read until the tree desctiptions

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notes.tex

@@ -0,0 +1,98 @@
+%https://ics.uci.edu/~eppstein/pubs/Epp-SJC-98.pdf%
+
+\documentclass[a4paper,10pt,ngerman]{scrartcl}
+\usepackage{babel}
+\usepackage[T1]{fontenc}
+\usepackage[utf8x]{inputenc}
+\usepackage[a4paper,margin=2.5cm,footskip=0.5cm]{geometry}
+\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}
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+\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}