initial commit
This commit is contained in:
commit
1258d768d2
301
.gitignore
vendored
Normal file
301
.gitignore
vendored
Normal file
|
|
@ -0,0 +1,301 @@
|
|||
## Core latex/pdflatex auxiliary files:
|
||||
*.aux
|
||||
*.lof
|
||||
*.log
|
||||
*.lot
|
||||
*.fls
|
||||
*.out
|
||||
*.toc
|
||||
*.fmt
|
||||
*.fot
|
||||
*.cb
|
||||
*.cb2
|
||||
.*.lb
|
||||
|
||||
## Intermediate documents:
|
||||
*.dvi
|
||||
*.xdv
|
||||
*-converted-to.*
|
||||
# these rules might exclude image files for figures etc.
|
||||
*.ps
|
||||
*.eps
|
||||
*.pdf
|
||||
|
||||
## Generated if empty string is given at "Please type another file name for output:"
|
||||
.pdf
|
||||
|
||||
## Bibliography auxiliary files (bibtex/biblatex/biber):
|
||||
*.bbl
|
||||
*.bcf
|
||||
*.blg
|
||||
*-blx.aux
|
||||
*-blx.bib
|
||||
*.run.xml
|
||||
|
||||
## Build tool auxiliary files:
|
||||
*.fdb_latexmk
|
||||
*.synctex
|
||||
*.synctex(busy)
|
||||
*.synctex.gz
|
||||
*.synctex.gz(busy)
|
||||
*.pdfsync
|
||||
|
||||
## Build tool directories for auxiliary files
|
||||
# latexrun
|
||||
latex.out/
|
||||
|
||||
## Auxiliary and intermediate files from other packages:
|
||||
# algorithms
|
||||
*.alg
|
||||
*.loa
|
||||
|
||||
# achemso
|
||||
acs-*.bib
|
||||
|
||||
# amsthm
|
||||
*.thm
|
||||
|
||||
# beamer
|
||||
*.nav
|
||||
*.pre
|
||||
*.snm
|
||||
*.vrb
|
||||
|
||||
# changes
|
||||
*.soc
|
||||
|
||||
# comment
|
||||
*.cut
|
||||
|
||||
# cprotect
|
||||
*.cpt
|
||||
|
||||
# elsarticle (documentclass of Elsevier journals)
|
||||
*.spl
|
||||
|
||||
# endnotes
|
||||
*.ent
|
||||
|
||||
# fixme
|
||||
*.lox
|
||||
|
||||
# feynmf/feynmp
|
||||
*.mf
|
||||
*.mp
|
||||
*.t[1-9]
|
||||
*.t[1-9][0-9]
|
||||
*.tfm
|
||||
|
||||
#(r)(e)ledmac/(r)(e)ledpar
|
||||
*.end
|
||||
*.?end
|
||||
*.[1-9]
|
||||
*.[1-9][0-9]
|
||||
*.[1-9][0-9][0-9]
|
||||
*.[1-9]R
|
||||
*.[1-9][0-9]R
|
||||
*.[1-9][0-9][0-9]R
|
||||
*.eledsec[1-9]
|
||||
*.eledsec[1-9]R
|
||||
*.eledsec[1-9][0-9]
|
||||
*.eledsec[1-9][0-9]R
|
||||
*.eledsec[1-9][0-9][0-9]
|
||||
*.eledsec[1-9][0-9][0-9]R
|
||||
|
||||
# glossaries
|
||||
*.acn
|
||||
*.acr
|
||||
*.glg
|
||||
*.glo
|
||||
*.gls
|
||||
*.glsdefs
|
||||
*.lzo
|
||||
*.lzs
|
||||
*.slg
|
||||
*.slo
|
||||
*.sls
|
||||
|
||||
# uncomment this for glossaries-extra (will ignore makeindex's style files!)
|
||||
# *.ist
|
||||
|
||||
# gnuplot
|
||||
*.gnuplot
|
||||
*.table
|
||||
|
||||
# gnuplottex
|
||||
*-gnuplottex-*
|
||||
|
||||
# gregoriotex
|
||||
*.gaux
|
||||
*.glog
|
||||
*.gtex
|
||||
|
||||
# htlatex
|
||||
*.4ct
|
||||
*.4tc
|
||||
*.idv
|
||||
*.lg
|
||||
*.trc
|
||||
*.xref
|
||||
|
||||
# hyperref
|
||||
*.brf
|
||||
|
||||
# knitr
|
||||
*-concordance.tex
|
||||
# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
|
||||
# *.tikz
|
||||
*-tikzDictionary
|
||||
|
||||
# listings
|
||||
*.lol
|
||||
|
||||
# luatexja-ruby
|
||||
*.ltjruby
|
||||
|
||||
# makeidx
|
||||
*.idx
|
||||
*.ilg
|
||||
*.ind
|
||||
|
||||
# minitoc
|
||||
*.maf
|
||||
*.mlf
|
||||
*.mlt
|
||||
*.mtc[0-9]*
|
||||
*.slf[0-9]*
|
||||
*.slt[0-9]*
|
||||
*.stc[0-9]*
|
||||
|
||||
# minted
|
||||
_minted*
|
||||
*.pyg
|
||||
|
||||
# morewrites
|
||||
*.mw
|
||||
|
||||
# newpax
|
||||
*.newpax
|
||||
|
||||
# nomencl
|
||||
*.nlg
|
||||
*.nlo
|
||||
*.nls
|
||||
|
||||
# pax
|
||||
*.pax
|
||||
|
||||
# pdfpcnotes
|
||||
*.pdfpc
|
||||
|
||||
# sagetex
|
||||
*.sagetex.sage
|
||||
*.sagetex.py
|
||||
*.sagetex.scmd
|
||||
|
||||
# scrwfile
|
||||
*.wrt
|
||||
|
||||
# svg
|
||||
svg-inkscape/
|
||||
|
||||
# sympy
|
||||
*.sout
|
||||
*.sympy
|
||||
sympy-plots-for-*.tex/
|
||||
|
||||
# pdfcomment
|
||||
*.upa
|
||||
*.upb
|
||||
|
||||
# pythontex
|
||||
*.pytxcode
|
||||
pythontex-files-*/
|
||||
|
||||
# tcolorbox
|
||||
*.listing
|
||||
|
||||
# thmtools
|
||||
*.loe
|
||||
|
||||
# TikZ & PGF
|
||||
*.dpth
|
||||
*.md5
|
||||
*.auxlock
|
||||
|
||||
# titletoc
|
||||
*.ptc
|
||||
|
||||
# todonotes
|
||||
*.tdo
|
||||
|
||||
# vhistory
|
||||
*.hst
|
||||
*.ver
|
||||
|
||||
# easy-todo
|
||||
*.lod
|
||||
|
||||
# xcolor
|
||||
*.xcp
|
||||
|
||||
# xmpincl
|
||||
*.xmpi
|
||||
|
||||
# xindy
|
||||
*.xdy
|
||||
|
||||
# xypic precompiled matrices and outlines
|
||||
*.xyc
|
||||
*.xyd
|
||||
|
||||
# endfloat
|
||||
*.ttt
|
||||
*.fff
|
||||
|
||||
# Latexian
|
||||
TSWLatexianTemp*
|
||||
|
||||
## Editors:
|
||||
# WinEdt
|
||||
*.bak
|
||||
*.sav
|
||||
|
||||
# Texpad
|
||||
.texpadtmp
|
||||
|
||||
# LyX
|
||||
*.lyx~
|
||||
|
||||
# Kile
|
||||
*.backup
|
||||
|
||||
# gummi
|
||||
.*.swp
|
||||
|
||||
# KBibTeX
|
||||
*~[0-9]*
|
||||
|
||||
# TeXnicCenter
|
||||
*.tps
|
||||
|
||||
# auto folder when using emacs and auctex
|
||||
./auto/*
|
||||
*.el
|
||||
|
||||
# expex forward references with \gathertags
|
||||
*-tags.tex
|
||||
|
||||
# standalone packages
|
||||
*.sta
|
||||
|
||||
# Makeindex log files
|
||||
*.lpz
|
||||
|
||||
# xwatermark package
|
||||
*.xwm
|
||||
|
||||
# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
|
||||
# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
|
||||
# Uncomment the next line to have this generated file ignored.
|
||||
#*Notes.bib
|
||||
14
LICENSE
Normal file
14
LICENSE
Normal file
|
|
@ -0,0 +1,14 @@
|
|||
DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
|
||||
Version 2, December 2004
|
||||
|
||||
Copyright (C) 2004 Sam Hocevar <sam@hocevar.net>
|
||||
|
||||
Everyone is permitted to copy and distribute verbatim or modified
|
||||
copies of this license document, and changing it is allowed as long
|
||||
as the name is changed.
|
||||
|
||||
DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
|
||||
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
|
||||
|
||||
0. You just DO WHAT THE FUCK YOU WANT TO.
|
||||
|
||||
10
Makefile
Normal file
10
Makefile
Normal file
|
|
@ -0,0 +1,10 @@
|
|||
MAIN = sheet
|
||||
FLAGS = -pdf
|
||||
|
||||
dev:
|
||||
latexmk $(FLAGS) -pvc $(MAIN)
|
||||
all:
|
||||
latexmk $(FLAGS) $(MAIN)
|
||||
clean:
|
||||
latexmk -C
|
||||
|
||||
429
sheet.tex
Normal file
429
sheet.tex
Normal file
|
|
@ -0,0 +1,429 @@
|
|||
\documentclass[11pt, a4paper, twoside]{article}
|
||||
\usepackage[
|
||||
a4paper,
|
||||
headsep=5mm,
|
||||
footskip=0mm,
|
||||
top=12mm,
|
||||
left=10mm,
|
||||
right=10mm,
|
||||
bottom=10mm
|
||||
]{geometry}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{multicol}
|
||||
\usepackage[noend]{algorithm2e}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{fancyhdr}
|
||||
\usepackage{hyperref}
|
||||
\hypersetup{
|
||||
colorlinks=true,
|
||||
linkcolor=blue,
|
||||
filecolor=magenta,
|
||||
urlcolor=cyan,
|
||||
pdftitle={Overleaf Example},
|
||||
pdfpagemode=FullScreen,
|
||||
}
|
||||
|
||||
\setlength{\algomargin}{0pt}
|
||||
|
||||
\begin{document}
|
||||
\pagestyle{fancy}
|
||||
\fancyhead{}
|
||||
\fancyhead[L]{Theoretische Grundlagen der Informatik}
|
||||
\fancyhead[R]{Gero Beckmann - \url{https://github.com/Geronymos/}}
|
||||
\fancyfoot{}
|
||||
\fancyfoot[R]{\thepage}
|
||||
\section{Laufzeit}
|
||||
\hspace*{-.5cm}
|
||||
\begin{tabular}{ l l l l }
|
||||
Notations & Asymptotischer Vergleich & Formale Definition & Grenzen \\
|
||||
$f(n) \in \omega(g(n))$&
|
||||
$f(n)$ wächst schneller als $g(n)$ &
|
||||
$\forall c \exists n_0 \forall n > n_0 f(n) > c \cdot g(n)$ &
|
||||
$$$\lim\sup\limits_{n \rightarrow \infty}\frac{f}{g} = \infty$$$ \\
|
||||
|
||||
$f(n) \in \Omega(g(n))$ &
|
||||
$f(n)$ wächst min. so schnell wie $g(n)$ &
|
||||
$\exists c \exists n_0 \forall n > n_0 c \cdot f(n) \leq g(n)$ &
|
||||
$$$0 < \liminf\limits_{n \rightarrow \infty}\frac{f}{g} \leq \infty$$$ \\
|
||||
|
||||
\( f(n) \in \Theta(g(n)) \) &
|
||||
$f(n)$ und $g(n)$ wachsen gleich schnell &
|
||||
$f(n) \in \mathcal{O}(g(n)) \wedge f(n) \in \Omega(g(n))$ &
|
||||
$$$0 < \lim\limits_{n \rightarrow \infty}\frac{f}{g} < \infty$$$ \\
|
||||
|
||||
\( f(n) \in \mathcal{O}(g(n)) \) &
|
||||
$f(n)$ wächst max. so schnell wie $g(n)$ &
|
||||
$\exists c \exists n_0 \forall n > n_0 f(n) \leq c \cdot g(n)$ &
|
||||
$$$0 \leq \limsup\limits_{n \rightarrow \infty}\frac{f}{g} < \infty$$$ \\
|
||||
|
||||
\( f(n) \in o(g(n)) \) &
|
||||
$f(n)$ wächst langsamer als $g(n)$ &
|
||||
$\forall c \exists n_0 \forall n > n_0 c \cdot f(n) < g(n)$ &
|
||||
$$$\lim\limits_{n \rightarrow \infty} \frac{f}{g} = \infty$$$ \\
|
||||
|
||||
\end{tabular}
|
||||
|
||||
\subsection{Vergleich}
|
||||
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
|
||||
$1$ & $\log^*n$ & $\log n$ & $\log^2n$ & $\sqrt[3]{n}$ &
|
||||
$\sqrt{n}$ & $n$ & $n^2$ & $n^3$ & $n^{\log n}$ &
|
||||
$2^{\sqrt{n}}$ & $2^n$ & $3^n$ & $4^n$ & $n!$ & $2^{n^2}$
|
||||
\end{tabular}
|
||||
|
||||
\begin{multicols}{3}
|
||||
|
||||
\subsubsection*{Transitivität}
|
||||
|
||||
$f_1(n) \in \mathcal{O}(f_2(n)) \wedge f_2(n) \in\mathcal{O}(f_3(n))$ \\
|
||||
$\Rightarrow f_1(n) \in \mathcal{O}(f_3(n))$
|
||||
|
||||
\subsubsection*{Summen}
|
||||
|
||||
$f_1(n) \in \mathcal{O}f_3(n)) \wedge f_2(n) \in \mathcal{O}(f_3(n))$ \\
|
||||
$\Rightarrow f_1(n) + f_2(n) \in \mathcal{O}(f_3(n))$
|
||||
|
||||
\subsubsection*{Produkte}
|
||||
|
||||
$f_1(n) \in \mathcal{O}(g_1(n)) \wedge f_2(n) \in \mathcal{O}(g_2(n))$ \\
|
||||
$\Rightarrow f_1(n) \cdot f_2(n) \in \mathcal{O}(g_1(n) \cdot g_2(n))$
|
||||
|
||||
|
||||
\columnbreak
|
||||
|
||||
\subsection{Master-Theorem}
|
||||
|
||||
Sei $T(n) = a \cdot T(\frac{n}{b}) + f(n)$ mit $f(n) \in \Theta(n^c)$ und i
|
||||
$T(1) \in \Theta(1)$. Dann gilt
|
||||
$
|
||||
T(n) \in \begin{cases}
|
||||
\Theta(n^c) &\text{wenn } a < b^c, \\
|
||||
\Theta(n^c \log n) &\text{wenn } a = b^c, \\
|
||||
\Theta(n^{\log_b(a)}) &\text{wenn } a > b^c.
|
||||
\end{cases}
|
||||
$
|
||||
|
||||
\subsubsection{Monome}
|
||||
|
||||
\begin{itemize}
|
||||
\item $a \leq b \Rightarrow n^a \in \mathcal{O}(n^b)$
|
||||
\item $n^a \in \Theta(n^b) \Leftrightarrow a = b$
|
||||
\item $\sum_{v \in V}deg(v) = \Theta(m)$
|
||||
\item $\forall n \in \mathbb{N}: \sum^n_{k=0}k = \frac{n(n+1)}{2}$
|
||||
\item $
|
||||
\sum^b_{i=a}c^i \in \begin{cases}
|
||||
\Theta(c^a) &\text{wenn } c < 1, \\
|
||||
\Theta(c^b) &\text{wenn } c > 1, \\
|
||||
\Theta(b-a) &\text{wenn } c = 1.
|
||||
\end{cases}
|
||||
$
|
||||
\item $\log(ab) = \log(a) + \log(b)$
|
||||
\item $\log(\frac{a}{b}) = \log(a) - \log(b)$
|
||||
\item $a^{\log_a(b)} = b$
|
||||
\item $a^x = e^{ln(a) \cdot x}$
|
||||
\item $\log(a^b) = b \cdot \log(a)$
|
||||
\item $\log_b(n) = \frac{\log_a(n)}{\log_a(b)}$
|
||||
\end{itemize}
|
||||
|
||||
%\subsubsection{Konstante Faktoren}
|
||||
%
|
||||
%$a \cdot f(n) \in \Theta(f(n))$
|
||||
|
||||
|
||||
\end{multicols}
|
||||
|
||||
\begin{minipage}{0.7\textwidth}
|
||||
|
||||
\section{Sortieren}
|
||||
|
||||
\begin{tabular}[t]{c || c | c | c | c}
|
||||
Algorithmus & best case & average & worst & Stabilität \\
|
||||
\hline
|
||||
Insertion-Sort &
|
||||
$\mathcal{O}(n)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & stabil\\
|
||||
Bubble-Sort &
|
||||
$\mathcal{O}(n)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & stabil\\
|
||||
Merge-Sort &
|
||||
$\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & stabil\\
|
||||
Quick-Sort &
|
||||
$\mathcal{O}(n \log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & i.A. nicht stabil\\
|
||||
Heap-Sort &
|
||||
$\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & $\mathcal{O}(n\log n)$ & nicht stabil\\
|
||||
\hline
|
||||
Bucket-Sort &
|
||||
$\Theta(n+m)$ & $\Theta(n+m)$ & $\Theta(n+m)$ &
|
||||
stabil $e \in [0, m)$\\
|
||||
Radix-Sort &
|
||||
$\Theta(c \cdot n)$ & $\Theta(c\cdot n)$ & $\Theta(c\cdot n)$ &
|
||||
stabil $e \in [0, n^c)$\\
|
||||
\end{tabular}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\subsection{Heaps}
|
||||
|
||||
\begin{tabular}[t]{c || c}
|
||||
Bin.-Heap & Laufzeit \\
|
||||
\hline
|
||||
push(x) & $\mathcal{O}(\log n)$ \\
|
||||
popMin() & $\mathcal{O}(\log n)$ \\
|
||||
decPrio(x, x') & $\mathcal{O}(\log n)$ \\
|
||||
build([$\mathbb{N}$; n]) & $\mathcal{O}(n)$
|
||||
\end{tabular}
|
||||
|
||||
\begin{itemize}
|
||||
\item linkes Kind: $2v + 1$
|
||||
\item rechts Kind: $2v + 2$
|
||||
\item Elternknoten: $ \lfloor \frac{v - 1}{2} \rfloor $
|
||||
\end{itemize}
|
||||
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\begin{multicols}{2}
|
||||
|
||||
\section{Datenstrukturen}
|
||||
|
||||
\subsection{Listen}
|
||||
|
||||
\begin{tabular}{c || c | c | c || c}
|
||||
Operation & DLL & SLL & Array & Erklärung(*) \\
|
||||
\hline
|
||||
first & 1 & 1 & 1 & \\
|
||||
last & 1 & 1 & 1 & \\
|
||||
insert & 1 & 1* & n & nur insertAfter \\
|
||||
remove & 1 & 1* & n & nur removeAfter \\
|
||||
pushBack & 1 & 1 & 1* & amortisiert \\
|
||||
pushFront & 1 & 1 & n & \\
|
||||
popBack & 1 & n & 1* & amortisiert \\
|
||||
popFront & 1 & 1 & n & \\
|
||||
concat & 1 & 1 & n & \\
|
||||
splice & 1 & 1 & n \\
|
||||
findNext & n & n & n
|
||||
|
||||
\end{tabular}
|
||||
|
||||
\subsection{Hash-Tabelle}
|
||||
$\mathcal{H}$ heißt \textbf{universell}, wenn für ein zufälliges gewähltes
|
||||
$h \in \mathcal{H}$ gilt: $U \rightarrow \{0, ..., m-1\}$ \\
|
||||
$\forall k, l \in U, k \neq l: Pr[h(k) = h(l)] = \frac{1}{m}$ \\
|
||||
$h_{a,b}(k) = ((a\cdot k + b) \mod p) \mod m$
|
||||
|
||||
\subsection{Graphen}
|
||||
|
||||
\begin{tabular}{c || c}
|
||||
Algorithmus & Laufzeit \\
|
||||
\hline
|
||||
BFS/DFS & $\Theta(n+m)$\\
|
||||
topoSort & $\Theta(n)$\\
|
||||
Kruskal & $\Theta(m \log n)$\\
|
||||
Prim & $\Theta((n+m)\log n)$ \\
|
||||
Dijksta & $\Theta((n + m) \log n)$\\
|
||||
Bellmann-Ford & $\Theta(nm)$\\
|
||||
Floyd-Warshall & $\Theta(n^3)$ \\
|
||||
\end{tabular}
|
||||
|
||||
\end{multicols}
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{multicols}{2}
|
||||
|
||||
\subsubsection{DFS}
|
||||
|
||||
\begin{tabular}{c || c | c}
|
||||
Kante & DFS & FIN \\
|
||||
\hline
|
||||
Vorkante & klein $\rightarrow$ groß & groß $\rightarrow$ klein \\
|
||||
Rückkante & groß $\rightarrow$ klein & klein $\rightarrow$ groß \\
|
||||
Querkante & groß $\rightarrow$ klein & groß $\rightarrow$ klein \\
|
||||
Baumkante & klein $\rightarrow$ groß & groß $\rightarrow$ klein \\
|
||||
\end{tabular}
|
||||
\subsection{Bäume}
|
||||
\subsubsection{Heap}
|
||||
Priorität eines Knotens $\geq (\leq)$ Priorität der Kinder.
|
||||
\textbf{BubbleUp}, \textbf{SinkDown}. \textbf{Build} mit \textbf{sinkDown}
|
||||
beginnend mit letztem Knoten der vorletzten Ebene weiter nach oben.
|
||||
\textbf{decPrio} entweder updaten, Eigenschaft wiederherstellen; löschen,
|
||||
mit neuer Prio einfügen oder Lazy Evaluation.
|
||||
|
||||
\subsubsection{(ab)-Baum}
|
||||
Balanciert. \textbf{find}, \textbf{insert}, \textbf{remove} in
|
||||
$\Theta(log n)$. Zu wenig Kinder: \textbf{rebalance} / \textbf{fuse}.
|
||||
Zu viele Kinder: \textbf{split}.
|
||||
|
||||
Linker Teilbaum $\leq$ Schlüssel k $<$ rechter Teilbaum
|
||||
|
||||
Unendlich-Trick, für Invarianten.
|
||||
|
||||
\subsection{Union-Find}
|
||||
Rang: höhe des Baums, damit ist die Höhe h mind. $2^h$ Knoten, h $\in
|
||||
\mathcal{O}(\log n)$.
|
||||
Union hängt niedrigen Baum an höherrängigen Baum. Pfadkompression hängt alle
|
||||
Knoten bei einem \textbf{find} an die Wurzel.
|
||||
|
||||
|
||||
\columnbreak
|
||||
\section{Amortisierte Analyse}
|
||||
|
||||
\subsection{Aggregation}
|
||||
Summiere die Kosten für alle Operationen. Teile Gesamtkkosten durch Anzahl
|
||||
Operationen.
|
||||
|
||||
\subsection{Charging}
|
||||
Verteile Kosten-Tokens von teuren zu günstigen Operationen (Charging). Zeige:
|
||||
jede Operation hat am Ende nur wenige Tokens.
|
||||
|
||||
\subsection{Konto}
|
||||
Günstige Operationen bezahlen mehr als sie tatsächlich kosten (ins Konto
|
||||
einzahlen). Teure Operationen bezahlen tatsächliche Kosten zum Teil mit
|
||||
Guthaben aus dem Konto. \textbf{Beachte: Konto darf nie negativ sein!}
|
||||
|
||||
\subsection{Potential (Umgekehrte Kontomethode)}
|
||||
Definiere Kontostand abhängig vom Zustand der Datenstruktur
|
||||
(Potentialfunktion)
|
||||
|
||||
amortisierten Kosten = tatsächliche Kosten
|
||||
$+ \Phi(S_\text{nach}) -\Phi(S_\text{vor})$
|
||||
|
||||
\end{multicols}
|
||||
|
||||
\section{Pseudocode}
|
||||
\scriptsize
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
DFS(Graph G, Node v) \\
|
||||
mark v \\
|
||||
dfs[v] := dfsCounter++ \\
|
||||
low[v] := dfs[v] \\
|
||||
\For{u $\in$ N(v)}{
|
||||
\eIf{not marked u}{
|
||||
dist[u] := dist[v] + 1 \\
|
||||
par[u] := v \\
|
||||
DFS(G, u) \\
|
||||
low[v] := min(low[v], low[u]) \\
|
||||
}{low[v] := min(low[v], dfs[u])}
|
||||
}
|
||||
fin[v] := fin++ \\
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
topoSort(Graph G) \\
|
||||
fin := [$\infty$; n] \\
|
||||
curr := 0 \\
|
||||
\For{Node v in V}{
|
||||
\If{v is colored}{DFS(G,v)}
|
||||
}
|
||||
return V sorted by decreasing fin \\
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Kruskal(Graph G) \\
|
||||
U := Union-Find(G.v) \\
|
||||
PriorityQueue Q := empty \\
|
||||
\For{Edge e in E}{Q.push(e, len(e))}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
e := Q.popMin() \\
|
||||
\If{U.find(v) $\neq$ U.find(u)}{
|
||||
L.add(e) \\
|
||||
U.union(v, u) \\
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Prim(Graph G) \\
|
||||
Priority Queue Q := empty \\
|
||||
p := [0; n] \\
|
||||
\For{Node v in V}{
|
||||
Q.push(v, $\infty$) \\
|
||||
}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.popMin() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{v $\in$ Q $\wedge$ (len(u, v) $<$ Q.prio(v))}{
|
||||
p[v] = u \\
|
||||
Q.decPrio(v, len(u, v) \\
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
BFS(Graph G, Start s, Goal z) \\
|
||||
Queue Q := empty queue \\
|
||||
Q.push(s) \\
|
||||
s.layer = 0 \\
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.pop() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{v.layer = $-\infty$}{
|
||||
Q.push(v) \\
|
||||
v.layer = u.layer + 1
|
||||
}
|
||||
\If{v = z}{
|
||||
return z.layer
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
Dijkstra(Graph G, Node s) \\
|
||||
d := [$\infty$; n] \\
|
||||
d[s] := 0 \\
|
||||
PriorityQueue Q := empty priority queue \\
|
||||
\For{Node v in V}{
|
||||
Q.push(v, d[v])
|
||||
}
|
||||
\While{Q $\neq \emptyset$}{
|
||||
u := Q.popMin() \\
|
||||
\For{Node v in N(u)}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
d[v] := d[u] + len(u, v) \\
|
||||
Q.decPrio(v, d[v]) \\
|
||||
}
|
||||
}
|
||||
}
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
BellManFord(Graph G, Node s) \\
|
||||
d := [$\infty$, n] \\
|
||||
d[s] := 0 \\
|
||||
\For{n-1 iterations}{
|
||||
\For{(u, v) $\in$ E}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
d[v] := d[u] + len(u, v)
|
||||
}
|
||||
}
|
||||
}
|
||||
\For{(u, v) $\in$ E}{
|
||||
\If{d[v] $>$ d[u] + len(u, v)}{
|
||||
return negative cycle
|
||||
}
|
||||
}
|
||||
return d
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\begin{minipage}{.25\linewidth}
|
||||
\begin{algorithm}[H]
|
||||
FloydWarshall(Graph G) \\
|
||||
D := [$\infty$, n $\times$ n] \\
|
||||
\For{(u, v) $\in$ E}{D[u][v] := len(u, v)}
|
||||
\For{v $\in$ V}{D[v][v] := 0}
|
||||
\For{i $\in 1,...,n$}{
|
||||
\For{(u,v) $\in V \times V$}{
|
||||
D[u][v] := min(D[u][v], D[u][$v_i$] + D[$v_i$][v]) \\
|
||||
}
|
||||
}
|
||||
return D
|
||||
\end{algorithm}
|
||||
\end{minipage}
|
||||
\end{document}
|
||||
Loading…
Reference in a new issue