Added "applications" section
This commit is contained in:
parent
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@ -6,59 +6,8 @@
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solutions
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]{ormc_handout}
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\usepackage{subfiles}
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\usetikzlibrary{arrows.meta}
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\usetikzlibrary{shapes.geometric}
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% We put nodes in a separate layer, so we can
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% slightly overlap with paths for a perfect fit
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\pgfdeclarelayer{nodes}
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\pgfdeclarelayer{path}
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\pgfsetlayers{main,nodes}
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% Layer settings
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\tikzset{
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% Layer hack, lets us write
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% later = * in scopes.
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layer/.style = {
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execute at begin scope={\pgfonlayer{#1}},
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execute at end scope={\endpgfonlayer}
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},
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%
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% Arrowhead tweaks
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>={Latex[ width=2mm, length=2mm ]},
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label/.style = {
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circle,
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% For automatic red background in solutions
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fill = \ORMCbgcolor,
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draw = none
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},
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%
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% Nodes
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main/.style = {
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draw,
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circle,
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fill = white
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},
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%
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% Flow annotations
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flow/.style = {
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opacity = 1,
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thin,
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inner xsep = 2.5mm,
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inner ysep = 2.5mm
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},
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%
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% Paths
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path/.style = {
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line width = 4mm,
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draw = black,
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% Lengthen paths so they're
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% completely under nodes.
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line cap = rect,
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opacity = 0.3
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}
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}
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\input{tikxset}
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\begin{document}
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@ -69,109 +18,14 @@
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{Prepared by Mark on \today}
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\subfile{parts/00 review}
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\subfile{parts/01 flow}
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\subfile{parts/02 residual}
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\subfile{parts/03 fulkerson}
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\input{parts/00 review}
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\input{parts/01 flow}
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\input{parts/02 residual}
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\input{parts/03 fulkerson}
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\input{parts/04 applications}
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\problem{Maximum Cardinality Matching}
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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. \\
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A matching is \textit{maximal} if it has more edges than any other matching.
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\vspace{5mm}
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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Initial Graph \\
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\vspace{2mm}
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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] (A1) at (0mm, 24mm) {};
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\node[main] (A2) at (0mm, 18mm) {};
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\node[main] (A3) at (0mm, 12mm) {};
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\node[main] (A4) at (0mm, 6mm) {};
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\node[main] (A5) at (0mm, 0mm) {};
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\node[main] (B1) at (20mm, 24mm) {};
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\node[main] (B2) at (20mm, 18mm) {};
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\node[main] (B3) at (20mm, 12mm) {};
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\node[main] (B4) at (20mm, 6mm) {};
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\node[main] (B5) at (20mm, 0mm) {};
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\end{scope}
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% Edges
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\draw
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(A1) edge (B2)
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(A1) edge (B3)
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(A2) edge (B1)
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(A2) edge (B4)
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(A4) edge (B3)
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(A2) edge (B3)
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(A5) edge (B3)
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(A5) edge (B4)
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;
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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Maximal Matching \\
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\vspace{2mm}
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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] (A1) at (0mm, 24mm) {};
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\node[main] (A2) at (0mm, 18mm) {};
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\node[main] (A3) at (0mm, 12mm) {};
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\node[main] (A4) at (0mm, 6mm) {};
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\node[main] (A5) at (0mm, 0mm) {};
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\node[main] (B1) at (20mm, 24mm) {};
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\node[main] (B2) at (20mm, 18mm) {};
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\node[main] (B3) at (20mm, 12mm) {};
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\node[main] (B4) at (20mm, 6mm) {};
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\node[main] (B5) at (20mm, 0mm) {};
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\end{scope}
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% Edges
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\draw[opacity = 0.4]
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(A1) edge (B2)
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(A1) edge (B3)
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(A2) edge (B1)
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(A2) edge (B4)
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(A4) edge (B3)
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(A4) edge (B3)
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(A5) edge (B3)
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(A5) edge (B4)
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;
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\draw
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(A1) edge (B2)
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(A2) edge (B1)
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(A4) edge (B3)
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(A5) edge (B4)
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;
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\vspace{5mm}
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Devise an algorithm to find a maximal matching in any bipartite graph. \\
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Find an upper bound for its runtime.
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\begin{solution}
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Turn this into a maximum flow problem and use FF. \\
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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.
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\vspace{2ex}
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Just like FF, this algorithm will take at most $\min(\# \text{ left nodes}, \# \text{ right nodes})$ iterations.
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\end{solution}
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\end{document}
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@ -1,91 +1,84 @@
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\documentclass[../main.tex]{subfiles}
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\section{Review}
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\definition{}
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A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
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A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
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A \textit{weighted graph} is a graph that features weights on its edges. \\
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A weighted directed graph is shown below.
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\begin{document}
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\begin{center}
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\begin{tikzpicture}[node distance = 20mm]
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% Nodes
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\begin{scope}
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\node[main] (A) {$A$};
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\node[main] (B) [below right of = A] {$B$};
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\node[main] (C) [below left of = A] {$C$};
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\end{scope}
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\section{Review}
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% Edges
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\draw[->]
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(A) edge[bend right] node[label] {$4$} (B)
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(B) edge node[label] {$2$} (C)
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(C) edge node[label] {$2$} (A)
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(B) edge[bend right] node[label] {$1$} (A)
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;
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\end{tikzpicture}
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\end{center}
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\definition{}
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A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
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A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
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A \textit{weighted graph} is a graph that features weights on its edges. \\
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A weighted directed graph is shown below.
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\vfill
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\begin{center}
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\begin{tikzpicture}[node distance = 20mm]
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% Nodes
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\begin{scope}
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\node[main] (A) {$A$};
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\node[main] (B) [below right of = A] {$B$};
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\node[main] (C) [below left of = A] {$C$};
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\end{scope}
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\definition{}
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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. \\
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The following graph is bipartite:
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% Edges
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\draw[->]
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(A) edge[bend right] node[label] {$4$} (B)
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(B) edge node[label] {$2$} (C)
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(C) edge node[label] {$2$} (A)
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(B) edge[bend right] node[label] {$1$} (A)
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;
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\end{tikzpicture}
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\end{center}
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\begin{center}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}
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\node[main] (A) at (0mm, 0mm) {$A$};
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\node[main] (B) at (0mm, -10mm) {$B$};
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\node[main] (C) at (0mm, -20mm) {$C$};
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\vfill
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\node[main] (D) at (20mm, 0mm) {$D$};
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\node[main] (E) at (20mm, -10mm) {$E$};
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\node[main] (F) at (20mm, -20mm) {$F$};
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\end{scope}
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\definition{}
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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. \\
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The following graph is bipartite:
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% Edges
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\draw
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(A) edge (D)
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(A) edge (E)
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(B) edge (F)
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(C) edge (E)
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(C) edge (D)
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;
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\end{tikzpicture}
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\end{center}
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\begin{center}
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\begin{tikzpicture}
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% Nodes
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\begin{scope}
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\node[main] (A) at (0mm, 0mm) {$A$};
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\node[main] (B) at (0mm, -10mm) {$B$};
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\node[main] (C) at (0mm, -20mm) {$C$};
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\vfill
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\node[main] (D) at (20mm, 0mm) {$D$};
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\node[main] (E) at (20mm, -10mm) {$E$};
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\node[main] (F) at (20mm, -20mm) {$F$};
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\end{scope}
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\definition{}
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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.\\
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% Edges
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\draw
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(A) edge (D)
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(A) edge (E)
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(B) edge (F)
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(C) edge (E)
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(C) edge (D)
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;
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\end{tikzpicture}
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\end{center}
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The bipartite graph above and the directed graph below are not connected.
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\vfill
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\begin{center}
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\begin{tikzpicture}[node distance = 20mm]
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% Nodes
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\begin{scope}
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\node[main] (A) {$A$};
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\node[main] (B) [below right of = A] {$B$};
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\node[main] (C) [below left of = A] {$C$};
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\end{scope}
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\definition{}
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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.\\
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% Edges
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\draw[->]
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(A) edge[bend right] (B)
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(B) edge[bend right] (A)
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(B) edge (C)
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;
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\end{tikzpicture}
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\end{center}
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The bipartite graph above and the directed graph below are not connected.
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\begin{center}
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\begin{tikzpicture}[node distance = 20mm]
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% Nodes
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\begin{scope}
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\node[main] (A) {$A$};
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\node[main] (B) [below right of = A] {$B$};
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\node[main] (C) [below left of = A] {$C$};
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\end{scope}
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% Edges
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\draw[->]
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(A) edge[bend right] (B)
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(B) edge[bend right] (A)
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(B) edge (C)
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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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\end{document}
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\vfill
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\pagebreak
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@ -1,363 +1,356 @@
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\documentclass[../main.tex]{subfiles}
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\section{Network Flow}
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\generic{Networks}
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Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc.
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\begin{document}
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\vspace{1ex}
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\section{Network Flow}
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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:
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\generic{Networks}
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Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc.
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\begin{itemize}
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\item The weight of each edge represents its capacity, the number of lanes in the highway.
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\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{}
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\item Node $S$ is a \textit{source}: it produces flow.
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\item Node $T$ is a \textit{sink}: it consumes flow.
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\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out.
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\end{itemize}
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\vspace{1ex}
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\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.}
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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:
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Here is an example of a such a graph:
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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 (+00mm, +00mm) {$S$};
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\node[main] (A) at (+15mm, +15mm) {$A$};
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\node[main] (B) at (+15mm, -15mm) {$B$};
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\node[main] (T) at (+30mm, +00mm) {$T$};
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\end{scope}
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% Edges
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\draw[->]
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(S) edge node[label] {$1$} (A)
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(A) edge node[label] {$4$} (T)
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(B) edge node[label] {$2$} (A)
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(S) edge node[label] {$2$} (B)
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(B) edge node[label] {$1$} (T)
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;
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\end{tikzpicture}
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\end{center}
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\hrule{}
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\generic{Flow}
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In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway:
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\vspace{2ex}
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\begin{minipage}{0.33\textwidth}
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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 (+00mm, +00mm) {$S$};
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\node[main] (A) at (+15mm, +15mm) {$A$};
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\node[main] (B) at (+15mm, -15mm) {$B$};
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\node[main] (T) at (+30mm, +00mm) {$T$};
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\end{scope}
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% Edges
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\draw[->]
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(S) edge node[label] {$1$} (A)
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(A) edge node[label] {$4$} (T)
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(B) edge node[label] {$2$} (A)
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(S) edge node[label] {$2$} (B)
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(B) edge node[label] {$1$} (T)
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;
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% Flow
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\draw[path]
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(S) -- node[above left, flow] {$(1)$} (A)
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-- node[above right, flow] {$(1)$} (T)
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;
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\begin{minipage}{0.65\textwidth}
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There are a few things to notice here:
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\begin{itemize}
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\item The weight of each edge represents its capacity, the number of lanes in the highway.
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\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{}
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\item Node $S$ is a \textit{source}: it produces flow.
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\item Node $T$ is a \textit{sink}: it consumes flow.
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\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out.
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\item Highlighted edges carry flow.
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\item Numbers in parentheses tell us how much flow each edge carries.
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\item The flow along an edge is always positive or zero.
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\item Flow comes from $S$ and goes towards $T$.
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\item Flow is conserved: all flow produced by $S$ enters $T$.
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\end{itemize}
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\end{minipage}
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\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.}
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\vspace{1ex}
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Here is an example of a such a graph:
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\begin{center}
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\begin{tikzpicture}
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The \textit{magnitude} of a flow\footnotemark{} is the number of \say{flow-units} that go from $S$ to $T$. \\
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (S) at (+00mm, +00mm) {$S$};
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\node[main] (A) at (+15mm, +15mm) {$A$};
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\node[main] (B) at (+15mm, -15mm) {$B$};
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\node[main] (T) at (+30mm, +00mm) {$T$};
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\end{scope}
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We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can push from $S$ to $T$?
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% Edges
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||||
\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}
|
||||
\footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.}
|
||||
|
||||
\hrule{}
|
||||
\problem{}
|
||||
What is the magnitude of the flow above?
|
||||
|
||||
\generic{Flow}
|
||||
In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway:
|
||||
\vfill
|
||||
\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}
|
||||
|
||||
\begin{minipage}{0.33\textwidth}
|
||||
\begin{center}
|
||||
\begin{tikzpicture}
|
||||
\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}
|
||||
|
||||
% 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)
|
||||
;
|
||||
|
||||
% 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}
|
||||
\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}
|
||||
% 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}
|
||||
|
||||
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{}
|
||||
What is the magnitude of the flow above?
|
||||
\vspace{2ex}
|
||||
\hrule
|
||||
\vspace{2ex}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
\problem{}
|
||||
Combine the following flows and ensure that the flow along all edges remains within capacity.
|
||||
|
||||
\problem{}
|
||||
Find a flow with magnitude 2 on the graph below.
|
||||
\vspace{2ex}
|
||||
|
||||
\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}
|
||||
\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] {$1$} (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}
|
||||
% 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)
|
||||
;
|
||||
|
||||
\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{}
|
||||
Find a maximal flow on the graph below. \\
|
||||
\hint{The total capacity coming out of $S$ is 3, so any flow must have magnitude $\leq 3$.}
|
||||
% 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)
|
||||
;
|
||||
|
||||
\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}
|
||||
% 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}
|
||||
|
||||
% 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}
|
||||
|
||||
\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}
|
||||
\vfill
|
||||
\pagebreak
|
||||
@ -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}
|
||||
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}
|
||||
|
||||
The first thing we'll need to construct such an algorithm is a \textit{residual graph}.
|
||||
|
||||
\vspace{2ex}
|
||||
\hrule
|
||||
\vspace{2ex}
|
||||
\hrule
|
||||
|
||||
\begin{center}
|
||||
\begin{minipage}[t]{0.48\textwidth}
|
||||
We'll start with the following network and flow:
|
||||
\begin{center}
|
||||
\begin{minipage}[t]{0.48\textwidth}
|
||||
We'll start with the following network and flow:
|
||||
\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}
|
||||
\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[->]
|
||||
(S) edge node[label] {$1$} (A)
|
||||
(A) edge node[label] {$3$} (T)
|
||||
(B) edge node[label] {$2$} (A)
|
||||
(S) edge node[label] {$2$} (B)
|
||||
(B) edge node[label] {$1$} (T)
|
||||
;
|
||||
% Edges
|
||||
\draw[->]
|
||||
(S) edge node[label] {$1$} (A)
|
||||
(A) edge node[label] {$3$} (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)
|
||||
;
|
||||
% 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}
|
||||
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{tikzpicture}
|
||||
\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}
|
||||
\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}
|
||||
|
||||
\hrule
|
||||
\vspace{3ex}
|
||||
\hrule
|
||||
|
||||
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}
|
||||
\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}
|
||||
|
||||
\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
|
||||
\pagebreak
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
\problem{}<FindResidual>
|
||||
Construct the residual of this flow.
|
||||
\problem{}<FindResidual>
|
||||
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{tikzpicture}[node distance = 25mm]
|
||||
% Nodes
|
||||
@ -154,66 +179,34 @@
|
||||
|
||||
% Edges
|
||||
\draw[->]
|
||||
(S) edge node[label] {$2$} (A)
|
||||
(A) edge node[label] {$2$} (S)
|
||||
(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)
|
||||
(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)
|
||||
(T) edge node[label] {$2$} (B)
|
||||
;
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\end{solution}
|
||||
|
||||
\begin{solution}
|
||||
\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}
|
||||
\vfill
|
||||
\problem{}
|
||||
Is the flow in \ref{FindResidual} maximal? \\
|
||||
If it isn't, find a maximal flow. \\
|
||||
\hint{Look at the residual graph. Can we add flow along another path?}
|
||||
|
||||
% Edges
|
||||
\draw[->]
|
||||
(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
|
||||
\pagebreak
|
||||
|
||||
\vfill
|
||||
\problem{}
|
||||
Is the flow in \ref{FindResidual} maximal? \\
|
||||
If it isn't, find a maximal flow. \\
|
||||
\hint{Look at the residual graph. Can we add flow along another path?}
|
||||
\problem{}
|
||||
Show that...
|
||||
\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}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
\footnotetext{Integral = \say{integer} as an adjective.}
|
||||
|
||||
\problem{}
|
||||
Show that...
|
||||
\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}
|
||||
\vfill
|
||||
\pagebreak
|
||||
@ -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}
|
||||
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{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}
|
||||
|
||||
\problem{}
|
||||
Run the Ford-Fulkerson algorithm on the following graph. \\
|
||||
There is extra space on the next page.
|
||||
% Edges
|
||||
\draw[->]
|
||||
(S) edge node[label] {$8$} (A)
|
||||
(S) edge node[label] {$7$} (B)
|
||||
(S) edge node[label] {$4$} (C)
|
||||
|
||||
\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}
|
||||
(A) edge node[label] {$2$} (B)
|
||||
(B) edge node[label] {$5$} (C)
|
||||
|
||||
% Edges
|
||||
\draw[->]
|
||||
(S) edge node[label] {$8$} (A)
|
||||
(S) edge node[label] {$7$} (B)
|
||||
(S) edge node[label] {$4$} (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)
|
||||
|
||||
(A) edge node[label] {$2$} (B)
|
||||
(B) edge node[label] {$5$} (C)
|
||||
(E) edge node[label] {$3$} (D)
|
||||
(E) edge node[label] {$4$} (F)
|
||||
|
||||
(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)
|
||||
(D) edge node[label] {$9$} (T)
|
||||
(E) edge node[label] {$5$} (T)
|
||||
(F) edge node[label] {$8$} (T)
|
||||
;
|
||||
|
||||
(E) edge node[label] {$3$} (D)
|
||||
(E) edge node[label] {$4$} (F)
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
(D) edge node[label] {$9$} (T)
|
||||
(E) edge node[label] {$5$} (T)
|
||||
(F) edge node[label] {$8$} (T)
|
||||
;
|
||||
\begin{solution}
|
||||
The maximum flow is $17$.
|
||||
\end{solution}
|
||||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\vspace{5mm}
|
||||
|
||||
\begin{solution}
|
||||
The maximum flow is $17$.
|
||||
\end{solution}
|
||||
\pagebreak
|
||||
|
||||
\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}
|
||||
\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}
|
||||
(A) edge node[label] {$2$} (B)
|
||||
(B) edge node[label] {$5$} (C)
|
||||
|
||||
% Edges
|
||||
\draw[->]
|
||||
(S) edge node[label] {$8$} (A)
|
||||
(S) edge node[label] {$7$} (B)
|
||||
(S) edge node[label] {$4$} (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)
|
||||
|
||||
(A) edge node[label] {$2$} (B)
|
||||
(B) edge node[label] {$5$} (C)
|
||||
(E) edge node[label] {$3$} (D)
|
||||
(E) edge node[label] {$4$} (F)
|
||||
|
||||
(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)
|
||||
(D) edge node[label] {$9$} (T)
|
||||
(E) edge node[label] {$5$} (T)
|
||||
(F) edge node[label] {$8$} (T)
|
||||
;
|
||||
|
||||
(E) edge node[label] {$3$} (D)
|
||||
(E) edge node[label] {$4$} (F)
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
(D) edge node[label] {$9$} (T)
|
||||
(E) edge node[label] {$5$} (T)
|
||||
(F) edge node[label] {$8$} (T)
|
||||
;
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\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?
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
\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.
|
||||
|
||||
\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?
|
||||
\vspace{2ex}
|
||||
|
||||
\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.
|
||||
A simpler answer could only count the flow on $S$.
|
||||
|
||||
\vspace{2ex}
|
||||
\end{solution}
|
||||
|
||||
A simpler answer could only count the flow on $S$.
|
||||
|
||||
\end{solution}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
\end{document}
|
||||
\vfill
|
||||
\pagebreak
|
||||
187
Advanced/Graph Algorithms/parts/04 applications.tex
Executable file
187
Advanced/Graph Algorithms/parts/04 applications.tex
Executable file
@ -0,0 +1,187 @@
|
||||
|
||||
\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
52
Advanced/Graph Algorithms/tikxset.tex
Normal file
@ -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
|
||||
}
|
||||
}
|
||||
Loading…
x
Reference in New Issue
Block a user