406 lines
12 KiB
TeX
406 lines
12 KiB
TeX
|
\documentclass[11pt, a4paper, twoside]{article}
|
||
|
|
\usepackage[a4paper, margin=1cm]{geometry}
|
||
|
|
\usepackage{amsmath}
|
||
|
|
\usepackage{amsfonts}
|
||
|
|
\usepackage{multicol}
|
||
|
|
\usepackage[noend]{algorithm2e}
|
||
|
|
\usepackage[utf8]{inputenc}
|
||
|
|
|
||
|
|
\setlength{\algomargin}{0pt}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
\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} \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() & $\mathcal{O}(\log n)$ \\
|
||
|
|
devPrio(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}y in
|
||
|
|
$\Theta(log n)$. Zu viele 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 Kosen-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}
|