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158
Advanced/Graph Algorithms/main.tex
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@ -6,59 +6,8 @@
solutions solutions
]{ormc_handout} ]{ormc_handout}
\usepackage{subfiles}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{shapes.geometric}
% We put nodes in a separate layer, so we can \input{tikxset}
% slightly overlap with paths for a perfect fit
\pgfdeclarelayer{nodes}
\pgfdeclarelayer{path}
\pgfsetlayers{main,nodes}
% Layer settings
\tikzset{
% Layer hack, lets us write
% later = * in scopes.
layer/.style = {
execute at begin scope={\pgfonlayer{#1}},
execute at end scope={\endpgfonlayer}
},
%
% Arrowhead tweaks
>={Latex[ width=2mm, length=2mm ]},
label/.style = {
circle,
% For automatic red background in solutions
fill = \ORMCbgcolor,
draw = none
},
%
% Nodes
main/.style = {
draw,
circle,
fill = white
},
%
% Flow annotations
flow/.style = {
opacity = 1,
thin,
inner xsep = 2.5mm,
inner ysep = 2.5mm
},
%
% Paths
path/.style = {
line width = 4mm,
draw = black,
% Lengthen paths so they're
% completely under nodes.
line cap = rect,
opacity = 0.3
}
}
\begin{document} \begin{document}
@ -69,109 +18,14 @@
{Prepared by Mark on \today} {Prepared by Mark on \today}
\subfile{parts/00 review} \input{parts/00 review}
\subfile{parts/01 flow} \input{parts/01 flow}
\subfile{parts/02 residual} \input{parts/02 residual}
\subfile{parts/03 fulkerson} \input{parts/03 fulkerson}
\input{parts/04 applications}
\problem{Maximum Cardinality Matching}
A \textit{matching} is a subset of edges in a bipartite graph. Nodes in a matching must not have more than one edge connected to them. \\
A matching is \textit{maximal} if it has more edges than any other matching.
\vspace{5mm}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Initial Graph \\
\vspace{2mm}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (A1) at (0mm, 24mm) {};
\node[main] (A2) at (0mm, 18mm) {};
\node[main] (A3) at (0mm, 12mm) {};
\node[main] (A4) at (0mm, 6mm) {};
\node[main] (A5) at (0mm, 0mm) {};
\node[main] (B1) at (20mm, 24mm) {};
\node[main] (B2) at (20mm, 18mm) {};
\node[main] (B3) at (20mm, 12mm) {};
\node[main] (B4) at (20mm, 6mm) {};
\node[main] (B5) at (20mm, 0mm) {};
\end{scope}
% Edges
\draw
(A1) edge (B2)
(A1) edge (B3)
(A2) edge (B1)
(A2) edge (B4)
(A4) edge (B3)
(A2) edge (B3)
(A5) edge (B3)
(A5) edge (B4)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Maximal Matching \\
\vspace{2mm}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (A1) at (0mm, 24mm) {};
\node[main] (A2) at (0mm, 18mm) {};
\node[main] (A3) at (0mm, 12mm) {};
\node[main] (A4) at (0mm, 6mm) {};
\node[main] (A5) at (0mm, 0mm) {};
\node[main] (B1) at (20mm, 24mm) {};
\node[main] (B2) at (20mm, 18mm) {};
\node[main] (B3) at (20mm, 12mm) {};
\node[main] (B4) at (20mm, 6mm) {};
\node[main] (B5) at (20mm, 0mm) {};
\end{scope}
% Edges
\draw[opacity = 0.4]
(A1) edge (B2)
(A1) edge (B3)
(A2) edge (B1)
(A2) edge (B4)
(A4) edge (B3)
(A4) edge (B3)
(A5) edge (B3)
(A5) edge (B4)
;
\draw
(A1) edge (B2)
(A2) edge (B1)
(A4) edge (B3)
(A5) edge (B4)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\vspace{5mm}
Devise an algorithm to find a maximal matching in any bipartite graph. \\
Find an upper bound for its runtime.
\begin{solution}
Turn this into a maximum flow problem and use FF. \\
Connect a node $S$ to all nodes in the left group and a node $T$ to all nodes in the right group. All edges have capacity 1.
\vspace{2ex}
Just like FF, this algorithm will take at most $\min(\# \text{ left nodes}, \# \text{ right nodes})$ iterations.
\end{solution}
\end{document} \end{document}

147
Advanced/Graph Algorithms/parts/00 review.tex
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@ -1,91 +1,84 @@
\documentclass[../main.tex]{subfiles} \section{Review}
\definition{}
A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
A \textit{weighted graph} is a graph that features weights on its edges. \\
A weighted directed graph is shown below.
\begin{document} \begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}
\node[main] (A) {$A$};
\node[main] (B) [below right of = A] {$B$};
\node[main] (C) [below left of = A] {$C$};
\end{scope}
\section{Review} % Edges
\draw[->]
(A) edge[bend right] node[label] {$4$} (B)
(B) edge node[label] {$2$} (C)
(C) edge node[label] {$2$} (A)
(B) edge[bend right] node[label] {$1$} (A)
;
\end{tikzpicture}
\end{center}
\definition{} \vfill
A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
A \textit{weighted graph} is a graph that features weights on its edges. \\
A weighted directed graph is shown below.
\begin{center} \definition{}
\begin{tikzpicture}[node distance = 20mm] We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$ so that no two nodes in the same group are connected by an edge. \\
% Nodes The following graph is bipartite:
\begin{scope}
\node[main] (A) {$A$};
\node[main] (B) [below right of = A] {$B$};
\node[main] (C) [below left of = A] {$C$};
\end{scope}
% Edges \begin{center}
\draw[->] \begin{tikzpicture}
(A) edge[bend right] node[label] {$4$} (B) % Nodes
(B) edge node[label] {$2$} (C) \begin{scope}
(C) edge node[label] {$2$} (A) \node[main] (A) at (0mm, 0mm) {$A$};
(B) edge[bend right] node[label] {$1$} (A) \node[main] (B) at (0mm, -10mm) {$B$};
; \node[main] (C) at (0mm, -20mm) {$C$};
\end{tikzpicture}
\end{center}
\vfill \node[main] (D) at (20mm, 0mm) {$D$};
\node[main] (E) at (20mm, -10mm) {$E$};
\node[main] (F) at (20mm, -20mm) {$F$};
\end{scope}
\definition{} % Edges
We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$ so that no two nodes in the same group are connected by an edge. \\ \draw
The following graph is bipartite: (A) edge (D)
(A) edge (E)
(B) edge (F)
(C) edge (E)
(C) edge (D)
;
\end{tikzpicture}
\end{center}
\begin{center} \vfill
\begin{tikzpicture}
% Nodes
\begin{scope}
\node[main] (A) at (0mm, 0mm) {$A$};
\node[main] (B) at (0mm, -10mm) {$B$};
\node[main] (C) at (0mm, -20mm) {$C$};
\node[main] (D) at (20mm, 0mm) {$D$}; \definition{}
\node[main] (E) at (20mm, -10mm) {$E$}; We say two nodes $A$ ane $B$ are \textit{connected} if we can reach $A$ from $B$ and $B$ from $A$ by walking along (possibly directed) edges. We say a graph is connected if all its nodes are connected to each other.\\
\node[main] (F) at (20mm, -20mm) {$F$};
\end{scope}
% Edges The bipartite graph above and the directed graph below are not connected.
\draw
(A) edge (D)
(A) edge (E)
(B) edge (F)
(C) edge (E)
(C) edge (D)
;
\end{tikzpicture}
\end{center}
\vfill \begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}
\node[main] (A) {$A$};
\node[main] (B) [below right of = A] {$B$};
\node[main] (C) [below left of = A] {$C$};
\end{scope}
\definition{} % Edges
We say two nodes $A$ ane $B$ are \textit{connected} if we can reach $A$ from $B$ and $B$ from $A$ by walking along (possibly directed) edges. We say a graph is connected if all its nodes are connected to each other.\\ \draw[->]
(A) edge[bend right] (B)
(B) edge[bend right] (A)
(B) edge (C)
;
\end{tikzpicture}
\end{center}
The bipartite graph above and the directed graph below are not connected. \vfill
\pagebreak
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}
\node[main] (A) {$A$};
\node[main] (B) [below right of = A] {$B$};
\node[main] (C) [below left of = A] {$C$};
\end{scope}
% Edges
\draw[->]
(A) edge[bend right] (B)
(B) edge[bend right] (A)
(B) edge (C)
;
\end{tikzpicture}
\end{center}
\vfill
\pagebreak
\end{document}

651
Advanced/Graph Algorithms/parts/01 flow.tex
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@ -1,363 +1,356 @@
\documentclass[../main.tex]{subfiles} \section{Network Flow}
\generic{Networks}
Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc.
\begin{document} \vspace{1ex}
\section{Network Flow} We'll represent our network with a connected directed weighted graph. If we take a city, edges will be highways and cities will be nodes. There are a few conditions for a valid network graph:
\generic{Networks} \begin{itemize}
Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc. \item The weight of each edge represents its capacity, the number of lanes in the highway.
\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{}
\item Node $S$ is a \textit{source}: it produces flow.
\item Node $T$ is a \textit{sink}: it consumes flow.
\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out.
\end{itemize}
\vspace{1ex} \footnotetext{An edge with capacity zero is equivalent to an edge that does not exist; An edge with negative capacity is equivalent to an edge in the opposite direction.}
We'll represent our network with a connected directed weighted graph. If we take a city, edges will be highways and cities will be nodes. There are a few conditions for a valid network graph: Here is an example of a such a graph:
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\hrule{}
\generic{Flow}
In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway:
\vspace{2ex}
\begin{minipage}{0.33\textwidth}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S) -- node[above left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{0.65\textwidth}
There are a few things to notice here:
\begin{itemize} \begin{itemize}
\item The weight of each edge represents its capacity, the number of lanes in the highway. \item Highlighted edges carry flow.
\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{} \item Numbers in parentheses tell us how much flow each edge carries.
\item Node $S$ is a \textit{source}: it produces flow. \item The flow along an edge is always positive or zero.
\item Node $T$ is a \textit{sink}: it consumes flow. \item Flow comes from $S$ and goes towards $T$.
\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out. \item Flow is conserved: all flow produced by $S$ enters $T$.
\end{itemize} \end{itemize}
\end{minipage}
\footnotetext{An edge with capacity zero is equivalent to an edge that does not exist; An edge with negative capacity is equivalent to an edge in the opposite direction.} \vspace{1ex}
Here is an example of a such a graph: The \textit{magnitude} of a flow\footnotemark{} is the number of \say{flow-units} that go from $S$ to $T$. \\
\begin{center}
\begin{tikzpicture}
% Nodes We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can push from $S$ to $T$?
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges \footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.}
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\hrule{} \problem{}
What is the magnitude of the flow above?
\generic{Flow} \vfill
In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway: \pagebreak
\problem{}
Find a flow with magnitude 2 on the graph below.
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\vfill
\problem{}
Find a maximal flow on the graph below. \\
\hint{The total capacity coming out of $S$ is 3, so any flow must have magnitude $\leq 3$.}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\vfill
\pagebreak
\section{Combining Flows}
It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. For example, consider the following flows:
\vspace{2ex}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S) -- node[above left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S)
-- node[below left, flow] {$(1)$} (B)
-- node[left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\vspace{1cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Combining these, we get the following:
\vspace{2ex} \vspace{2ex}
\begin{minipage}{0.33\textwidth} \begin{tikzpicture}
\begin{center} % Nodes
\begin{tikzpicture} \begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Nodes % Edges
\begin{scope}[layer = nodes] \draw[->]
\node[main] (S) at (+00mm, +00mm) {$S$}; (S) edge node[label] {$1$} (A)
\node[main] (A) at (+15mm, +15mm) {$A$}; (A) edge node[label] {$2$} (T)
\node[main] (B) at (+15mm, -15mm) {$B$}; (B) edge node[label] {$2$} (A)
\node[main] (T) at (+30mm, +00mm) {$T$}; (S) edge node[label] {$2$} (B)
\end{scope} (B) edge node[label] {$1$} (T)
;
% Edges % Flow
\draw[->] \draw[path]
(S) edge node[label] {$1$} (A) (S)
(A) edge node[label] {$4$} (T) -- node[below left, flow] {$(1)$} (B)
(B) edge node[label] {$2$} (A) -- node[left, flow] {$(1)$} (A)
(S) edge node[label] {$2$} (B) -- node[above right, flow] {$(2) = (1) + (1)$} (T)
(B) edge node[label] {$1$} (T) (S)
; -- node[above left, flow] {$(1)$} (A)
;
% Flow \end{tikzpicture}
\draw[path] \end{center}
(S) -- node[above left, flow] {$(1)$} (A) \end{minipage}
-- node[above right, flow] {$(1)$} (T) \hfill
; \begin{minipage}[t]{0.48\textwidth}
\end{tikzpicture} \raggedright
\end{center} When adding flows, we must respect edge capacities.
\end{minipage}
\begin{minipage}{0.65\textwidth}
There are a few things to notice here:
\begin{itemize}
\item Highlighted edges carry flow.
\item Numbers in parentheses tell us how much flow each edge carries.
\item The flow along an edge is always positive or zero.
\item Flow comes from $S$ and goes towards $T$.
\item Flow is conserved: all flow produced by $S$ enters $T$.
\end{itemize}
\end{minipage}
\vspace{1ex} \vspace{1ex}
The \textit{magnitude} of a flow\footnotemark{} is the number of \say{flow-units} that go from $S$ to $T$. \\ For example, we could not add these graphs if the magnitude of flow in the right graph above was 2.
We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can push from $S$ to $T$? \vspace{1ex}
\footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.} This is because the capacity of the top-right edge is 2, and $2 + 1 > 2$.
\end{minipage}
\problem{} \vspace{2ex}
What is the magnitude of the flow above? \hrule
\vspace{2ex}
\vfill \problem{}
\pagebreak Combine the following flows and ensure that the flow along all edges remains within capacity.
\problem{} \vspace{2ex}
Find a flow with magnitude 2 on the graph below.
\begin{center} \begin{minipage}[t]{0.48\textwidth}
\begin{tikzpicture} \begin{center}
% Nodes \begin{tikzpicture}[node distance = 20mm]
\begin{scope}[layer = nodes] % Nodes
\node[main] (S) at (+00mm, +00mm) {$S$}; \begin{scope}[layer = nodes]
\node[main] (A) at (+15mm, +15mm) {$A$}; \node[main] (S) {$S$};
\node[main] (B) at (+15mm, -15mm) {$B$}; \node[main] (A) [above right of = S] {$A$};
\node[main] (T) at (+30mm, +00mm) {$T$}; \node[main] (B) [below right of = S] {$B$};
\end{scope} \node[main] (C) [right of = A] {$C$};
\node[main] (D) [right of = B] {$D$};
\node[main] (T) [above right of = D] {$T$};
\end{scope}
% Edges % Edges
\draw[->] \draw[->]
(S) edge node[label] {$1$} (A) (S) edge node[label] {$5$} (A)
(A) edge node[label] {$4$} (T) (A) edge node[label] {$3$} (C)
(B) edge node[label] {$2$} (A) (C) edge node[label] {$2$} (T)
(S) edge node[label] {$2$} (B) (A) edge node[label] {$4$} (D)
(B) edge node[label] {$1$} (T) (S) edge node[label] {$4$} (B)
; (B) edge node[label] {$1$} (D)
\end{tikzpicture} (D) edge node[label] {$2$} (T)
\end{center} ;
\vfill % Flow
\draw[path]
(S)
-- node[above left, flow] {$(2)$} (A)
-- node[above, flow] {$(2)$} (C)
-- node[above right, flow] {$(2)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (C) [right of = A] {$C$};
\node[main] (D) [right of = B] {$D$};
\node[main] (T) [above right of = D] {$T$};
\end{scope}
\problem{} % Edges
Find a maximal flow on the graph below. \\ \draw[->]
\hint{The total capacity coming out of $S$ is 3, so any flow must have magnitude $\leq 3$.} (S) edge node[label] {$5$} (A)
(A) edge node[label] {$3$} (C)
(C) edge node[label] {$2$} (T)
(A) edge node[label] {$4$} (D)
(S) edge node[label] {$4$} (B)
(B) edge node[label] {$1$} (D)
(D) edge node[label] {$2$} (T)
;
\begin{center} % Flow
\begin{tikzpicture} \draw[path]
% Nodes (S)
\begin{scope}[layer = nodes] -- node[above left, flow] {$(2)$} (A)
\node[main] (S) at (+00mm, +00mm) {$S$}; -- node[above right, flow] {$(2)$} (D)
\node[main] (A) at (+15mm, +15mm) {$A$}; -- node[below right, flow] {$(2)$} (T)
\node[main] (B) at (+15mm, -15mm) {$B$}; ;
\node[main] (T) at (+30mm, +00mm) {$T$}; \end{tikzpicture}
\end{scope} \end{center}
\end{minipage}
% Edges \vfill
\draw[->] \pagebreak
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$4$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\vfill
\pagebreak
\section{Combining Flows}
It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. For example, consider the following flows:
\vspace{2ex}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S) -- node[above left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S)
-- node[below left, flow] {$(1)$} (B)
-- node[left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\vspace{1cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Combining these, we get the following:
\vspace{2ex}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) at (+00mm, +00mm) {$S$};
\node[main] (A) at (+15mm, +15mm) {$A$};
\node[main] (B) at (+15mm, -15mm) {$B$};
\node[main] (T) at (+30mm, +00mm) {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$1$} (A)
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
% Flow
\draw[path]
(S)
-- node[below left, flow] {$(1)$} (B)
-- node[left, flow] {$(1)$} (A)
-- node[above right, flow] {$(2) = (1) + (1)$} (T)
(S)
-- node[above left, flow] {$(1)$} (A)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\raggedright
When adding flows, we must respect edge capacities.
\vspace{1ex}
For example, we could not add these graphs if the magnitude of flow in the right graph above was 2.
\vspace{1ex}
This is because the capacity of the top-right edge is 2, and $2 + 1 > 2$.
\end{minipage}
\vspace{2ex}
\hrule
\vspace{2ex}
\problem{}
Combine the following flows and ensure that the flow along all edges remains within capacity.
\vspace{2ex}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (C) [right of = A] {$C$};
\node[main] (D) [right of = B] {$D$};
\node[main] (T) [above right of = D] {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$5$} (A)
(A) edge node[label] {$3$} (C)
(C) edge node[label] {$2$} (T)
(A) edge node[label] {$4$} (D)
(S) edge node[label] {$4$} (B)
(B) edge node[label] {$1$} (D)
(D) edge node[label] {$2$} (T)
;
% Flow
\draw[path]
(S)
-- node[above left, flow] {$(2)$} (A)
-- node[above, flow] {$(2)$} (C)
-- node[above right, flow] {$(2)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (C) [right of = A] {$C$};
\node[main] (D) [right of = B] {$D$};
\node[main] (T) [above right of = D] {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$5$} (A)
(A) edge node[label] {$3$} (C)
(C) edge node[label] {$2$} (T)
(A) edge node[label] {$4$} (D)
(S) edge node[label] {$4$} (B)
(B) edge node[label] {$1$} (D)
(D) edge node[label] {$2$} (T)
;
% Flow
\draw[path]
(S)
-- node[above left, flow] {$(2)$} (A)
-- node[above right, flow] {$(2)$} (D)
-- node[below right, flow] {$(2)$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\vfill
\pagebreak
\end{document}

349
Advanced/Graph Algorithms/parts/02 residual.tex
View File

@ -1,147 +1,172 @@
\documentclass[../main.tex]{subfiles} \section{Residual Graphs}
As our network gets bigger, finding a maximum flow by hand becomes much more difficult. It will be convenient to have an algorithm that finds a maximal flow in any network.
\vspace{1ex}
\begin{document} The first thing we'll need to construct such an algorithm is a \textit{residual graph}.
\section{Residual Graphs} \vspace{2ex}
As our network gets bigger, finding a maximum flow by hand becomes much more difficult. It will be convenient to have an algorithm that finds a maximal flow in any network. \hrule
\vspace{1ex}
The first thing we'll need to construct such an algorithm is a \textit{residual graph}.
\vspace{2ex}
\hrule
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
We'll start with the following network and flow:
\begin{center} \begin{center}
\begin{minipage}[t]{0.48\textwidth} \begin{tikzpicture}[node distance = 20mm]
We'll start with the following network and flow: % Nodes
\begin{center} \begin{scope}[layer = nodes]
\begin{tikzpicture}[node distance = 20mm] \node[main] (S) {$S$};
% Nodes \node[main] (A) [above right of = S] {$A$};
\begin{scope}[layer = nodes] \node[main] (B) [below right of = S] {$B$};
\node[main] (S) {$S$}; \node[main] (T) [above right of = B] {$T$};
\node[main] (A) [above right of = S] {$A$}; \end{scope}
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges % Edges
\draw[->] \draw[->]
(S) edge node[label] {$1$} (A) (S) edge node[label] {$1$} (A)
(A) edge node[label] {$3$} (T) (A) edge node[label] {$3$} (T)
(B) edge node[label] {$2$} (A) (B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B) (S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T) (B) edge node[label] {$1$} (T)
; ;
% Flow % Flow
\draw[path] \draw[path]
(S) -- node[above left, flow] {$(1)$} (A) (S) -- node[above left, flow] {$(1)$} (A)
-- node[above right, flow] {$(1)$} (T) -- node[above right, flow] {$(1)$} (T)
; ;
\end{tikzpicture} \end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
First, we'll copy all nodes and \say{unused} edges:
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges
\draw[->]
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\end{center} \end{center}
\hrule
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
Then, we'll add the unused capacity of \say{used} edges: (Note that $3 - 1 = 2$)
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges
\draw[->]
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
Finally, we'll add \say{used} capacity as edges in the opposite direction:
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges
\draw[->]
(A) edge node[label] {$1$} (S)
(T) edge [bend right] node[label] {$1$} (A)
(A) edge [bend right] node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
This graph is the residual of the original flow.
\end{minipage} \end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
First, we'll copy all nodes and \say{unused} edges:
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges
\draw[->]
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center} \end{center}
\end{minipage}
\end{center}
\hrule \hrule
\vspace{3ex}
You can think of the residual graph as a \say{list of possible changes} to the original flow. \\ \begin{center}
There are two ways we can change a flow: \begin{minipage}[t]{0.48\textwidth}
\begin{itemize} Then, we'll add the unused capacity of \say{used} edges: (Note that $3 - 1 = 2$)
\item We can add flow along a path \begin{center}
\item We can remove flow along another path \begin{tikzpicture}[node distance = 20mm]
\end{itemize} % Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
\vspace{1ex} % Edges
\draw[->]
(A) edge node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
Finally, we'll add \say{used} capacity as edges in the opposite direction:
\begin{center}
\begin{tikzpicture}[node distance = 20mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
A residual graph captures both of these actions, showing us where we can add flow (forward edges) and where we can remove it (reverse edges). Note that \say{removing} flow along an edge is equivalent to adding flow in the opposite direction. % Edges
\draw[->]
(A) edge node[label] {$1$} (S)
(T) edge [bend right] node[label] {$1$} (A)
(A) edge [bend right] node[label] {$2$} (T)
(B) edge node[label] {$2$} (A)
(S) edge node[label] {$2$} (B)
(B) edge node[label] {$1$} (T)
;
\end{tikzpicture}
\end{center}
This graph is the residual of the original flow.
\end{minipage}
\end{center}
\hrule
\vspace{3ex}
You can think of the residual graph as a \say{list of possible changes} to the original flow. \\
There are two ways we can change a flow:
\begin{itemize}
\item We can add flow along a path
\item We can remove flow along another path
\end{itemize}
\vspace{1ex}
A residual graph captures both of these actions, showing us where we can add flow (forward edges) and where we can remove it (reverse edges). Note that \say{removing} flow along an edge is equivalent to adding flow in the opposite direction.
\vfill \vfill
\pagebreak \pagebreak
\problem{}<FindResidual> \problem{}<FindResidual>
Construct the residual of this flow. Construct the residual of this flow.
\begin{center}
\begin{tikzpicture}[node distance = 25mm]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges
\draw[->]
(S) edge node[label] {$2$} (A)
(A) edge node[label] {$1$} (T)
(A) edge node[label] {$3$} (B)
(S) edge node[label] {$1$} (B)
(B) edge node[label] {$2$} (T)
;
% Flow
\draw[path]
(S)
-- node[above left, flow] {$(2)$} (A)
-- node[left, flow] {$(2)$} (B)
-- node[below right, flow] {$(2)$} (T)
;
\end{tikzpicture}
\end{center}
\begin{solution}
\begin{center} \begin{center}
\begin{tikzpicture}[node distance = 25mm] \begin{tikzpicture}[node distance = 25mm]
% Nodes % Nodes
@ -154,66 +179,34 @@
% Edges % Edges
\draw[->] \draw[->]
(S) edge node[label] {$2$} (A) (A) edge node[label] {$2$} (S)
(A) edge node[label] {$1$} (T) (A) edge node[label] {$1$} (T)
(A) edge node[label] {$3$} (B) (A) edge[out=295,in=65] node[label] {$1$} (B)
(B) edge[out=115,in=245] node[label] {$2$} (A)
(S) edge node[label] {$1$} (B) (S) edge node[label] {$1$} (B)
(B) edge node[label] {$2$} (T) (T) edge node[label] {$2$} (B)
;
% Flow
\draw[path]
(S)
-- node[above left, flow] {$(2)$} (A)
-- node[left, flow] {$(2)$} (B)
-- node[below right, flow] {$(2)$} (T)
; ;
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
\end{solution}
\begin{solution} \vfill
\begin{center} \problem{}
\begin{tikzpicture}[node distance = 25mm] Is the flow in \ref{FindResidual} maximal? \\
% Nodes If it isn't, find a maximal flow. \\
\begin{scope}[layer = nodes] \hint{Look at the residual graph. Can we add flow along another path?}
\node[main] (S) {$S$};
\node[main] (A) [above right of = S] {$A$};
\node[main] (B) [below right of = S] {$B$};
\node[main] (T) [above right of = B] {$T$};
\end{scope}
% Edges \vfill
\draw[->] \pagebreak
(A) edge node[label] {$2$} (S)
(A) edge node[label] {$1$} (T)
(A) edge[out=295,in=65] node[label] {$1$} (B)
(B) edge[out=115,in=245] node[label] {$2$} (A)
(S) edge node[label] {$1$} (B)
(T) edge node[label] {$2$} (B)
;
\end{tikzpicture}
\end{center}
\end{solution}
\vfill \problem{}
\problem{} Show that...
Is the flow in \ref{FindResidual} maximal? \\ \begin{enumerate}
If it isn't, find a maximal flow. \\ \item A maximal flow exists in every network with integral\footnotemark{} edge weights.
\hint{Look at the residual graph. Can we add flow along another path?} \item Every edge in this flow carries an integral amount of flow
\end{enumerate}
\vfill \footnotetext{Integral = \say{integer} as an adjective.}
\pagebreak
\problem{} \vfill
Show that... \pagebreak
\begin{enumerate}
\item A maximal flow exists in every network with integral\footnotemark{} edge weights.
\item Every edge in this flow carries an integral amount of flow
\end{enumerate}
\footnotetext{Integral = \say{integer} as an adjective.}
\vfill
\end{document}

199
Advanced/Graph Algorithms/parts/03 fulkerson.tex
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@ -1,128 +1,123 @@
\documentclass[../main.tex]{subfiles}
\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}
\begin{document} \problem{}
Run the Ford-Fulkerson algorithm on the following graph. \\
There is extra space on the next page.
\section{The Ford-Fulkerson Algorithm} \begin{center}
We now have all the tools we need to construct an algorithm that finds a maximal flow. \\ \begin{tikzpicture}
It works as follows: % Nodes
\begin{enumerate} \begin{scope}[layer = nodes]
\item[\texttt{00}] Take a weighted directed graph $G$. \node[main] (S) at (-5mm, 0mm) {$S$};
\item[\texttt{01}] Find any flow $F$ in $G$ \node[main] (A) at (20mm, 20mm) {$A$};
\item[\texttt{02}] Calculate $R$, the residual of $F$. \node[main] (B) at (20mm, 0mm) {$B$};
\item[\texttt{03}] ~~~~If $S$ and $T$ are not connected in $R$, $F$ is a maximal flow. \texttt{HALT}. \node[main] (C) at (20mm, -20mm) {$C$};
\item[\texttt{04}] Otherwise, find another flow $F_0$ in $R$. \node[main] (D) at (50mm, 20mm) {$D$};
\item[\texttt{05}] Add $F_0$ to $F$ \node[main] (E) at (50mm, 0mm) {$E$};
\item[\texttt{06}] \texttt{GOTO 02} \node[main] (F) at (50mm, -20mm) {$F$};
\end{enumerate} \node[main] (T) at (75mm, 0mm) {$T$};
\end{scope}
\problem{} % Edges
Run the Ford-Fulkerson algorithm on the following graph. \\ \draw[->]
There is extra space on the next page. (S) edge node[label] {$8$} (A)
(S) edge node[label] {$7$} (B)
(S) edge node[label] {$4$} (C)
\begin{center} (A) edge node[label] {$2$} (B)
\begin{tikzpicture} (B) edge node[label] {$5$} (C)
% 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 (A) edge node[label] {$3$} (D)
\draw[->] (A) edge node[label] {$9$} (E)
(S) edge node[label] {$8$} (A) (B) edge node[label] {$6$} (E)
(S) edge node[label] {$7$} (B) (C) edge node[label] {$7$} (E)
(S) edge node[label] {$4$} (C) (C) edge node[label] {$2$} (F)
(A) edge node[label] {$2$} (B) (E) edge node[label] {$3$} (D)
(B) edge node[label] {$5$} (C) (E) edge node[label] {$4$} (F)
(A) edge node[label] {$3$} (D) (D) edge node[label] {$9$} (T)
(A) edge node[label] {$9$} (E) (E) edge node[label] {$5$} (T)
(B) edge node[label] {$6$} (E) (F) edge node[label] {$8$} (T)
(C) edge node[label] {$7$} (E) ;
(C) edge node[label] {$2$} (F)
(E) edge node[label] {$3$} (D) \end{tikzpicture}
(E) edge node[label] {$4$} (F) \end{center}
(D) edge node[label] {$9$} (T) \begin{solution}
(E) edge node[label] {$5$} (T) The maximum flow is $17$.
(F) edge node[label] {$8$} (T) \end{solution}
;
\end{tikzpicture} \vspace{5mm}
\end{center}
\begin{solution} \pagebreak
The maximum flow is $17$.
\end{solution}
\vspace{5mm} \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}
\pagebreak % Edges
\draw[->]
(S) edge node[label] {$8$} (A)
(S) edge node[label] {$7$} (B)
(S) edge node[label] {$4$} (C)
\begin{center} (A) edge node[label] {$2$} (B)
\begin{tikzpicture} (B) edge node[label] {$5$} (C)
% 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 (A) edge node[label] {$3$} (D)
\draw[->] (A) edge node[label] {$9$} (E)
(S) edge node[label] {$8$} (A) (B) edge node[label] {$6$} (E)
(S) edge node[label] {$7$} (B) (C) edge node[label] {$7$} (E)
(S) edge node[label] {$4$} (C) (C) edge node[label] {$2$} (F)
(A) edge node[label] {$2$} (B) (E) edge node[label] {$3$} (D)
(B) edge node[label] {$5$} (C) (E) edge node[label] {$4$} (F)
(A) edge node[label] {$3$} (D) (D) edge node[label] {$9$} (T)
(A) edge node[label] {$9$} (E) (E) edge node[label] {$5$} (T)
(B) edge node[label] {$6$} (E) (F) edge node[label] {$8$} (T)
(C) edge node[label] {$7$} (E) ;
(C) edge node[label] {$2$} (F)
(E) edge node[label] {$3$} (D) \end{tikzpicture}
(E) edge node[label] {$4$} (F) \end{center}
(D) edge node[label] {$9$} (T) \vfill
(E) edge node[label] {$5$} (T) \pagebreak
(F) edge node[label] {$8$} (T)
;
\end{tikzpicture} \problem{}
\end{center} 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?
\vfill \begin{solution}
\pagebreak 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.
\problem{} \vspace{2ex}
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} A simpler answer could only count the flow on $S$.
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} \end{solution}
A simpler answer could only count the flow on $S$. \vfill
\pagebreak
\end{solution}
\vfill
\pagebreak
\end{document}

187
Advanced/Graph Algorithms/parts/04 applications.tex Executable file
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\section{Applications}
\problem{Maximum Cardinality Matching}
A \textit{matching} is a subset of edges in a bipartite graph. Nodes in a matching must not have more than one edge connected to them. \\
A matching is \textit{maximal} if it has more edges than any other matching.
\vspace{5mm}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Initial Graph \\
\vspace{2mm}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (A1) at (0mm, 24mm) {};
\node[main] (A2) at (0mm, 18mm) {};
\node[main] (A3) at (0mm, 12mm) {};
\node[main] (A4) at (0mm, 6mm) {};
\node[main] (A5) at (0mm, 0mm) {};
\node[main] (B1) at (20mm, 24mm) {};
\node[main] (B2) at (20mm, 18mm) {};
\node[main] (B3) at (20mm, 12mm) {};
\node[main] (B4) at (20mm, 6mm) {};
\node[main] (B5) at (20mm, 0mm) {};
\end{scope}
% Edges
\draw
(A1) edge (B2)
(A1) edge (B3)
(A2) edge (B1)
(A2) edge (B4)
(A4) edge (B3)
(A2) edge (B3)
(A5) edge (B3)
(A5) edge (B4)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
Maximal Matching \\
\vspace{2mm}
\begin{tikzpicture}
% Nodes
\begin{scope}[layer = nodes]
\node[main] (A1) at (0mm, 24mm) {};
\node[main] (A2) at (0mm, 18mm) {};
\node[main] (A3) at (0mm, 12mm) {};
\node[main] (A4) at (0mm, 6mm) {};
\node[main] (A5) at (0mm, 0mm) {};
\node[main] (B1) at (20mm, 24mm) {};
\node[main] (B2) at (20mm, 18mm) {};
\node[main] (B3) at (20mm, 12mm) {};
\node[main] (B4) at (20mm, 6mm) {};
\node[main] (B5) at (20mm, 0mm) {};
\end{scope}
% Edges
\draw[opacity = 0.4]
(A1) edge (B2)
(A1) edge (B3)
(A2) edge (B1)
(A2) edge (B4)
(A4) edge (B3)
(A4) edge (B3)
(A5) edge (B3)
(A5) edge (B4)
;
\draw
(A1) edge (B2)
(A2) edge (B1)
(A4) edge (B3)
(A5) edge (B4)
;
\end{tikzpicture}
\end{center}
\end{minipage}
\vspace{5mm}
Devise an algorithm to find a maximal matching in any bipartite graph. \\
Find an upper bound for its runtime.
\begin{solution}
Turn this into a maximum flow problem and use FF. \\
Connect a node $S$ to all nodes in the left group and a node $T$ to all nodes in the right group. All edges have capacity 1.
\vspace{2ex}
Just like FF, this algorithm will take at most $\min(\# \text{ left nodes}, \# \text{ right nodes})$ iterations.
\end{solution}
\vfill
\pagebreak
\problem{Circulations with Demand}
Say we have a network of cities and power stations. Stations produce power; cities consume it.
Each station produces a limited amount of power, and each city has limited demand.
\vspace{2ex}
We can represent this power grid as a graph, with cities and stations as nodes and transmission lines as edges.
\vspace{2ex}
A simple example is below. There are two cities ($2$ and $4$) and two stations (both $-3$).
We'll represent station capacity with a negative number, since they \textit{consume} a negative amount of energy.
\begin{center}
\begin{tikzpicture}[
node distance = 25mm,
main/.style = {
draw,
circle,
fill = white,
minimum size = 8mm
},
]
% Nodes
\begin{scope}[layer = nodes]
\node[main] (S1) {$-3$};
\node[main] (S2) [below left of = S1] {$-3$};
\node[main] (C1) [below right of = S1] {$2$};
\node[main] (C2) [below right of = S2] {$4$};
\end{scope}
% Edges
\draw[->]
(S1) edge node[label] {$3$} (S2)
(S1) edge node[label] {$3$} (C1)
(S2) edge node[label] {$2$} (C1)
(S2) edge node[label] {$2$} (C2)
(C1) edge node[label] {$2$} (C2)
;
\end{tikzpicture}
\end{center}
We'd like to know if there exists a \textit{feasible circulation} in this network---that is, can we supply our cities with the energy they need without exceeding the capacity of power plants or transmission lines?
\vspace{2ex}
\textbf{Your job:} Devise an algorithm that solve this problem.
\vspace{2ex}
\note{\textbf{Bonus:} Say certain edges have a lower bound on their capacity, meaning that we must send \textit{at least} that much flow down the edge. Modify your algorithm to account for these additional constraints.}
\begin{solution}
Create a source node $S$, and connect it to each station with an edge. Set the capacity of that edge to the capacity of the station. \\
Create a sink node $T$ and do the same for cities.
\vspace{2ex}
This is now a maximum-flow problem with one source and one sink. Apply FF.
\linehack{}
To solve the bonus problem, we'll modify the network before running the algorithm above.
\vspace{2ex}
Say an edge from $A$ to $B$ has minimum capacity $l$ and maximum capacity $u \geq l$. Apply the following transformations:
\begin{itemize}
\item Add $l$ to the capacity of $A$
\item Subtract $l$ from the capacity of $B$
\item Subtract $l$ from the total capacity of the edge.
\end{itemize}
Do this for every edge that has a lower bound then apply the algorithm above.
\end{solution}
\vfill
\pagebreak

52
Advanced/Graph Algorithms/tikxset.tex Normal file
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@ -0,0 +1,52 @@
\usetikzlibrary{arrows.meta}
\usetikzlibrary{shapes.geometric}
% We put nodes in a separate layer, so we can
% slightly overlap with paths for a perfect fit
\pgfdeclarelayer{nodes}
\pgfdeclarelayer{path}
\pgfsetlayers{main,nodes}
% Layer settings
\tikzset{
% Layer hack, lets us write
% later = * in scopes.
layer/.style = {
execute at begin scope={\pgfonlayer{#1}},
execute at end scope={\endpgfonlayer}
},
%
% Arrowhead tweaks
>={Latex[ width=2mm, length=2mm ]},
label/.style = {
circle,
% For automatic red background in solutions
fill = \ORMCbgcolor,
draw = none
},
%
% Nodes
main/.style = {
draw,
circle,
fill = white
},
%
% Flow annotations
flow/.style = {
opacity = 1,
thin,
inner xsep = 2.5mm,
inner ysep = 2.5mm
},
%
% Paths
path/.style = {
line width = 4mm,
draw = black,
% Lengthen paths so they're
% completely under nodes.
line cap = rect,
opacity = 0.3
}
}