123 lines
3.2 KiB
TeX
Executable File
123 lines
3.2 KiB
TeX
Executable File
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\section{The Ford-Fulkerson Algorithm}
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We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
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It works as follows:
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\begin{enumerate}
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\item[\texttt{00}] Take a weighted directed graph $G$.
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\item[\texttt{01}] Find any flow $F$ in $G$
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\item[\texttt{02}] Calculate $R$, the residual of $F$.
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\item[\texttt{03}] ~~~~If $S$ and $T$ are not connected in $R$, $F$ is a maximal flow. \texttt{HALT}.
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\item[\texttt{04}] Otherwise, find another flow $F_0$ in $R$.
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\item[\texttt{05}] Add $F_0$ to $F$
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\item[\texttt{06}] \texttt{GOTO 02}
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\end{enumerate}
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\problem{}
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Run the Ford-Fulkerson algorithm on the following graph. \\
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There is extra space on the next page.
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\begin{center}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (S) at (-5mm, 0mm) {$S$};
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\node[main] (A) at (20mm, 20mm) {$A$};
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\node[main] (B) at (20mm, 0mm) {$B$};
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\node[main] (C) at (20mm, -20mm) {$C$};
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\node[main] (D) at (50mm, 20mm) {$D$};
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\node[main] (E) at (50mm, 0mm) {$E$};
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\node[main] (F) at (50mm, -20mm) {$F$};
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\node[main] (T) at (75mm, 0mm) {$T$};
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\end{scope}
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% Edges
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\draw[->]
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(S) edge node[label] {$8$} (A)
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(S) edge node[label] {$7$} (B)
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(S) edge node[label] {$4$} (C)
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(A) edge node[label] {$2$} (B)
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(B) edge node[label] {$5$} (C)
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(A) edge node[label] {$3$} (D)
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(A) edge node[label] {$9$} (E)
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(B) edge node[label] {$6$} (E)
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(C) edge node[label] {$7$} (E)
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(C) edge node[label] {$2$} (F)
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(E) edge node[label] {$3$} (D)
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(E) edge node[label] {$4$} (F)
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(D) edge node[label] {$9$} (T)
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(E) edge node[label] {$5$} (T)
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(F) edge node[label] {$8$} (T)
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;
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\end{tikzpicture}
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\end{center}
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\begin{solution}
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The maximum flow is $17$.
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\end{solution}
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\vspace{5mm}
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\pagebreak
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\begin{center}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (S) at (-5mm, 0mm) {$S$};
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\node[main] (A) at (20mm, 20mm) {$A$};
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\node[main] (B) at (20mm, 0mm) {$B$};
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\node[main] (C) at (20mm, -20mm) {$C$};
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\node[main] (D) at (50mm, 20mm) {$D$};
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\node[main] (E) at (50mm, 0mm) {$E$};
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\node[main] (F) at (50mm, -20mm) {$F$};
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\node[main] (T) at (75mm, 0mm) {$T$};
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\end{scope}
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% Edges
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\draw[->]
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(S) edge node[label] {$8$} (A)
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(S) edge node[label] {$7$} (B)
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(S) edge node[label] {$4$} (C)
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(A) edge node[label] {$2$} (B)
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(B) edge node[label] {$5$} (C)
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(A) edge node[label] {$3$} (D)
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(A) edge node[label] {$9$} (E)
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(B) edge node[label] {$6$} (E)
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(C) edge node[label] {$7$} (E)
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(C) edge node[label] {$2$} (F)
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(E) edge node[label] {$3$} (D)
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(E) edge node[label] {$4$} (F)
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(D) edge node[label] {$9$} (T)
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(E) edge node[label] {$5$} (T)
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(F) edge node[label] {$8$} (T)
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;
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\problem{}
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You are given a large network. How can you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?
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\begin{solution}
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Each iteration adds at least one unit of flow. So, we will find a maximum flow in at most $\min(\text{flow out of } S,~\text{flow into } T)$ iterations.
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\vspace{2ex}
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A simpler answer could only count the flow on $S$.
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\end{solution}
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\vfill
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\pagebreak |