmr/m/n/12 Reksten-Monsen, Christian: Distance Tables Part 1 - Defi- [] -Underfull \hbox (badness 10000) in paragraph at lines 329--330 +Underfull \hbox (badness 10000) in paragraph at lines 330--331 \TU/lmr/m/n/12 ning the Problem, []$[][][][][] [] [] [] [][][][][][][][][] [] [ ][][][][][] [] [][] [] [][][][] [] [][] [] [][] [] [] -Underfull \hbox (badness 10000) in paragraph at lines 330--331 +Underfull \hbox (badness 10000) in paragraph at lines 331--332 []\TU/lmr/m/n/12 Reksten-Monsen, Christian: Distance Tables Part 2 - Lee’s Visi bi- [] -Underfull \hbox (badness 10000) in paragraph at lines 330--331 +Underfull \hbox (badness 10000) in paragraph at lines 331--332 \TU/lmr/m/n/12 lity Graph Algorithm, []$[][][][][] [] [] [] [][][][][][][][][] [] [][][][][][] [] [][] [] [][][][] [] [][] [] [][] [] [] -Underfull \hbox (badness 5652) in paragraph at lines 332--333 +Underfull \hbox (badness 5652) in paragraph at lines 333--334 []\TU/lmr/m/n/12 Bygi, Mojtaba Nouri; Ghodsi, Mohammad (Sharif University of Te chno- [] -Underfull \hbox (badness 10000) in paragraph at lines 332--333 +Underfull \hbox (badness 10000) in paragraph at lines 333--334 \TU/lmr/m/n/12 logy): 3D Visibility Graph, []$[][][][][] [] [] [] [][][][] [] [ ][][][][][][][][][][][][][][] [] [][][] [] [][][][] [] [] [10] -Underfull \hbox (badness 10000) in paragraph at lines 334--335 +Underfull \hbox (badness 10000) in paragraph at lines 335--336 [] [] [][][][] [] [][][][][][][][][][][][][][][] [] [][][] [] [][][][] [] [][][ ][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][][] [] [] -Underfull \hbox (badness 4995) in paragraph at lines 335--336 +Underfull \hbox (badness 4995) in paragraph at lines 336--337 []\TU/lmr/m/n/12 Skript der Uni Freiburg zur Minkowski-Summe, []$[][][][] [] [] [] [][][][] [] [][][][][][][][][][] [] [] -Underfull \hbox (badness 3601) in paragraph at lines 336--337 +Underfull \hbox (badness 3601) in paragraph at lines 337--338 []\TU/lmr/m/n/12 Wikipedia-Artikel zur NP-Schwere des dreidimensionalen GSP, [] $[][][][][] [] [] [] [][] [] [] -Underfull \hbox (badness 10000) in paragraph at lines 341--343 +Underfull \hbox (badness 10000) in paragraph at lines 342--344 [] -Underfull \hbox (badness 10000) in paragraph at lines 341--343 +Underfull \hbox (badness 10000) in paragraph at lines 342--344 [] [11] -Underfull \hbox (badness 10000) in paragraph at lines 363--365 +Underfull \hbox (badness 10000) in paragraph at lines 364--366 [] -Overfull \hbox (6.72527pt too wide) in paragraph at lines 363--365 +Overfull \hbox (6.72527pt too wide) in paragraph at lines 364--366 \TU/lmr/m/n/12 In der Dokumentation sind nur Teile des Quellcodes aus \TU/lmtt/ m/n/12 vis_graph.py\TU/lmr/m/n/12 , \TU/lmtt/m/n/12 shortest_path.py\TU/lmr/m/n /12 , [] LaTeX Font Info: Font shape `TU/lmtt/bx/n' in size <12> not available -(Font) Font shape `TU/lmtt/b/n' tried instead on input line 365. +(Font) Font shape `TU/lmtt/b/n' tried instead on input line 366. 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@@ -4,7 +4,6 @@ \subtitle{37. Bundeswettbewerb Informatik 2018/19 - 2. Runde \\~\\} \date{29. April 2019} \usepackage[ngerman]{babel} -%\usepackage[utf8]{inputenc} \usepackage{graphicx} \usepackage{wrapfig} \usepackage{color} @@ -16,7 +15,7 @@ \usepackage{mathtools} \usepackage{amssymb} \usepackage{amsthm} -\usepackage{fancyhdr} +\usepackage{scrlayer-scrpage} \usepackage{lastpage} \usepackage{algorithm} \usepackage{algpseudocode} @@ -33,10 +32,12 @@ \usepackage{longtable} \usepackage{tabu} -\pagestyle{fancy} -\lhead{Lukas Rost, Teilnahme-ID: 48125} -\rhead{Aufgabe 1, Seite \thepage ~von \pageref{LastPage}} -\cfoot{ } +\KOMAoptions{ + headsepline = true, +} +\clearpairofpagestyles +\ihead{Lukas Rost, Teilnahme-ID: 48125} +\ohead{Aufgabe 1, Seite \thepage ~von \pageref{LastPage}} \newenvironment{longlisting}{\captionsetup{type=listing}}{} @@ -83,7 +84,7 @@ Für eine Polygonmenge $S$ mit Knotenmenge $V(S)$ sei dieser definiert als $G_{vis}(S) = (V(S),E_{vis}(S))$. Dabei ist $E_{vis}(S)$ die Menge der Knotenpaare von $S$, sodass die dazwischenliegende Strecke kein Polygon (bzw. das Innere eines Polygons) schneidet. Mathematisch ausgedrückt bedeutet das $E_{vis}(S) = \{ uv | u,v \in V(S)$ und $\overline{uv} \subset C_{free} = \mathbb{R}^{2} \setminus \bigcup S \}$. Das Gewicht einer Kante sei dabei als euklidischer Abstand der beiden Endpunkte definiert. \\ \\ Wenn man $S^{*}$ als $S \cup \{s\}$ definiert (bei normalen Sichtbarkeitsgraphen wird auch $t$ hinzugefügt) und den Sichtbarkeitsgraphen dafür analog, kann man damit das GSP-Problem lösen. Nun entspricht der kürzeste $s$-$t$-Weg, der Hindernisse vermeidet, einem kürzesten Weg in $G_{vis}(S^{*})$. \subsubsection{Naiver Ansatz} - $G_{vis}(S^{*})$ lässt sich nun naiv in $\mathcal{O}(n^3)$ berechnen, wobei $n = |V(S^*)|$. Dazu bestimmt man für jeden Knoten $u \in V(S*)$ die von ihm sichtbaren Knoten $v$. Dabei muss man für jede Strecke $\overline{uv}$ prüfen, ob sie eine der $|E_{vis}(S^*)| = \mathcal{O}(n)$ in Frage kommenden Hinderniskanten schneidet. Die Bestimmung der von $u$ sichtbaren Knoten ist somit in $\mathcal{O}(n^2)$ durchführbar und insgesamt ergibt sich eine Laufzeit von $O(n^3)$. \\ \\ + $G_{vis}(S^{*})$ lässt sich nun naiv in $\mathcal{O}(n^3)$ berechnen, wobei $n = |V(S^*)|$. Dazu bestimmt man für jeden Knoten $u \in V(S^*)$ die von ihm sichtbaren Knoten $v$. Dabei muss man für jede Strecke $\overline{uv}$ prüfen, ob sie eine der $|E_{vis}(S^*)| = \mathcal{O}(n)$ in Frage kommenden Hinderniskanten schneidet. Die Bestimmung der von $u$ sichtbaren Knoten ist somit in $\mathcal{O}(n^2)$ durchführbar und insgesamt ergibt sich eine Laufzeit von $O(n^3)$. \\ \\ Die Funktionsweise des naiven Algorithmus wird auch in folgendem Pseudocode deutlich: \begin{algorithm}[H] \begin{algorithmic} @@ -125,7 +126,7 @@ \end{algorithmic} \caption{Der Algorithmus von Lee} \end{algorithm} - Der Algorithmus in der Methode \texttt{visible\_vertices} benutzt dabei eine sogenannte rotierende \emph{Sweep line} beziehungsweise \emph{Scan line}. Diese Sweep line fegt (\glqq sweept\grqq) den zweidimensionalen Raum aus, wobei sie durch den gesamten Raum bewegt wird, bis alle Objekte (in diesem Fall die Knoten) besucht und verarbeitet wurden. Im Falle von Lees Algorithmus rotiert die Sweep line (mathematisch gesehen ein Strahl) um dem Startpunkt $v$ gegen den Uhrzeigersinn. \\ \\ + Der Algorithmus in der Methode \texttt{visible\_vertices} benutzt dabei eine sogenannte rotierende \emph{Sweep line} beziehungsweise \emph{Scan line}. Diese Sweep line fegt (\glqq sweept\grqq) den zweidimensionalen Raum aus, wobei sie durch den gesamten Raum bewegt wird, bis alle Objekte (in diesem Fall die Knoten) besucht und verarbeitet wurden. Im Falle von Lees Algorithmus rotiert die Sweep line (mathematisch gesehen ein Strahl) um den Startpunkt $v$ gegen den Uhrzeigersinn. \\ \\ Am Anfang zeigt die Sweep line nach rechts (parallel zur x-Achse) und rotiert so lange gegen den Uhrzeigersinn, bis sie einen Punkt trifft, der auf Sichtbarkeit überprüft werden muss. Dazu wird eine Liste der offenen Kanten geführt, mit denen sich die Sweep line aktuell überhaupt schneiden kann und die damit relevant sind.\cite{Src:pyvistwo} Anfangs werden dabei einmal alle Kanten daraufhin überprüft, ob sie sich mit der Sweep line (in der Anfangsstellung) schneiden. Entsprechend wird dann diese Liste initialisiert. \\ \\ Trifft die Sweep line auf einen Punkt $a$, werden deshalb alle Kanten überprüft, deren Endpunkt $a$ ist. Liegt eine Kante bezüglich der Sweep line im Uhrzeigersinn (\emph{clockwise side}), kann sie, sofern sie enthalten ist, aus der Liste der offenen Kanten entfernt werden, da die Sweep line bei Bewegung gegen den Uhrzeigersinn nie wieder mit ihr in Berührung kommt. Liegt die Kante dagegen entgegen des Uhrzeigersinns (\emph{counter-clockwise side}), so muss sie zu den offenen Kanten hinzugefügt werden, da sich die Sweep line im Folgenden mit ihr schneiden kann. \\ \\ So werden in diesem Beispiel (siehe nächste Seite) am Punkt $a$ die Kanten $\overline{ab}$ und $\overline{ac}$ hinzugefügt, da sie bezüglich der Sweep line gegen den Uhrzeigersinn (auf ihrer linken Seite) liegen. In Punkt $b$ kann $\overline{ab}$ dagegen wieder entfernt werden, da sie rechts der Sweep line liegt. Hier wird dann jedoch $\overline{bc}$ hinzugefügt. @@ -174,7 +175,7 @@ \end{algorithmic} \caption{Bestimmung der sichtbaren Knoten bzw. Punkte im Lee-Algorithmus} \end{algorithm} -Das Sortieren der Knoten erfolgt dabei anhand ihrer Polarkoordinaten relativ zu $v$. Dabei wird zuerst nach kleinerem Winkel zu $s$ und dann, falls dieser gleich sein sollte nach kleinerem Abstand zu $v$ sortiert. Die Formulierung \emph{links} im Pseudocode bedeutet entgegengesetzt dem Uhrzeigersinn und \emph{rechts} bedeutet im Uhrzeigersinn. \\ +Das Sortieren der Knoten erfolgt dabei anhand ihrer Polarkoordinaten relativ zu $v$. Dabei wird zuerst nach kleinerem Winkel zu $s$ und dann, falls dieser gleich sein sollte, nach kleinerem Abstand zu $v$ sortiert. Die Formulierung \emph{links} im Pseudocode bedeutet entgegengesetzt dem Uhrzeigersinn und \emph{rechts} bedeutet im Uhrzeigersinn. \\ \begin{algorithm}[H] \begin{algorithmic} \Function{visible}{$w_i$} @@ -255,7 +256,7 @@ b^2 &= \frac{v_L^2}{v_B^2} \cdot \frac{1}{(1 - \frac{v_L^2}{v_B^2})} \cdot x^2 \\ b &= \frac{v_L}{v_B} \cdot \frac{1}{\sqrt{1 - \frac{v_L^2}{v_B^2}}} \cdot x \end{align} - Für den optimalen Zielpunkt $C$ gilt somit $x_c = 0$ und $y_c = y_p + \frac{v_L}{v_B} \cdot \frac{1}{\sqrt{1 - \frac{v_L^2}{v_B^2}}} \cdot x_p$. \footnote{Ich habe keine Ahnung, warum das wie der Lorentzfaktor aus der Relativitätstheorie aussieht.} Setzt man die in der Aufgabenstellung gegebenen Geschwindigkeiten ein. ergibt sich: $y_c = y_p + \frac{\sqrt{3}}{3} \cdot x_p$ + Für den optimalen Zielpunkt $C$ gilt somit $x_c = 0$ und $y_c = y_p + \frac{v_L}{v_B} \cdot \frac{1}{\sqrt{1 - \frac{v_L^2}{v_B^2}}} \cdot x_p$. \footnote{Ich habe keine Ahnung, warum das wie der Lorentzfaktor aus der Relativitätstheorie aussieht.} Setzt man die in der Aufgabenstellung gegebenen Geschwindigkeiten ein, ergibt sich: $y_c = y_p + \frac{\sqrt{3}}{3} \cdot x_p$ \subsection{Kombination der Ansätze} Um das Problem nun lösen zu können, muss man beide Ansätze (mit und ohne Hindernisse) kombinieren. Dazu bestimmt man zu jedem Knoten/Punkt $p$ in der Eingabe einen (von mir so bezeichneten) \textit{Companion}-Punkt $c$, indem man die im vorigen Abschnitt berechnete Gleichung auf diesen Punkt anwendet. Dieser Punkt $c$ ist dann der optimale Punkt, um von $p$ aus die y-Achse zu erreichen und dabei möglichst spät starten zu müssen. \\ \\ Da es jedoch sein kann, dass $c$ von $p$ nicht sichtbar ist (ein Hindernis liegt dazwischen), muss man eine Sichtbarkeitsprüfung durchführen. Dazu betrachtet man $c$ einfach bei der Erstellung des Sichtbarkeitsgraphen, genauer bei der Ermittlung der sichtbaren Punkte von $p$ aus (und nur dabei, denn für alle anderen Punkte hat $c$ keine Relevanz). \\ \\ @@ -272,9 +273,9 @@ \begin{proof}[Algorithmus von Lee] Die Funktion \texttt{visible\_vertices} im Algorithmus von Lee läuft offensichtlich in $\mathcal{O}(n \cdot \log n)$. Der Grund dafür ist, dass die einzelnen Schritte folgende Laufzeiten haben: \begin{itemize} - \item \textbf{Sortieren der Knoten}: kann in $\mathcal{O}(n \cdot \log n)$ erfolgen (mit z.B. Quicksort), wobei $\Omega(n \cdot \log n)$ wie allgemein bekannt auch die Untergrenze für die Worst-Case-Laufzeit eines Sortieralgorithmus ist. + \item \textbf{Sortieren der Knoten}: kann in $\mathcal{O}(n \cdot \log n)$ erfolgen (mit z.B. Mergesort oder Quicksort), wobei $\Omega(n \cdot \log n)$ wie allgemein bekannt auch die Untergrenze für die Worst-Case-Laufzeit eines Sortieralgorithmus ist. \item \textbf{Schnittprüfung von Sweep line und Kanten}: Diese muss anfangs für $n$ Polygonkanten durchgeführt werden, hier ergibt sich also $\mathcal{O}(n)$. - \item \textbf{Überprüfen aller Knoten}: Dies wird für alle $\mathcal{O}(n)$ Knoten durchgeführt. Dabei benötigt die Sichtbarkeitsprüfung $\mathcal{O}(\log n)$ (durch die Verwendung eines Suchbaumes). Danach müssen noch konstant viele Kanten dem Baum hinzugefügt bzw. entfernt werden\footnote{Normalerweise höchstens 2, in Sonderfällen können es aber auch bis zu 4 sein. Diese Sonderfälle treten dann auf, wenn sich zwei Polygone eine Kante oder einen Knoten teilen.} Das Hinzufügen bzw. Entfernen benötigt wieder $\mathcal{O}(\log n)$. Insgesamt ergibt sich also auch hier wieder $\mathcal{O}(n \cdot \log n)$. + \item \textbf{Überprüfen aller Knoten}: Dies wird für alle $\mathcal{O}(n)$ Knoten durchgeführt. Dabei benötigt die Sichtbarkeitsprüfung $\mathcal{O}(\log n)$ (durch die Verwendung eines Suchbaumes). Danach müssen noch konstant viele Kanten dem Baum hinzugefügt bzw. entfernt werden.\footnote{Normalerweise höchstens 2, in Sonderfällen können es aber auch bis zu 4 sein. Diese Sonderfälle treten dann auf, wenn sich zwei Polygone eine Kante oder einen Knoten teilen.} Das Hinzufügen bzw. Entfernen benötigt wieder $\mathcal{O}(\log n)$. Insgesamt ergibt sich also auch hier wieder $\mathcal{O}(n \cdot \log n)$. \end{itemize} Wie eindeutig sichtbar ist, dominiert hier $\mathcal{O}(n \cdot \log n)$. Da die Funktion für alle $n$ Knoten aufgerufen wird, ergibt sich für den Gesamtalgorithmus eine Laufzeit in $\mathcal{O}(n^2 \cdot \log n)$ \end{proof} @@ -282,7 +283,7 @@ Der Dijkstra-Algorithmus hat bei der allgemein übliche Implementierung mit einer Priority Queue eine Laufzeit von $\mathcal{O}((|V|+|E|) \cdot \log |V|)$, wobei $V$ die Knotenmenge und $E$ die Kantenmenge ist. Da in diesem Fall $|V| \in \mathcal{O}(n)$ und $|E| \in \mathcal{O}(n^2)$ gilt, ergibt sich hier eine Laufzeit in $\mathcal{O}(n^2 \cdot \log n)$. \end{proof} \begin{proof}[Kombination der Ansätze] - Da bei der Berechnung des Companion-Punktes nur einfache Grundrechenoperationen durchgeführt werden, kann dieser in konstanter Zeit, also $\mathcal{O}(1)$ berechnet werden. Für die $n$ Punkte bzw. Knoten ergibt sich dann $\mathcal{O}(n)$. Alle weiteren Schritte aus dem Abschnitt \emph{Kombination der Ansätze} sind in die Algorithmen von Dijkstra bzw. Lee integriert, tragen also nicht weiter zur Laufzeit bei. Nur die Berechnung der Startzeit muss für jeden Punkt noch separat erfolgen, hier ergibt sich wieder $\mathcal{O}(n)$. + Da bei der Berechnung des Companion-Punktes nur einfache Rechenoperationen durchgeführt werden, kann dieser in konstanter Zeit, also $\mathcal{O}(1)$ berechnet werden. Für die $n$ Punkte bzw. Knoten ergibt sich dann $\mathcal{O}(n)$. Alle weiteren Schritte aus dem Abschnitt \emph{Kombination der Ansätze} sind in die Algorithmen von Dijkstra bzw. Lee integriert, tragen also nicht weiter zur Laufzeit bei. Nur die Berechnung der Startzeit muss für jeden Punkt noch separat erfolgen, hier ergibt sich wieder $\mathcal{O}(n)$. \end{proof} Insgesamt ergibt sich für die Laufzeit aller drei Algorithmen also $\mathcal{O}(n^2 \cdot \log n + n^2 \cdot \log n + n) = \mathcal{O}(n^2 \cdot \log n)$. \subsection{Mögliche und implementierte Erweiterungen} @@ -296,7 +297,7 @@ A + B := \{a+b\,|\,a \in A, b \in B\} \end{equation} Sie ist also die Menge aller Elemente, die Summen von je einem Element aus $A$ und einem aus $B$ sind.\cite{Src:minkowski} Dies entspricht im Prinzip dem Entlangfahren des Lisa-Polygons am Rand des Hindernispolygons. Dabei wird das Hindernis so verbreitert, dass Lisa nun wieder als punktförmig angenommen und der ursprüngliche Algorithmus ausgeführt werden kann. \\ \\ - Seien nun das Hindernispolygon $P = (v_1,...,v_k)$ und der Kopie des Lisa-Polygons $L = (w_1,...,w_m)$ mit $v_i$ und $w_i$ als Eckpunkten. Für diese zwei Polygone im $\mathbb{R}^{2}$ kann die Minkowski-Summe (bzw. deren konvexe Hülle) wie folgt berechnet werden: \\ \\ + Seien nun das Hindernispolygon $P = (v_1,...,v_k)$ und die Kopie des Lisa-Polygons $L = (w_1,...,w_m)$ mit $v_i$ und $w_i$ als Eckpunkten. Für diese zwei Polygone im $\mathbb{R}^{2}$ kann die Minkowski-Summe (bzw. deren konvexe Hülle) wie folgt berechnet werden: \\ \\ Zunächst berechnet man die Minkowski-Summe $S = P + L$ nach der oben angegebenen Definition. Anschließend kann man deren konvexe Hülle mit einem Algorithmus wie dem Graham Scan berechnen: \begin{algorithm}[H] \begin{algorithmic} @@ -315,7 +316,7 @@ \end{algorithmic} \caption{Bestimmung der konvexen Hülle der Minkowski-Summe} \end{algorithm} -\hspace*{-1em}Dabei gibt die Funktion $ccw(a,b,c)$ an, ob $c$ links von $AB$ ist (Rückgabe ist positiv) oder rechts von $AB$ (Rückgabe ist negativ). Der Graham-Scan-Algorithmus läuft für $n$ Punkte in $\mathcal{O}(n \cdot \log n)$. Da die Minkowski-Summe nach Definition $\mathcal{O}(k \cdot m)$ Punkte hat, ergibt sich in diesem Fall $\mathcal{O}(km \cdot \log km)$. Für alle Polygone zusammen ergibt sich eine Laufzeit von $\mathcal{O}(k_{max}m \cdot \log k_{max}m)$ mit $k_{max}$ als maximaler Eckpunktanzahl. Da $k_{max} \in \mathcal{O}(n)$ mit $n$ als Anzahl aller Eckpunkte, hat man eine Laufzeit von $\mathcal{O}(nm \cdot \log nm)$. Diese Erweiterung wurde \textbf{implementiert}. +\hspace*{-1em}Dabei gibt die Funktion $ccw(a,b,c)$ an, ob $c$ links von $\overrightarrow{AB}$ ist (Rückgabe ist positiv) oder rechts von $\overrightarrow{AB}$ (Rückgabe ist negativ). Der Graham-Scan-Algorithmus läuft für $n$ Punkte in $\mathcal{O}(n \cdot \log n)$. Da die Minkowski-Summe nach Definition $\mathcal{O}(k \cdot m)$ Punkte hat, ergibt sich in diesem Fall $\mathcal{O}(km \cdot \log km)$. Für alle Polygone zusammen ergibt sich eine Laufzeit von $\mathcal{O}(k_{max}m \cdot \log k_{max}m)$ mit $k_{max}$ als maximaler Eckpunktanzahl. Da $k_{max} \in \mathcal{O}(n)$ mit $n$ als Anzahl aller Eckpunkte, hat man eine Laufzeit von $\mathcal{O}(nm \cdot \log nm)$. Diese Erweiterung wurde \textbf{implementiert}. \subsubsection{Nicht-polygonale Hindernisse} Es wäre auch möglich, die Art der Hindernisse auf Flächen, die keine Polygone sind, zu erweitern. Dies könnten beispielsweise Ovale oder Kreise sein. Eine Möglichkeit, den Sichtbarkeitsgraph darauf zu erweitern, ist in \cite{Src:arcvis} beschrieben. Dabei muss man anstelle des klassischen Sichtbarkeitsgraphen einen Tangenten-Sichtbarkeitsgraphen verwenden. Da dies jedoch relativ kompliziert ist, wird an dieser Stelle nicht genauer darauf eingegangen. Außerdem lässt sich eine gute Näherung dafür erreichen, wenn man anstelle eines Kreises ein Polygon mit sehr vielen Ecken verwendet (siehe Beispiele). \\ \\ Diese Erweiterung wurde \textbf{nicht implementiert}. @@ -390,7 +391,7 @@ \subsubsection{Die Datei \texttt{visible\_vertices.py}} \texttt{angle()} & Wird für das Sortieren der Kanten am Anfang des Algorithmus genutzt. Berechnet den Winkel, den eine Kante $\overline{cp}$ zur positiven x-Achse besitzt.\\ \hline \texttt{angle2()} & Berechnet mithilfe des Cosinussatzes einen gesuchten Winkel in einem Dreieck.\\ \hline \texttt{ccw()} & Überprüft, ob ein Punkt $c$ gegen den Uhrzeigersinn bezüglich eines Strahls $ab$ liegt. Dazu wird eine Version der Gaußschen Trapezformel (Shoelace formula) verwendet.\\ \hline - \texttt{on\_segment()} & Überprüft, ob ein Punkt $q$ auf einer Strecke $pr$ liegt. Das ist genau dann der Fall, wenn sowohl der x-Wert als auch der y-Wet von $q$ zwischen den x- bzw. y-Werten der beiden anderen Punkte liegen.\\ \hline + \texttt{on\_segment()} & Überprüft, ob ein Punkt $q$ auf einer Strecke $pr$ liegt. Das ist genau dann der Fall, wenn sowohl der x-Wert als auch der y-Wert von $q$ zwischen den x- bzw. y-Werten der beiden anderen Punkte liegen.\\ \hline \texttt{edge\_intersect()} & Überprüft, ob zwei Strecken sich schneiden. \\ \hline \texttt{insort()} und \texttt{bisect()} & Diese beiden Funktionen sind für das Einfügen in den bzw. Suchen im binären Suchbaum, der die offenen Kanten beinhaltet, zuständig. 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wrapfig 2003/01/31 v 3.6 ) (/usr/share/texmf-dist/tex/latex/graphics/color.sty @@ -359,7 +351,7 @@ Package: geometry 2018/04/16 v5.8 Page Geometry \Gm@layoutheight=\dimen117 \Gm@layouthoffset=\dimen118 \Gm@layoutvoffset=\dimen119 -\Gm@dimlist=\toks18 +\Gm@dimlist=\toks16 ) (/usr/share/texmf-dist/tex/latex/fancyvrb/fancyvrb.sty Package: fancyvrb 2019/01/15 @@ -421,7 +413,7 @@ Package: amstext 2000/06/29 v2.01 AMS text (/usr/share/texmf-dist/tex/latex/amsmath/amsgen.sty File: amsgen.sty 1999/11/30 v2.0 generic functions -\@emptytoks=\toks19 +\@emptytoks=\toks17 \ex@=\dimen129 )) (/usr/share/texmf-dist/tex/latex/amsmath/amsbsy.sty @@ -451,21 +443,21 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 730. \dotsspace@=\muskip11 \c@parentequation=\count122 \dspbrk@lvl=\count123 -\tag@help=\toks20 +\tag@help=\toks18 \row@=\count124 \column@=\count125 \maxfields@=\count126 -\andhelp@=\toks21 +\andhelp@=\toks19 \eqnshift@=\dimen132 \alignsep@=\dimen133 \tagshift@=\dimen134 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LaTeX Info: Redefining \] on input line 2845. ) @@ -495,46 +487,39 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' )) (/usr/share/texmf-dist/tex/latex/amscls/amsthm.sty Package: amsthm 2017/10/31 v2.20.4 -\thm@style=\toks24 -\thm@bodyfont=\toks25 -\thm@headfont=\toks26 -\thm@notefont=\toks27 -\thm@headpunct=\toks28 +\thm@style=\toks22 +\thm@bodyfont=\toks23 +\thm@headfont=\toks24 +\thm@notefont=\toks25 +\thm@headpunct=\toks26 \thm@preskip=\skip65 \thm@postskip=\skip66 \thm@headsep=\skip67 -\dth@everypar=\toks29 +\dth@everypar=\toks27 ) (/usr/share/texmf-dist/tex/latex/was/gensymb.sty Package: gensymb 2003/07/02 v1.0 (WaS) ) +(/usr/share/texmf-dist/tex/latex/koma-script/scrlayer-scrpage.sty +Package: scrlayer-scrpage 2019/02/01 v3.26b KOMA-Script package (end user inter +face for scrlayer) -Class scrartcl Warning: Usage of package `fancyhdr' together -(scrartcl) with a KOMA-Script class is not recommended. -(scrartcl) I'd suggest to use -(scrartcl) package `scrlayer' or `scrlayer-scrpage', because -(scrartcl) they support KOMA-Script classes. -(scrartcl) With `fancyhdr' several features of class `scrartcl' -(scrartcl) like options `headsepline', `footsepline' or command -(scrartcl) `\MakeMarkcase' and the commands `\setkomafont' and -(scrartcl) `\addtokomafont' for the page style elements need -(scrartcl) explicite user intervention to work. -(scrartcl) Nevertheless, using requested -(scrartcl) package `fancyhdr' on input line 21. - -(/usr/share/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty -Package: fancyhdr 2019/01/31 v3.10 Extensive control of page headers and footer -s -\f@nch@headwidth=\skip68 -\f@nch@O@elh=\skip69 -\f@nch@O@erh=\skip70 -\f@nch@O@olh=\skip71 -\f@nch@O@orh=\skip72 -\f@nch@O@elf=\skip73 -\f@nch@O@erf=\skip74 -\f@nch@O@olf=\skip75 -\f@nch@O@orf=\skip76 +(/usr/share/texmf-dist/tex/latex/koma-script/scrlayer.sty +Package: scrlayer 2019/02/01 v3.26b KOMA-Script package (defining layers and pa +ge styles) +Package scrlayer Info: Using already defined \footheight +(scrlayer) hoping, that this is a length and +(scrlayer) not only a macro on input line 109. +Package scrlayer Info: patching LaTeX kernel macro \pagestyle on input line 208 +3. ) +Package scrlayer-scrpage Info: Makeing stand-alone element `pagehead' from +(scrlayer-scrpage) alias to `pageheadfoot' on input line 1241. +Package scrlayer-scrpage Info: deactivating warning for font element +(scrlayer-scrpage) `pagefoot' on input line 1268. +) +Package scrlayer-scrpage Info: auto-selection of `pagestyleset=KOMA-Script'. + (/usr/share/texmf-dist/tex/latex/lastpage/lastpage.sty Package: lastpage 2015/03/29 v1.2m Refers to last page's name (HMM; JPG) ) @@ -597,12 +582,12 @@ Package: expl3 2019-02-15 L3 programming layer (code) \l_tmpb_dim=\dimen148 \g_tmpa_dim=\dimen149 \g_tmpb_dim=\dimen150 -\c_zero_skip=\skip77 -\c_max_skip=\skip78 -\l_tmpa_skip=\skip79 -\l_tmpb_skip=\skip80 -\g_tmpa_skip=\skip81 -\g_tmpb_skip=\skip82 +\c_zero_skip=\skip68 +\c_max_skip=\skip69 +\l_tmpa_skip=\skip70 +\l_tmpb_skip=\skip71 +\g_tmpa_skip=\skip72 +\g_tmpb_skip=\skip73 \c_zero_muskip=\muskip12 \c_max_muskip=\muskip13 \l_tmpa_muskip=\muskip14 @@ -772,7 +757,7 @@ LaTeX Info: Redefining \emph on input line 4114. )) (/usr/share/texmf-dist/tex/latex/microtype/microtype.sty Package: microtype 2018/01/14 v2.7a Micro-typographical refinements (RS) -\MT@toks=\toks30 +\MT@toks=\toks28 \MT@count=\count293 LaTeX Info: Redefining \textls on input line 793. \MT@outer@kern=\dimen180 @@ -792,7 +777,7 @@ File: microtype.cfg 2018/01/14 v2.7a microtype main configuration file (RS) (/usr/share/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty (/usr/share/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex -\pgfutil@everybye=\toks31 +\pgfutil@everybye=\toks29 \pgfutil@tempdima=\dimen181 \pgfutil@tempdimb=\dimen182 @@ -815,11 +800,11 @@ Package: pgf 2019/02/02 v3.1.1 (3.1.1) Package: pgfsys 2019/02/02 v3.1.1 (3.1.1) (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex -\pgfkeys@pathtoks=\toks32 -\pgfkeys@temptoks=\toks33 +\pgfkeys@pathtoks=\toks30 +\pgfkeys@temptoks=\toks31 (/usr/share/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex -\pgfkeys@tmptoks=\toks34 +\pgfkeys@tmptoks=\toks32 )) \pgf@x=\dimen183 \pgf@y=\dimen184 @@ -835,9 +820,9 @@ Package: pgfsys 2019/02/02 v3.1.1 (3.1.1) \c@pgf@countb=\count296 \c@pgf@countc=\count297 \c@pgf@countd=\count298 -\t@pgf@toka=\toks35 -\t@pgf@tokb=\toks36 -\t@pgf@tokc=\toks37 +\t@pgf@toka=\toks33 +\t@pgf@tokb=\toks34 +\t@pgf@tokc=\toks35 \pgf@sys@id@count=\count299 (/usr/share/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg @@ -874,9 +859,9 @@ Package: pgfcore 2019/02/02 v3.1.1 (3.1.1) \pgfmath@dimen=\dimen191 \pgfmath@count=\count303 \pgfmath@box=\box54 -\pgfmath@toks=\toks38 -\pgfmath@stack@operand=\toks39 -\pgfmath@stack@operation=\toks40 +\pgfmath@toks=\toks36 +\pgfmath@stack@operand=\toks37 +\pgfmath@stack@operation=\toks38 ) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) @@ -995,8 +980,8 @@ Package: pgffor 2019/02/02 v3.1.1 (3.1.1) (/usr/share/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) \pgffor@iter=\dimen274 \pgffor@skip=\dimen275 -\pgffor@stack=\toks41 -\pgffor@toks=\toks42 +\pgffor@stack=\toks39 +\pgffor@toks=\toks40 )) (/usr/share/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex Package: tikz 2019/02/02 v3.1.1 (3.1.1) @@ -1071,9 +1056,9 @@ Package: array 2018/12/30 v2.4k Tabular extension package (FMi) \col@sep=\dimen284 \ar@mcellbox=\box64 \extrarowheight=\dimen285 -\NC@list=\toks43 -\extratabsurround=\skip83 -\backup@length=\skip84 +\NC@list=\toks41 +\extratabsurround=\skip74 +\backup@length=\skip75 \ar@cellbox=\box65 ) (/usr/share/texmf-dist/tex/latex/varwidth/varwidth.sty @@ -1098,20 +1083,20 @@ Package: varwidth 2009/03/30 ver 0.92; 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(/usr/share/texmf-dist/tex/latex/minted/minted.sty Package: minted 2017/07/19 v2.5 Yet another Pygments shim for LaTeX @@ -1229,7 +1214,7 @@ Package textcomp Info: Setting ptmj sub-encoding to TS1/4 on input line 409. Package: lineno 2005/11/02 line numbers on paragraphs v4.41 Invalid UTF-8 byte or sequence at line 296 replaced by U+FFFD. \linenopenalty=\count329 -\output=\toks47 +\output=\toks45 \linenoprevgraf=\count330 \linenumbersep=\dimen298 \linenumberwidth=\dimen299 @@ -1247,7 +1232,7 @@ LaTeX Info: Redefining \\ on input line 3056. \c@FV@TrueTabGroupLevel=\count336 \c@FV@TrueTabCounter=\count337 \FV@TabBox@Group=\box76 -\FV@TmpLength=\skip90 +\FV@TmpLength=\skip81 \c@FV@HighlightLinesStart=\count338 \c@FV@HighlightLinesStop=\count339 \FV@LoopCount=\count340 @@ -1412,7 +1397,7 @@ Package: xstring 2019/02/06 v1.83 String manipulations (CT) ) (/usr/share/texmf-dist/tex/latex/framed/framed.sty Package: framed 2011/10/22 v 0.96: framed or shaded text with page breaks -\OuterFrameSep=\skip91 +\OuterFrameSep=\skip82 \fb@frw=\dimen309 \fb@frh=\dimen310 \FrameRule=\dimen311 @@ -1421,9 +1406,9 @@ Package: framed 2011/10/22 v 0.96: framed or shaded text with page breaks (/usr/share/texmf-dist/tex/latex/float/float.sty Package: float 2001/11/08 v1.3d Float enhancements (AL) \c@float@type=\count349 -\float@exts=\toks48 +\float@exts=\toks46 \float@box=\box80 -\@float@everytoks=\toks49 +\@float@everytoks=\toks47 \@floatcapt=\box81 ) \minted@appexistsfile=\read4 @@ -1431,7 +1416,7 @@ Package: float 2001/11/08 v1.3d Float enhancements (AL) \minted@code=\write5 \c@minted@FancyVerbLineTemp=\count350 \c@minted@pygmentizecounter=\count351 -\@float@every@listing=\toks50 +\@float@every@listing=\toks48 \c@listing=\count352 ) runsystem(mkdir -p _minted-Aufgabe2-Dokumentation)...executed. @@ -1455,47 +1440,47 @@ Package zref Info: New property: abspage on input line 62. 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Invalid UTF-8 byte or sequence at line 11 replaced by U+FFFD. Package: algorithm 2009/08/24 v0.1 Document Style `algorithm' - floating enviro nment -\@float@every@algorithm=\toks51 +\@float@every@algorithm=\toks49 \c@algorithm=\count361 ) (/usr/share/texmf-dist/tex/latex/algorithmicx/algpseudocode.sty @@ -1563,13 +1548,13 @@ Document Style algorithmicx 1.2 - a greatly improved `algorithmic' style \c@ALG@line=\count362 \c@ALG@rem=\count363 \c@ALG@nested=\count364 -\ALG@tlm=\skip159 -\ALG@thistlm=\skip160 +\ALG@tlm=\skip150 +\ALG@thistlm=\skip151 \c@ALG@Lnr=\count365 \c@ALG@blocknr=\count366 \c@ALG@storecount=\count367 \c@ALG@tmpcounter=\count368 -\ALG@tmplength=\skip161 +\ALG@tmplength=\skip152 ) Document Style - pseudocode environments for use with the `algorithmicx' style ) @@ -1580,33 +1565,33 @@ runsystem(rm Aufgabe2-Dokumentation.aex)...executed. 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(./Aufgabe2-Dokumentation.out) (./Aufgabe2-Dokumentation.out) \@outlinefile=\write6 @@ -1630,7 +1615,7 @@ Class scrartcl Info: loading recommended package `bookmark'. (scrartcl) You can avoid loading `bookmark' with KOMA-Script option (scrartcl) `bookmarkpackage=false' before \begin{document} and (scrartcl) you can avoid this message with `\usepackage{bookmark}' -(scrartcl) before \begin{document} on input line 52. +(scrartcl) before \begin{document} on input line 54. 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Dadurch befi -nden sich „längere“ bzw. - [] - +Package caption Warning: The option `hypcap=true' will be ignored for this +(caption) particular \caption on input line 244. +See the caption package documentation for explanation. -Underfull \hbox (badness 10000) in paragraph at lines 213--215 +Overfull \hbox (8.91386pt too wide) in paragraph at lines 243--244 +[][] [] -[4] Package caption Warning: The option `hypcap=true' will be ignored for this -(caption) particular \caption on input line 246. +(caption) particular \caption on input line 280. 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Package svg Info: Last page of `./svg-inkscape/dreiecke1_svg-tex.pdf' is 1 on i -nput line 519. +nput line 681. (./svg-inkscape/dreiecke1_svg-tex.pdf_tex File: ./svg-inkscape/dreiecke1_svg-tex.pdf Graphic file (type pdf) ) -Package svg Info: Calling Inkscape on input line 540. +Package svg Info: Calling Inkscape on input line 702. runsystem(inkscape -z -D --export-latex --file="../Aufgabe2-Implementierung/ex amples/out/dreiecke2.svg" --export-pdf="dreiecke2_svg-tex.pdf" )...executed. @@ -2062,13 +2061,13 @@ runsystem(mv "dreiecke2_svg-tex.pdf_tex" "./svg-inkscape/dreiecke2_svg-tex.pdf_ tex")...executed. Package svg Info: Last page of `./svg-inkscape/dreiecke2_svg-tex.pdf' is 1 on i -nput line 540. +nput line 702. 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zs?bis(kkcAwB77g*P@82@yTD9Sa_TlNmok8H%&dV8$DlL4?!fYtz4R^Z}x2R, "$\alpha$", angle eccentricity=1.5] {angle = B--A--C};
+\path (C) -- (B) -- (A) pic[draw, ->, "$\beta$", angle eccentricity=1.5] {angle = C--B--A};
+\path (A) -- (C) -- (B) pic[draw, ->, "$\gamma$", angle eccentricity=1.5] {angle = A--C--B};
+\end{tikzpicture}
+\caption{Innenwinkel eines Dreiecks}
+\end{figure}
+\textbf{Berechnung des freien Winkels} \quad Auf der rechten Seite ist der freie Winkel $\alpha_R$ der Winkel des Vektors zwischen $R$ und einem Punkt eines der platzierten Dreiecke zur positiven x-Achse. Insbesondere ist $\alpha_R$ der kleinste solche Winkel. Es gilt also:
+\begin{equation}
+\alpha_R = \min_{P \in T_{pl}} atan2(\Delta y, \Delta x)
+\end{equation}
+Dabei ist $\Delta y$ der y-Anteil des Vektors $\overrightarrow{RP}$ und $\Delta x$ der x-Anteil. Für $\alpha_L$ wiederum gilt (wobei $\Delta y$ und $\Delta x$ über $\overrightarrow{LP}$ definiert werden):
+\begin{equation}
+\alpha_L = \min_{P \in T_{pl}} 180\degree - atan2(\Delta y, \Delta x)
+\end{equation}
+\begin{figure}[H]
+	\centering \begin{tikzpicture}
+	\coordinate (A) at (2,0);
+	\coordinate (B) at (1,2);
+	\coordinate (C) at (4,0);
+	\draw[->] (A) node[below]{$R$}
+	-- (B) node[right]{$P$};
+	\draw[->] (0,0) 
+	-> (C) node[right]{x};
+	\path (C) -- (A) -- (B) pic[draw, ->, "$\alpha$", angle eccentricity=1.5] {angle = C--A--B};
+	\end{tikzpicture}
+	\caption{Definition des atan-Winkels zur x-Achse}
+\end{figure}
+\textbf{Drehen eines Punkts um ein Zentrum} \quad Im Algorithmus taucht immer wieder die Notwendigkeit auf, einen Punkt um einen anderen Punkt zu rotieren. Um dies programmatisch umzusetzen, verwendet man sogenannte Drehmatrizen.\cite{Src:dreh} Multipliziert man nun diese mit dem Ortsvektor des Punktes, so wird dieser um den Winkel $\alpha$ gegen den Uhrzeigersinn um den Ursprung gedreht.
+\begin{align}
+\begin{pmatrix} x' \\ y' \end{pmatrix} &= \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{pmatrix} \cdot \begin{pmatrix} x \\y \end{pmatrix} \\
+x' &= x \cdot \cos \alpha - y \cdot \sin \alpha \\
+y' &= x \cdot \sin \alpha + y \cdot \cos \alpha
+\end{align}
+Da man jedoch meist nicht um den Ursprung, sondern um einen anderen Punkt drehen will, muss man vorher das Koordinatensystem so transformieren, dass das Drehzentrum dem Ursprung entspricht. Dazu zieht man die Koordinaten des Zentrums von denen des Punkts ab. Nach der Drehung muss dann eine Rücktransformation erfolgen, wozu man die Koordinaten des Zentrums wieder addiert. \\
+\textbf{Verschieben eines Punktes} \quad Das Verschieben eines Punktes $P$ um einen Vektor $\vec{x}$ lässt sich mithilfe von Vektorrechnung ebenfalls sehr einfach umsetzen. Es gilt (mit $\overrightarrow{OP}$ als Ortsvektor von $P$):
+\begin{equation}
+\overrightarrow{OP'} = \overrightarrow{OP} + \vec{x}
+\end{equation}
+\textbf{Rotieren eines Dreiecks an seine endgültige Position} \quad Sobald ein Dreieck so verschoben wurde, dass der Optimalpunkt auf der x-Achse liegt, muss es noch so um den Optimalpunkt gedreht werden, dass es an seine richtige Position gelangt und sich mit keinem anderen Dreieck überlappt. Das erfolgt in drei Schritten:
+\begin{enumerate}
+	\item \textit{Drehen auf die x-Achse}: Das Dreieck wird so gedreht, dass eine Seite auf der postiven x-Achse liegt und der Rest des Dreiecks über der x-Achse. Dazu wird jeder Punkt um folgenden Winkel (gegen den Uhrzeigersinn) gedreht.\footnote{atan2 - Dein Freund und Helfer!}
+	\begin{equation}
+	\beta = 360\degree - atan2(\Delta y, \Delta x)
+	\end{equation}
+	$\Delta y$ und $\Delta x$ beziehen sich auf den Vektor zwischen dem Optimalpunkt $A$ und dem Punkt $P$. $P$ ist dabei der Punkt, der im Uhrzeigersinn bezüglich des Vektors aus dem Optimalpunkt und dem dritten Punkt des Dreiecks liegt.
+	\item \textit{Überlappungen beseitigen}: Nach dem ersten Schritt überlappen sich die Dreiecke noch. Um dies zu beseitigen, rotiert man das Dreieck um die Summe aller $\varphi_i$ der Dreiecke des Subsets, die in der Längensortierung vor ihm kommen, also um
+	\begin{equation}
+	\beta_i = \sum_{d \in S, d < i} \varphi_d
+	\end{equation}
+	Fügt man links an, ändert sich dieser Ausdruck auf:
+	\begin{equation}
+	\beta_i = 180\degree - \sum_{d \in S, d \leq i} \varphi_d
+	\end{equation}
+	\item \textit{Herandrehen an die bisherige Anordnung}: Dazu dreht man alle Dreiecke des Subsets nun um folgenden Winkel:
+	\begin{equation}
+	\beta = \alpha - \sum_{d \in S} \varphi_d
+	\end{equation}
+	wobei $\alpha$ der freie Winkel ist und die Summe dem Winkel, der von diesem Subset insgesamt belegt wird, entspricht. Auf der linken Seite rotiert man das Dreieck um diesen Winkel in die andere Richtung (im Uhrzeigersinn).
+\end{enumerate}
+\begin{figure}[H]
+	\centering \begin{tikzpicture}
+	\coordinate (A) at (2,0);
+	\coordinate (B) at (4,0);
+	\coordinate (C) at (6,0);
+	\draw (4,0)
+	-- (6,0)
+	-- (4,2)
+	-- cycle;
+	\draw (4,0)
+	-- (2,0)
+	-- (4,-2)
+	-- cycle;
+	\draw[->] (0,0) 
+	-> (7,0) node[right]{x};
+	\path (A) -- (B) -- (C) pic[draw, ->, "\quad$\beta$", angle eccentricity=1.5] {angle = A--B--C};
+	\end{tikzpicture}
+	\caption{Berechnung der Drehwinkel, Schritt 1}
+\end{figure}
+\begin{figure}[H]
+	\centering \begin{tikzpicture}
+	\coordinate (A) at (6,0);
+	\coordinate (B) at (4,0);
+	\coordinate (C) at (6,1);
+	\draw (4,0)
+	-- (6,0)
+	-- (6,1)
+	-- cycle;
+	\draw (4,0)
+	-- (6,1)
+	-- (4,2)
+	-- cycle;
+	\draw[->] (0,0) 
+	-> (7,0) node[right]{x};
+	\path (A) -- (B) -- (C) pic[draw, ->, "\quad$\beta$", angle eccentricity=1.5] {angle = A--B--C};
+	\end{tikzpicture}
+	\caption{Berechnung der Drehwinkel, Schritt 2}
+\end{figure}
+\begin{figure}[H]
+	\centering \begin{tikzpicture}
+	\coordinate (A) at (4,2);
+	\coordinate (B) at (4,0);
+	\coordinate (C) at (2,2);
+	\draw (4,0)
+	-- (6,0)
+	-- (4,2)
+	-- cycle;
+	\draw[dashed] (4,0)
+	-- (0,0)
+	-- (0,2)
+	-- (2,2)
+	-- cycle;
+	\draw[->] (0,0) 
+	-> (7,0) node[right]{x};
+	\path (A) -- (B) -- (C) pic[draw, ->, "$\beta$", angle eccentricity=1.5] {angle = A--B--C};
+	\end{tikzpicture}
+	\caption{Berechnung der Drehwinkel, Schritt 3}
+\end{figure}
   \subsubsection{Der DP-Algorithmus für Subset Sum}
   Um anhand der Winkel der Dreiecke die jeweils im freien Winkel (anfangs z.B. in einem Halbkreis) zu platzierenden Dreiecke zu ermitteln, muss man das Subset-Sum-Problem (Teilsummenproblem) lösen. Dabei ist eine Menge von ganzen Zahlen $I = \{w_1,...,w_n\} (w_i \in \mathbb{Z})$ gegeben. Nun wird eine Teilmenge $S$ mit maximaler Summe gesucht, die aber nicht größer als eine obere Schranke $c$ (in diesem Fall z.B. $180\degree$ bzw. $\pi$) ist. Formal erfüllt $S$ also folgende Eigenschaften:
   \begin{equation}
@@ -322,16 +450,16 @@ \subsubsection{Geometrische Berechnungen}
   \sum_{w_j \in S} \leq c
   \end{equation}
   Leider ist das Subset-Sum-Problem aber NP-vollständig und somit grundsätzlich nicht effizient lösbar. Ist $c$ jedoch klein genug, existiert ein Dynamic-Programming-Algorithmus zur Lösung des Problems in pseudopolynomieller Zeit.\cite{Src:dpsum} Dazu lässt sich eine DP-Funktion definieren, die mittels einer DP-Tabelle effizient berechnet werden kann:
-  \begin{equation}
+  \begin{equation*}
   dp(i,sum) = 
   \begin{cases}
-  true & sum = 0 \vee (i = 0 \wedge sum = w_1) \\
+  true & sum = 0 \\
   false & sum > 0 \wedge i = 0 \\
   dp(i-1,sum) \vee dp(i-1,sum-w_i) & \, \text{sonst}
   \end{cases}
-  \end{equation}
+  \end{equation*}
   Diese Funktion gibt an, ob sich eine Summe von $sum$ mit den ersten $i$ Elementen der Menge erreichen lässt. Die Funktion baut darauf auf, dass es an jeder Stelle genau zwei Möglichkeiten gibt: Entweder das aktuelle Element wird nicht in das Subset aufgenommen (dann muss man die Summe mit den anderen $i-1$ Elementen erreichen) oder das Element wird in das Subset aufgenommen (dann muss man mit $i-1$ Elementen nur noch eine um $w_i$ verringerte Summe erreichen). \\ \\
-  Nun gibt $dp(n-1,c)$ an, ob es möglich ist, mit allen Elementen die Summe $c$ zu erreichen. Doch da diese Summe oft nicht exakt erreicht werden kann, muss man $c$ entsprechend oft dekrementieren, bis $dp(n-1,c) = true$ ist und es möglich ist, diese Summe zu erreichen. \\ \\
+  Nun gibt $dp(n,c)$ an, ob es möglich ist, mit allen Elementen die Summe $c$ zu erreichen. Doch da diese Summe oft nicht exakt erreicht werden kann, muss man $c$ entsprechend oft dekrementieren, bis $dp(n,c) = true$ ist und es möglich ist, diese Summe zu erreichen. \\ \\
   Um aus der DP-Tabelle nun das eigentliche Subset zu erhalten, muss man die Lösung backtracen.\footnote{Hiermit ist das Rückverfolgen einer Lösung bei DP-Algorithmen gemeint und nicht Backtracking.}\cite{Src:dpbacktrace} Dabei betrachtet man für jedes Element $w_i$ einerseits die Möglichkeit, dass das Element enthalten ist, und andererseits, dass das Element nicht enthalten ist. Wenn eine dieser Möglichkeiten laut DP-Tabelle möglich ist, kann man rekursiv eine Lösung für das entsprechende Feld für $i-1$ generieren und dann das aktuelle Element anhängen oder nicht (je nachdem). Am Ende erhält man dann ein mögliches Subset.
   \\ \\
   Da man eine DP-Tabelle mit $n \cdot c$ Elementen ausfüllt, ergibt sich für diesen Algorithmus somit auch eine Laufzeit in $\mathcal{O}(n \cdot c)$, also in pseudopolynomieller Zeit.
@@ -339,15 +467,49 @@ \subsubsection{Geometrische Berechnungen}
   Mittels dieses Subset-Sum-Algorithmus ist es möglich, Dreiecke so auszuwählen, dass ihre kleinsten Winkel $\varphi_i$ einen gegebenen freien Winkel (z.B. den Halbkreis oberhalb der x-Achse) möglichst gut ausnutzen. Dadurch können die Dreiecke relativ dicht gepackt und der Gesamtabstand verkleinert werden. Entsprechend sind für diese Aufgabe die $w_i = \varphi_i$. \\ \\
    Da es sich bei den Winkeln in der Realität jedoch um Gleitkommazahlen handelt, müssen diese zunächst in Festkommazahlen mit wenigen Nachkommastellen umgewandelt und anschließend mit einem festen Faktor (eine entsprechende Zehnerpotenz) multipliziert werden, damit man natürliche Zahlen erhält, die vom Algorithmus verarbeitet werden können.
   \subsubsection{Der Algorithmus zur Platzierung der Dreiecke}
-  Die in den vorherigen Abschnitten gewonnenen Erkenntnisse können in folgendem Algorithmus zusammengefasst werden:
-  %TODO richtigen Algorithmus einfügen
+  Die in den vorherigen Abschnitten gewonnenen Erkenntnisse können in folgendem Algorithmus zusammengefasst werden. Dabei gibt $\{\varphi_T\}$ die Menge aller $\varphi_i$ der Dreiecksmenge $T$ an.
   \begin{algorithm}[H]
+  	\caption{Algorithmus zur Dreiecksplatzierung (Teil 1)}
   	\begin{algorithmic}
-  		\Function{doAlgorithm}{$T$}
-  		\State \Return $42$
+  		\Function{doAlgorithm}{$D$}
+  		\State $T_{pl} \gets \emptyset$
+  		\ForAll{$d_i \in D$}
+  		\State Berechne den Optimalpunkt und $\varphi_i$
+  		\EndFor
+  		\State $S \gets$ \Call{subsetSum}{$\{\varphi_T\},180\degree$}
+  		\Comment{allererstes Subset (kann an beliebigem Punkt platziert werden)}
+  		\State Sortiere $S$ nach längster anliegender Seite
+  		\ForAll{$d_i \in S$}
+  		\State Verschiebe und rotiere $d_i$ wie beschrieben
+  		\EndFor
+  		\State $T \gets T\setminus S$ ; $T_{pl} \gets T_{pl} \cup S$
+  		\While{T $\neq \emptyset$}
+  		\State Finde die möglichen Anlegepunkte $L$ und $R$ sowie die Abstände $l$ und $r$
+  		\State Berechne die freien Winkel $\alpha_L$ und $\alpha_R$
+  		\If{$l < r$}
+  		\State $\alpha = \alpha_L$
+  		\Else
+  		\State $\alpha = \alpha_R$
+  		\EndIf
+  		\algstore{myalg}
+  	\end{algorithmic}
+\end{algorithm}
+\begin{algorithm}[H]
+	\ContinuedFloat
+	\begin{algorithmic}
+		\algrestore{myalg}
+  		\State $S \gets$ \Call{subsetSum}{$\{\varphi_T\},\alpha$}
+  		\State Sortiere $S$ nach längster anliegender Seite
+  		\ForAll{$d_i \in S$}
+  		\State Verschiebe und rotiere $d_i$ wie beschrieben
+  		\Comment{je nachdem links oder rechts}
+  		\EndFor
+  		\State $T \gets T\setminus S$ ; $T_{pl} \gets T_{pl} \cup S$
+  		\EndWhile
+  		\State \Return $T_{pl}$
   		\EndFunction
   	\end{algorithmic}
-  	\caption{Algorithmus zur Dreiecksplatzierung}
+  	\caption{Algorithmus zur Dreiecksplatzierung (Teil 2)}
   \end{algorithm}
   \subsection{Optimalität des Algorithmus und Verbesserungsmöglichkeiten}
   Bezüglich der Qualität des Verfahrens lässt sich feststellen, dass es viele typische Nachteile einer Heuristik besitzt. So muss die von diesem Verfahren erzeugte Dreiecksanordnung nicht unbedingt optimal sein. Um das zu erreichen, müsste man jedoch alle möglichen Lösungen mittels Backtracking durchprobieren. \\ \\
@@ -364,7 +526,7 @@ \subsubsection{Geometrische Berechnungen}
   	\item Theoretisch könnte es nicht nur ein Subset mit der maximalen Summe, sondern auch mehrere geben. Da sich die einzelnen ausgewählten Dreiecke in den Seitenlängen unterscheiden können, kann bei einem anderen Subset möglicherweise auch ein anderer Gesamtabstand herauskommen. Für eine bessere Lösung könnte man also alle möglichen maximalen Subsets ausprobieren. Da dadurch für das Backtracen jedoch im Extremfall eine exponentielle Laufzeit in $\mathcal{O}(2^n)$ entstehen könnte, habe ich dies nicht implementiert.
   	\item Im Algorithmus werden bisher keine Achsenspiegelungen betrachtet. Dadurch lässt sich jedoch möglicherweise ein besseres Ergebnis erzeugen. Insbesondere kann es nötig sein, ein Dreieck an der Seitenhalbierenden, welche durch den Optimalpunkt verläuft, zu spiegeln. \\ \\ Dadurch lässt sich in dem unten dargestellten Fall die längere Seite des schraffierten Dreiecks nach oben spiegeln und die kürzere nach unten. Gegebenenfalls lässt sich dadurch ein größerer freier Winkel für den nächsten Anlegeschritt erreichen. Die so erreichten Vorteile haben jedoch nur einen sehr geringen Einfluss, weshalb auch dieser Schritt nicht implementiert wurde.
   	\item Teilweise füllt der Algorithmus nicht alle Lücken zwischen Dreiecken auf. Dies ist insbesondere im unten dargestellten Beispiel zu sehen. Die gepunktete Lücke enthält kein Dreieck, obwohl dort Platz für eines wäre. Dies ist durch den verwendeten Algorithmus bedingt, welcher im Subset-Sum-Schritt die Seitenlängen der Dreiecke nicht berücksichtigt. So kann es vorkommen, dass ein Dreieck mit dem gleichen kleinsten Winkel wie das hier platzierte so lange Seiten hat, dass es überhaupt nicht in diese Lücke passen würde. Aus diesem Grund werden solche Lücken nicht aufgefüllt, was verbesserungswürdig ist.
-  	\item Es wäre denkbar, andere Arten von Heuristiken verwenden, insbesondere \textit{evolutionäre Algorithmen} oder \textit{selbstlernende neuronale Netzwerke}. Damit ließen sich möglicherweise noch bessere Ergebnisse erzielen. Diese beiden Arten von Heuristiken sind jedoch in etwa das Äquivalent von Magie in der Informatik. Man kann weder Aussagen darüber treffen, ob und warum die Heuristik vermutlich gute Ergebnisse produziert noch darüber, was während der Ausführung des Algorithmus genau passiert oder welche Laufzeit er (in Landau-Notation) hat. \\ \\
+  	\item Es wäre denkbar, andere Arten von Heuristiken verwenden, insbesondere \textit{evolutionäre Algorithmen} oder \textit{selbstlernende neuronale Netzwerke}. Damit ließen sich möglicherweise noch bessere Ergebnisse erzielen. Diese beiden Arten von Heuristiken sind jedoch in etwa das Äquivalent von Magie in der Informatik. Man kann weder Aussagen darüber treffen, ob und warum die Heuristik vermutlich gute Ergebnisse produziert noch darüber, was während der Ausführung des Algorithmus genau passiert oder welche Laufzeit er (in O-Notation) hat. \\ \\
   	Zudem variieren die Ergebnisse solcher Heuristiken zufällig abhängig von den verwendeten Startwerten. Da ich mit dieser Einsendung nicht am trimagischen Turnier, sondern am Bundeswettbewerb Informatik teilnehmen will, habe ich mich gegen die Implementierung einer solchen Heuristik entschieden.
   \end{itemize}
 \begin{minipage}{0.45\linewidth}
@@ -429,7 +591,7 @@ \subsubsection{Geometrische Berechnungen}
  \begin{equation}
 T(n) = \mathcal{O}(n) + \mathcal{O}(n \cdot \log n) + \mathcal{O}(n) = \mathcal{O}(n \cdot \log n)
 \end{equation}
-Nehmen wir an, dass es $m$ Schleifendurchläufe gibt, erhalten wir $\mathcal{O}(m \cdot n \cdot \log n)$. Da die Anzahl der Schleifendurchläufe von der jeweiligen Rückgabe des Subset-Sum-Algorithmus abhängt, kann man nur feststellen, dass $m = \mathcal{O}(n)$ und $m = \Omega(1)$ ist.\footnote{Die geringste Anzahl tritt auf, wenn die Summe aller kleinsten Winkel $\leq 180\degree$ ist.} \\ \\
+Nehmen wir an, dass es $m$ Schleifendurchläufe gibt, erhalten wir $\mathcal{O}(m \cdot n \cdot \log n)$. Da die Anzahl der Schleifendurchläufe von der jeweiligen Rückgabe des Subset-Sum-Algorithmus abhängt, kann man nur feststellen, dass $m = \mathcal{O}(n)$ und $m = \Omega(1)$ ist.\footnote{Die geringste Anzahl tritt auf, wenn die Summe aller kleinsten Winkel $\leq 180\degree$ ist.} \\
 Insgesamt erhält man für die Laufzeit also:\footnote{Da das $n$ nicht für jede Iteration der Schleife gleich ist, sondern kleiner wird, handelt es sich bei der Laufzeit in O-Notation nur um eine grobe Abschätzung nach oben. Die Laufzeit wird also in den meisten Fällen eher $\mathcal{O}(n \cdot \log n)$ sein.}
 \begin{equation}
 T_{gesamt}(n) = \mathcal{O}(n^2 \cdot \log n) \text{~und~} T_{gesamt}(n) = \Omega(n \cdot \log n)
@@ -437,7 +599,7 @@ \subsubsection{Geometrische Berechnungen}
  Eine Frage, die sich hierbei auch stellt, ist diejenige, ob es für dieses Problem einen in Polynomialzeit terminierenden Algorithmus geben kann, der eine optimale Lösung liefert. Dies entspricht der Frage, ob das Problem in der Klasse $NPC$\footnote{Genaugenommen ist diese Klasse nur für Entscheidungsprobleme definiert, daher handelt es sich bei diesem Suchproblem um NP-Äquivalenz.} (NP-vollständig bzw. NP-complete) liegt. Ich vermute, dass dies der Fall ist, kann es jedoch nicht beweisen. \\ \\
  Zum Beweis, dass ein Problem in $NPC$ liegt, werden zwei Voraussetzungen benötigt:
  \begin{enumerate}
- 	\item Eine deterministisch arbeitende Turingmaschine benötigt nur Polynomialzeit, um zu entscheiden, ob eine z.B. von einer Orakel-Turingmaschine vorgeschlagene Lösung tatsächlich eine Lösung des Problems ist. Dies ist hier der Fall, denn wenn eine Lösung vorgeschlagen wird, kann man in Polynomialzeit überprüfen, ob es sich dabei um eine erlaubte Dreiecksanordnung handelt. \\ \\ Dazu überprüft man alle vier Bedingungen dafür. Die ersten beiden Bedingungen lassen sich einfach für jedes Dreieck in konstanter Zeit, insgesamt also in $\mathcal{O}(n)$, überprüfen. Für die dritte Bedingung (Kongruenz) ist dies mithilfe von Kongruenzsätzen ebenfalls in linearer Zeit möglich. Bei der vierte Bedingung  (keine Überlappung) muss man alle Dreieckspaare, insgesamt also $\mathcal{O}(n^2)$, auf Überlappung überprüfen. Insgesamt erhält man mit $\mathcal{O}(n^2)$ also Polynomialzeit.
+ 	\item Eine deterministisch arbeitende Turingmaschine benötigt nur Polynomialzeit, um zu entscheiden, ob eine z.B. von einer Orakel-Turingmaschine vorgeschlagene Lösung tatsächlich eine Lösung des Problems ist. Dies ist hier der Fall, denn wenn eine Lösung vorgeschlagen wird, kann man in Polynomialzeit überprüfen, ob es sich dabei um eine erlaubte Dreiecksanordnung handelt. \\ \\ Dazu überprüft man alle vier Bedingungen dafür. Die ersten beiden Bedingungen lassen sich einfach für jedes Dreieck in konstanter Zeit, insgesamt also in $\mathcal{O}(n)$, überprüfen. Für die dritte Bedingung (Kongruenz) ist dies mithilfe von Kongruenzsätzen ebenfalls in linearer Zeit möglich. Bei der vierten Bedingung  (keine Überlappung) muss man alle Dreieckspaare, insgesamt also $\mathcal{O}(n^2)$, auf Überlappung überprüfen. Insgesamt erhält man mit $\mathcal{O}(n^2)$ also Polynomialzeit.
  	\item Das Problem ist NP-schwer. Das bedeutet, dass alle anderen NP-schweren Probleme auf dieses Problem in Polynomialzeit zurückgeführt werden können. Es ist also eine Polynomialzeitreduktion notwendig. Dabei ist ein Problem aus NPC als Ausgangsproblem nötig, wie z.B. 3-Satisfiability. Eine solche Reduktion zu vollziehen, ist mir jedoch nicht möglich.\footnote{Auch wenn eine Beziehung zwischen \textit{3}-SAT und \textit{Drei}ecken natürlich naheliegt.}
  \end{enumerate}
 Abschließend möchte ich bemerken, dass das Problem viel einfacher zu lösen wäre, wenn sich die Trianguläre einfach für rechteckige Grundstücksformen entscheiden würden. Bei diesen ließe sich einfach die kürzere Seite an die Küstenstraße anlegen. Die Beschreibung der Trianguläre als \glqq seltsam\grqq~ist also offensichtlich gerechtfertigt.
@@ -466,9 +628,9 @@ \subsubsection{Die Datei \texttt{triangles.cpp}}
 	\item Die \texttt{Vektor}-Klasse, die einen Vektor in ebendiesem Koordinatensystem repräsentiert.\footnote{Hier wurde nicht die englische Bezeichnung verwendet, um Verwechslungen mit der STL-Klasse \texttt{vector} auszuschließen.} Außerdem gibt es eine Funktion, die den Betrag des Vektors berechnet.
 	\item Die \texttt{Triangle}-Klasse, die ein Dreieck repräsentiert. Dabei werden eine ID für die Ausgabe, die drei Eckpunkte sowie die Vektoren zwischen ihnen gespeichert. Weiterhin kann mittels eine Funktion die längste an einem Punkt des Dreieck anliegende Seite bestimmt werden, was für das Sortieren der Dreiecke vor der Platzierung notwendig ist.
 \end{itemize}
-Weiterhin gibt es folgende Methoden:
-\begin{longtabu} to \linewidth {lX}
-	Funktion & Beschreibung \\ \hline \hline \endhead
+Weiterhin gibt es folgende Methoden: \\
+{\begin{tabu} to \linewidth {lX}
+	Funktion & Beschreibung \\ \hline
 	\texttt{addVektor()} & Addiert einen Vektor zu einem Punkt. \\ \hline
 	\texttt{dotProduct()} & Berechnet das Skalarprodukt zweier Vektoren. ($a \cdot b = a.x \cdot b.x + a.y \cdot b.y$) \\ \hline
 	\texttt{angle()} & Berechnet den Winkel zwischen zwei Vektoren wie in der Lösungsidee beschrieben.\\ \hline
@@ -476,12 +638,12 @@ \subsubsection{Die Datei \texttt{triangles.cpp}}
 	\texttt{rotate\_tri()} & Rotiert einen Punkt $p$ um einen Winkel $\alpha$ um das Drehzentrum $c$. (siehe Lösungsidee)\\ \hline
 	\texttt{atan\_angle()}& Berechnet den Winkel, den ein Vektor zwischen zwei Punkten zur positiven x-Achse besitzt. \texttt{atan180()} und \texttt{atan360()} sind Shortcuts für das Subtrahieren des Winkels von 180 bzw. 360 Grad. \\ \hline
 	\texttt{ccw()} & Berechnet, ob ein Punkt gegen den Uhrzeigersinn bezüglich des Vektors zwischen zwei anderen Punkten liegt (siehe Lösungsidee).\\ \hline
-	\texttt{findAngleCalcPoint()} & Findet den Punkt, der für die Bestimmung des Winkels zur x-Achse beim Drehen maßgeblich ist. Dies ist immer der Punkt, der gegen Uhrzeigersinn bezüglich des Vektors aus dem Optimalpunkt und dem dritten Punkt des Dreiecks liegt.\\
-\end{longtabu}
+	\texttt{findAngleCalcPoint()} & Findet den Punkt, der für die Bestimmung des Winkels zur x-Achse beim Drehen maßgeblich ist. Dies ist immer der Punkt, der im Uhrzeigersinn bezüglich des Vektors aus dem Optimalpunkt und dem dritten Punkt des Dreiecks liegt.\\
+\end{tabu}}
 \subsubsection{Die Datei \texttt{triangleAlgorithm.cpp}}
-\texttt{triangleAlgorithm.cpp} enthält den eigentlichen Algorithmus, der aus folgenden Methoden besteht:
-\begin{longtabu} to \linewidth {lX}
-	Funktion & Beschreibung \\ \hline \hline \endhead
+\texttt{triangleAlgorithm.cpp} enthält den eigentlichen Algorithmus, der aus folgenden Methoden besteht: \\ \\
+{\begin{tabu} to \linewidth {lX}
+	Funktion & Beschreibung \\ \hline
 	\texttt{getSubsetsRec()} & Ist für das Backtracen des maximalen Subsets aus dem DP-Array verantwortlich. \\ \hline
 	\texttt{subsetSum()} & Berechnet die Werte des DP-Arrays nach der in der Lösungsidee angegebenen Rekursionsgleichung. \\ \hline
 	\texttt{lengthSortFunc()} & Wird als Vergleichsfunktion beim Sortieren der Dreiecke eines Subsets verwendet. Dabei wird die Länge der längsten am Optimalpunkt anliegenden Seite verglichen.\\ \hline
@@ -492,7 +654,7 @@ \subsubsection{Die Datei \texttt{triangleAlgorithm.cpp}}
 	\texttt{calculateDistance()} & Berechnet den Gesamtabstand der Dreiecksanordnung. Dabei werden der linkeste und der rechteste Punkt auf der x-Achse, die für die Berechnung zählen, bestimmt.\\ \hline
 	\texttt{moveToRightOfY()} & Verschiebt die Anordnung so, dass der linkeste Punkt genau auf der y-Achse liegt.\\ \hline
 	\texttt{doAlgorithm()} & Setzt den in der Lösungsidee vorgeschlagenen Algorithmus um, wobei die Dreiecksanordnung vom willkürlich gewählten Punkt $(300|0)$ aufgebaut wird.\\
-\end{longtabu}
+\end{tabu}}
 Außerdem werden in dieser Datei mehrere globale Listen (\texttt{vector}) benutzt. Dies sind:
 \begin{itemize}
 	\item \texttt{bestPointIndex}: Speichert für jedes Dreieck den Index des Optimalpunktes in der internen Punkteliste des Dreiecks.
diff --git a/Aufgabe2-Dokumentation/Aufgabe2-Dokumentation.toc b/Aufgabe2-Dokumentation/Aufgabe2-Dokumentation.toc
index 2d11cc5..162ea24 100644
--- a/Aufgabe2-Dokumentation/Aufgabe2-Dokumentation.toc
+++ b/Aufgabe2-Dokumentation/Aufgabe2-Dokumentation.toc
@@ -6,20 +6,20 @@
 \contentsline {subsubsection}{\numberline {1.3.1}Beobachtungen bezüglich einer guten Lösung}{3}{subsubsection.1.3.1}% 
 \contentsline {subsubsection}{\numberline {1.3.2}Implementierte Verbesserungen}{4}{subsubsection.1.3.2}% 
 \contentsline {subsubsection}{\numberline {1.3.3}Geometrische Berechnungen}{5}{subsubsection.1.3.3}% 
-\contentsline {subsubsection}{\numberline {1.3.4}Der DP-Algorithmus für Subset Sum}{5}{subsubsection.1.3.4}% 
-\contentsline {subsubsection}{\numberline {1.3.5}Der Algorithmus zur Platzierung der Dreiecke}{6}{subsubsection.1.3.5}% 
-\contentsline {subsection}{\numberline {1.4}Optimalität des Algorithmus und Verbesserungsmöglichkeiten}{7}{subsection.1.4}% 
-\contentsline {subsection}{\numberline {1.5}Laufzeitbetrachtung und NP-Vollständigkeit}{8}{subsection.1.5}% 
-\contentsline {section}{\numberline {2}Umsetzung}{10}{section.2}% 
-\contentsline {subsection}{\numberline {2.1}Allgemeine Hinweise zur Benutzung}{10}{subsection.2.1}% 
-\contentsline {subsection}{\numberline {2.2}Struktur des Programms und Implementierung der Algorithmen}{11}{subsection.2.2}% 
-\contentsline {subsubsection}{\numberline {2.2.1}Die Datei \texttt {main.cpp}}{11}{subsubsection.2.2.1}% 
-\contentsline {subsubsection}{\numberline {2.2.2}Die Datei \texttt {triangles.cpp}}{11}{subsubsection.2.2.2}% 
-\contentsline {subsubsection}{\numberline {2.2.3}Die Datei \texttt {triangleAlgorithm.cpp}}{12}{subsubsection.2.2.3}% 
-\contentsline {section}{\numberline {3}Beispiele}{13}{section.3}% 
-\contentsline {subsection}{\numberline {3.1}Beispiel 1}{13}{subsection.3.1}% 
-\contentsline {subsection}{\numberline {3.2}Beispiel 2}{14}{subsection.3.2}% 
-\contentsline {subsection}{\numberline {3.3}Beispiel 3}{15}{subsection.3.3}% 
-\contentsline {subsection}{\numberline {3.4}Beispiel 4}{15}{subsection.3.4}% 
-\contentsline {subsection}{\numberline {3.5}Beispiel 5}{16}{subsection.3.5}% 
-\contentsline {section}{\numberline {4}Quellcode}{17}{section.4}% 
+\contentsline {subsubsection}{\numberline {1.3.4}Der DP-Algorithmus für Subset Sum}{8}{subsubsection.1.3.4}% 
+\contentsline {subsubsection}{\numberline {1.3.5}Der Algorithmus zur Platzierung der Dreiecke}{9}{subsubsection.1.3.5}% 
+\contentsline {subsection}{\numberline {1.4}Optimalität des Algorithmus und Verbesserungsmöglichkeiten}{10}{subsection.1.4}% 
+\contentsline {subsection}{\numberline {1.5}Laufzeitbetrachtung und NP-Vollständigkeit}{12}{subsection.1.5}% 
+\contentsline {section}{\numberline {2}Umsetzung}{13}{section.2}% 
+\contentsline {subsection}{\numberline {2.1}Allgemeine Hinweise zur Benutzung}{13}{subsection.2.1}% 
+\contentsline {subsection}{\numberline {2.2}Struktur des Programms und Implementierung der Algorithmen}{14}{subsection.2.2}% 
+\contentsline {subsubsection}{\numberline {2.2.1}Die Datei \texttt {main.cpp}}{14}{subsubsection.2.2.1}% 
+\contentsline {subsubsection}{\numberline {2.2.2}Die Datei \texttt {triangles.cpp}}{14}{subsubsection.2.2.2}% 
+\contentsline {subsubsection}{\numberline {2.2.3}Die Datei \texttt {triangleAlgorithm.cpp}}{15}{subsubsection.2.2.3}% 
+\contentsline {section}{\numberline {3}Beispiele}{16}{section.3}% 
+\contentsline {subsection}{\numberline {3.1}Beispiel 1}{16}{subsection.3.1}% 
+\contentsline {subsection}{\numberline {3.2}Beispiel 2}{17}{subsection.3.2}% 
+\contentsline {subsection}{\numberline {3.3}Beispiel 3}{18}{subsection.3.3}% 
+\contentsline {subsection}{\numberline {3.4}Beispiel 4}{18}{subsection.3.4}% 
+\contentsline {subsection}{\numberline {3.5}Beispiel 5}{19}{subsection.3.5}% 
+\contentsline {section}{\numberline {4}Quellcode}{20}{section.4}% 
diff --git a/Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/7B2F7C6D41DCE6772411798425F1F75884FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex b/Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/B09D3170810788694402390E3C916A2284FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex
similarity index 99%
rename from Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/7B2F7C6D41DCE6772411798425F1F75884FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex
rename to Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/B09D3170810788694402390E3C916A2284FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex
index 870caf8..df08aa4 100644
--- a/Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/7B2F7C6D41DCE6772411798425F1F75884FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex
+++ b/Aufgabe2-Dokumentation/_minted-Aufgabe2-Dokumentation/B09D3170810788694402390E3C916A2284FD9B1CDA6DC1D421A4ED8AF8FDFE8A.pygtex
@@ -177,7 +177,7 @@
 \PYG{p}{\PYGZcb{}}
 
 \PYG{c+c1}{// Lage Punkt c von ab aus}
-\PYG{c+c1}{// CCW: \PYGZlt{} 0, CW: \PYGZgt{}0}
+\PYG{c+c1}{// CCW: \PYGZgt{} 0, CW: \PYGZlt{} 0}
 \PYG{k+kt}{double} \PYG{n}{ccw}\PYG{p}{(}\PYG{n}{Point} \PYG{n}{a}\PYG{p}{,} \PYG{n}{Point} \PYG{n}{b}\PYG{p}{,} \PYG{n}{Point} \PYG{n}{c}\PYG{p}{)\PYGZob{}}
     \PYG{k}{return} \PYG{p}{(}\PYG{n}{b}\PYG{p}{.}\PYG{n}{x} \PYG{o}{\PYGZhy{}} \PYG{n}{a}\PYG{p}{.}\PYG{n}{x}\PYG{p}{)} \PYG{o}{*} \PYG{p}{(}\PYG{n}{c}\PYG{p}{.}\PYG{n}{y} \PYG{o}{\PYGZhy{}} \PYG{n}{a}\PYG{p}{.}\PYG{n}{y}\PYG{p}{)} \PYG{o}{\PYGZhy{}} \PYG{p}{(}\PYG{n}{b}\PYG{p}{.}\PYG{n}{y} \PYG{o}{\PYGZhy{}} \PYG{n}{a}\PYG{p}{.}\PYG{n}{y}\PYG{p}{)} \PYG{o}{*} \PYG{p}{(}\PYG{n}{c}\PYG{p}{.}\PYG{n}{x} \PYG{o}{\PYGZhy{}} \PYG{n}{a}\PYG{p}{.}\PYG{n}{x}\PYG{p}{);}
 \PYG{p}{\PYGZcb{}}
@@ -185,29 +185,29 @@
 \PYG{c+c1}{// Punkt finden, von dem aus Drehwinkel bestimmt wird}
 \PYG{c+c1}{// (beim Drehen auf die x\PYGZhy{}Achse)}
 \PYG{k+kt}{int} \PYG{n}{findAngleCalcPoint}\PYG{p}{(}\PYG{n}{Triangle} \PYG{o}{\PYGZam{}}\PYG{n}{t}\PYG{p}{,} \PYG{k+kt}{int} \PYG{n}{bestPoint}\PYG{p}{)\PYGZob{}}
-    \PYG{c+c1}{//just return the point that is counterclockwise from line between other points}
+    \PYG{c+c1}{//just return the point that is clockwise from line between other points}
     \PYG{k}{switch}\PYG{p}{(}\PYG{n}{bestPoint}\PYG{p}{)} \PYG{p}{\PYGZob{}}
         \PYG{k}{case} \PYG{l+m+mi}{0}\PYG{o}{:} \PYG{k}{if}\PYG{p}{(}\PYG{n}{ccw}\PYG{p}{(}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{2}\PYG{p}{])} \PYG{o}{\PYGZgt{}} \PYG{l+m+mi}{0}\PYG{p}{)\PYGZob{}}
-                    \PYG{c+c1}{//point 2 is clockwise}
+                    \PYG{c+c1}{//point 2 is counterclockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{1}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}} \PYG{k}{else} \PYG{p}{\PYGZob{}}
-                    \PYG{c+c1}{//point 2 is counterclockwise}
+                    \PYG{c+c1}{//point 2 is clockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{2}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}}
                 \PYG{k}{break}\PYG{p}{;}
         \PYG{k}{case} \PYG{l+m+mi}{1}\PYG{o}{:} \PYG{k}{if}\PYG{p}{(}\PYG{n}{ccw}\PYG{p}{(}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{2}\PYG{p}{])} \PYG{o}{\PYGZgt{}} \PYG{l+m+mi}{0}\PYG{p}{)\PYGZob{}}
-                    \PYG{c+c1}{//point 2 is clockwise}
+                    \PYG{c+c1}{//point 2 is counterclockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{0}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}} \PYG{k}{else} \PYG{p}{\PYGZob{}}
-                    \PYG{c+c1}{//point 2 is counterclockwise}
+                    \PYG{c+c1}{//point 2 is clockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{2}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}}
                 \PYG{k}{break}\PYG{p}{;}
         \PYG{k}{case} \PYG{l+m+mi}{2}\PYG{o}{:} \PYG{k}{if}\PYG{p}{(}\PYG{n}{ccw}\PYG{p}{(}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{2}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{],}\PYG{n}{t}\PYG{p}{.}\PYG{n}{points}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{])} \PYG{o}{\PYGZgt{}} \PYG{l+m+mi}{0}\PYG{p}{)\PYGZob{}}
-                    \PYG{c+c1}{//point 0 is clockwise}
+                    \PYG{c+c1}{//point 0 is counterclockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{1}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}} \PYG{k}{else} \PYG{p}{\PYGZob{}}
-                    \PYG{c+c1}{//point 0 is counterclockwise}
+                    \PYG{c+c1}{//point 0 is clockwise}
                     \PYG{k}{return} \PYG{l+m+mi}{0}\PYG{p}{;}
                 \PYG{p}{\PYGZcb{}}
                 \PYG{k}{break}\PYG{p}{;}
diff --git a/Aufgabe2-Dokumentation/svg-inkscape/dreiecke1_svg-tex.pdf b/Aufgabe2-Dokumentation/svg-inkscape/dreiecke1_svg-tex.pdf
index a5ae99566469a54d06f1a9bdf0541b1c2953f5ce..f1af4a54aaed6a6f8a81901ff65cfa020ab64f88 100644
GIT binary patch
delta 20
bcmdnNv4dm7FD7;iLrY^zLxatX%;t;$N1Fwb

delta 20
bcmdnNv4dm7FD7<#Lo*XI1GCMH%;t;$M}Gy8

diff --git a/Aufgabe2-Dokumentation/svg-inkscape/dreiecke2_svg-tex.pdf b/Aufgabe2-Dokumentation/svg-inkscape/dreiecke2_svg-tex.pdf
index 3ee55232890075ed5105fbeac5a0545b1e01b915..97a8cbc537174fd804d1ed875e2d9620bc8cebd6 100644
GIT binary patch
delta 20
bcmX@Xae`w57c;wsp{22BzT

diff --git a/Aufgabe2-Dokumentation/svg-inkscape/dreiecke5_svg-tex.pdf b/Aufgabe2-Dokumentation/svg-inkscape/dreiecke5_svg-tex.pdf
index 601891184cecc962e7c63b4d188c69b7c11d97de..74e5274eb59c2eff9f55aa4876ff0c0a1e1d2c38 100644
GIT binary patch
delta 20
bcmZ3+w~TMYb5?c>LrY^zL&MGQSS=U0
+// CCW: > 0, CW: < 0
 double ccw(Point a, Point b, Point c){
     return (b.x - a.x) * (c.y - a.y) - (b.y - a.y) * (c.x - a.x);
 }
@@ -184,29 +184,29 @@ double ccw(Point a, Point b, Point c){
 // Punkt finden, von dem aus Drehwinkel bestimmt wird
 // (beim Drehen auf die x-Achse)
 int findAngleCalcPoint(Triangle &t, int bestPoint){
-    //just return the point that is counterclockwise from line between other points
+    //just return the point that is clockwise from line between other points
     switch(bestPoint) {
         case 0: if(ccw(t.points[0],t.points[1],t.points[2]) > 0){
-                    //point 2 is clockwise
+                    //point 2 is counterclockwise
                     return 1;
                 } else {
-                    //point 2 is counterclockwise
+                    //point 2 is clockwise
                     return 2;
                 }
                 break;
         case 1: if(ccw(t.points[1],t.points[0],t.points[2]) > 0){
-                    //point 2 is clockwise
+                    //point 2 is counterclockwise
                     return 0;
                 } else {
-                    //point 2 is counterclockwise
+                    //point 2 is clockwise
                     return 2;
                 }
                 break;
         case 2: if(ccw(t.points[2],t.points[1],t.points[0]) > 0){
-                    //point 0 is clockwise
+                    //point 0 is counterclockwise
                     return 1;
                 } else {
-                    //point 0 is counterclockwise
+                    //point 0 is clockwise
                     return 0;
                 }
                 break;