From 420e09de9e30f4fc081aae2da2caa46dc93bdded Mon Sep 17 00:00:00 2001 From: DarianTr Date: Thu, 7 Dec 2023 19:16:54 +0100 Subject: [PATCH] read until the tree desctiptions --- notes.aux | 13 ++ notes.fdb_latexmk | 106 +++++++++ notes.fls | 573 ++++++++++++++++++++++++++++++++++++++++++++++ notes.log | 439 +++++++++++++++++++++++++++++++++++ notes.pdf | Bin 0 -> 106893 bytes notes.synctex.gz | Bin 0 -> 6240 bytes notes.tex | 98 ++++++++ 7 files changed, 1229 insertions(+) create mode 100644 notes.aux create mode 100644 notes.fdb_latexmk create mode 100644 notes.fls create mode 100644 notes.log create mode 100644 notes.pdf create mode 100644 notes.synctex.gz create mode 100644 notes.tex diff --git a/notes.aux b/notes.aux new file mode 100644 index 0000000..02cf68f --- /dev/null +++ b/notes.aux @@ -0,0 +1,13 @@ +\relax +\providecommand*\new@tpo@label[2]{} +\providecommand\babel@aux[2]{} +\@nameuse{bbl@beforestart} +\catcode `"\active +\babel@aux{ngerman}{} +\@writefile{toc}{\contentsline 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+\usepackage[section]{placeins} +% Die nächsten vier Felder bitte anpassen: + +\usepackage{lastpage} +% Kopf- und Fußzeilen +\usepackage{scrlayer-scrpage, lastpage} +\setkomafont{pageheadfoot}{\large\textrm} +\lohead{\Aufgabe} +\rohead{Team-ID: \TeamId} +\cfoot*{\thepage{}/\pageref{LastPage}} + +% Position des Titels +\usepackage{titling} +\setlength{\droptitle}{-1.0cm} + +% Für mathematische Befehle und Symbole +\usepackage{amsmath} +\usepackage{amssymb} + +% Für Bilder +\usepackage{graphicx} + +% Für Algorithmen +\usepackage{algpseudocode} + +% Für Quelltext +\usepackage{listings} +\usepackage{color} +\definecolor{mygreen}{rgb}{0,0.6,0} +\definecolor{mygray}{rgb}{0.5,0.5,0.5} +\definecolor{mymauve}{rgb}{0.58,0,0.82} +\lstset{ + keywordstyle=\color{blue},commentstyle=\color{mygreen}, + stringstyle=\color{mymauve},rulecolor=\color{black}, + basicstyle=\footnotesize\ttfamily,numberstyle=\tiny\color{mygray}, + captionpos=b, % sets the caption-position to bottom + keepspaces=true, % keeps spaces in text + numbers=left, numbersep=5pt, showspaces=false,showstringspaces=true, + showtabs=false, stepnumber=2, tabsize=2, title=\lstname +} +\lstdefinelanguage{JavaScript}{ % JavaScript ist als einzige Sprache noch nicht vordefiniert + keywords={break, case, catch, continue, debugger, default, delete, do, else, finally, for, function, if, in, instanceof, new, return, switch, this, throw, try, typeof, var, void, while, with}, + morecomment=[l]{//}, + morecomment=[s]{/*}{*/}, + morestring=[b]', + morestring=[b]", + sensitive=true +} + +% Diese beiden Pakete müssen zuletzt geladen werden +%\usepackage{hyperref} % Anklickbare Links im Dokument +\usepackage{cleveref} + +% Daten für die Titelseite +\title{ + K-Shortest-Path \\ +Eppsteins Algorithmus} +\author{Darian L. Tratar} +\date{\LARGE\today} + +\begin{document} +\maketitle + +\section {Verwendung} +\begin{itemize} + \item Erf\"ullen zus\"atzlicher Bedingungen, die schwer zu optimieren sind + \item L\"osung f\"ur ein NP-Schweres Problem (unterschiedlich markierte Kanten + jede Kante des Weges muss unterschiedlich markiert sein) (siehe Dreyfus und Lawler) + \item Evaluieren eines Modells: $k$ als Ma\ss{} + \item Vieles mehr (komplizierter) +\end{itemize} +\section {Laufzeiten} +$m$ ist die Anzahl der Kanten und $n$ ist die Anzahl der Knoten +\begin{itemize} + \item F\"ur ein Paar von Punkten: $O(m + n \log n + k)$ + \item F\"ur ein Punkt zu allen andern Punkten: $O(m + n \log n + kn)$ + \item Aufstellen des Shortest-Path-Trees: $O( n \log n)$ + \item Gleiche Laufzeit f\"ur alle Wege kleiner als eine gegebene Strecke + \item Dijkstra implementierung von Fox: $O(m + kn \log n)$ + \item Mit negativen Kanten ohne negativen Zyklus: $O(mn)$ oder $O(mn^{1/2}\log N)$ mit $N$ als gr\"o\ss{}ter Absolutbetrag der negativen Kanten +\end{itemize} + +\section {Basics des Algorithmus} +\begin{itemize} + \item Binary-Heap f\"ur jeden Knoten, der nicht zum shortest-Path-Tree geh\"ort und die von diesem ereicht werden k\"onnen + \item Heaps teilen sich gleiche Teile -> Graph mit $O(m + n\log n)$ Knoten + \item Verbesserung mit Fredericksons tree decomposition techniques +\end{itemize} + + + +\end{document}