\documentclass[../main.tex]{subfiles}


\begin{document}

	\section{The Ford-Fulkerson Algorithm}
	We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
	It works as follows:
	\begin{enumerate}
		\item[\texttt{00}] Take a weighted directed graph $G$.
		\item[\texttt{01}] Find any flow $F$ in $G$
		\item[\texttt{02}] Calculate $R$, the residual of $F$.
		\item[\texttt{03}] ~~~~If $S$ and $T$ are not connected in $R$, $F$ is a maximal flow. \texttt{HALT}.
		\item[\texttt{04}] Otherwise, find another flow $F_0$ in $R$.
		\item[\texttt{05}] Add $F_0$ to $F$
		\item[\texttt{06}] \texttt{GOTO 02}
	\end{enumerate}

	\problem{}
	Run the Ford-Fulkerson algorithm on the following graph. \\
	There is extra space on the next page.

	\begin{center}
		\begin{tikzpicture}
			% Nodes
			\begin{scope}[layer = nodes]
				\node[main] (S) at (-5mm, 0mm) {$S$};
				\node[main] (A) at (20mm, 20mm) {$A$};
				\node[main] (B) at (20mm, 0mm) {$B$};
				\node[main] (C) at (20mm, -20mm) {$C$};
				\node[main] (D) at (50mm, 20mm) {$D$};
				\node[main] (E) at (50mm, 0mm) {$E$};
				\node[main] (F) at (50mm, -20mm) {$F$};
				\node[main] (T) at (75mm, 0mm) {$T$};
			\end{scope}

			% Edges
			\draw[->]
				(S) edge node[label] {$8$} (A)
				(S) edge node[label] {$7$} (B)
				(S) edge node[label] {$4$} (C)

				(A) edge node[label] {$2$} (B)
				(B) edge node[label] {$5$} (C)

				(A) edge node[label] {$3$} (D)
				(A) edge node[label] {$9$} (E)
				(B) edge node[label] {$6$} (E)
				(C) edge node[label] {$7$} (E)
				(C) edge node[label] {$2$} (F)

				(E) edge node[label] {$3$} (D)
				(E) edge node[label] {$4$} (F)

				(D) edge node[label] {$9$} (T)
				(E) edge node[label] {$5$} (T)
				(F) edge node[label] {$8$} (T)
			;

		\end{tikzpicture}
	\end{center}

	\begin{solution}
		The maximum flow is $17$.
	\end{solution}

	\vspace{5mm}

	\pagebreak

	\begin{center}
		\begin{tikzpicture}
			% Nodes
			\begin{scope}[layer = nodes]
				\node[main] (S) at (-5mm, 0mm) {$S$};
				\node[main] (A) at (20mm, 20mm) {$A$};
				\node[main] (B) at (20mm, 0mm) {$B$};
				\node[main] (C) at (20mm, -20mm) {$C$};
				\node[main] (D) at (50mm, 20mm) {$D$};
				\node[main] (E) at (50mm, 0mm) {$E$};
				\node[main] (F) at (50mm, -20mm) {$F$};
				\node[main] (T) at (75mm, 0mm) {$T$};
			\end{scope}

			% Edges
			\draw[->]
				(S) edge node[label] {$8$} (A)
				(S) edge node[label] {$7$} (B)
				(S) edge node[label] {$4$} (C)

				(A) edge node[label] {$2$} (B)
				(B) edge node[label] {$5$} (C)

				(A) edge node[label] {$3$} (D)
				(A) edge node[label] {$9$} (E)
				(B) edge node[label] {$6$} (E)
				(C) edge node[label] {$7$} (E)
				(C) edge node[label] {$2$} (F)

				(E) edge node[label] {$3$} (D)
				(E) edge node[label] {$4$} (F)

				(D) edge node[label] {$9$} (T)
				(E) edge node[label] {$5$} (T)
				(F) edge node[label] {$8$} (T)
			;

		\end{tikzpicture}
	\end{center}

	\vfill
	\pagebreak

	\problem{}
	You are given a large network. How would you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?

	\begin{solution}
		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.

		\vspace{2ex}

		A simpler answer could only count the flow on $S$.

	\end{solution}

	\vfill
	\pagebreak
\end{document}