Added "applications" section
This commit is contained in:
@@ -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}
|
||||||
@@ -1,8 +1,3 @@
|
|||||||
\documentclass[../main.tex]{subfiles}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\section{Review}
|
\section{Review}
|
||||||
|
|
||||||
\definition{}
|
\definition{}
|
||||||
@@ -87,5 +82,3 @@
|
|||||||
|
|
||||||
\vfill
|
\vfill
|
||||||
\pagebreak
|
\pagebreak
|
||||||
|
|
||||||
\end{document}
|
|
||||||
@@ -1,8 +1,3 @@
|
|||||||
\documentclass[../main.tex]{subfiles}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\section{Network Flow}
|
\section{Network Flow}
|
||||||
|
|
||||||
\generic{Networks}
|
\generic{Networks}
|
||||||
@@ -359,5 +354,3 @@
|
|||||||
|
|
||||||
\vfill
|
\vfill
|
||||||
\pagebreak
|
\pagebreak
|
||||||
|
|
||||||
\end{document}
|
|
||||||
@@ -1,8 +1,3 @@
|
|||||||
\documentclass[../main.tex]{subfiles}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\section{Residual Graphs}
|
\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.
|
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.
|
||||||
|
|
||||||
@@ -214,6 +209,4 @@
|
|||||||
\footnotetext{Integral = \say{integer} as an adjective.}
|
\footnotetext{Integral = \say{integer} as an adjective.}
|
||||||
|
|
||||||
\vfill
|
\vfill
|
||||||
|
\pagebreak
|
||||||
|
|
||||||
\end{document}
|
|
||||||
@@ -1,7 +1,3 @@
|
|||||||
\documentclass[../main.tex]{subfiles}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{document}
|
|
||||||
|
|
||||||
\section{The Ford-Fulkerson Algorithm}
|
\section{The Ford-Fulkerson Algorithm}
|
||||||
We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
|
We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
|
||||||
@@ -125,4 +121,3 @@
|
|||||||
|
|
||||||
\vfill
|
\vfill
|
||||||
\pagebreak
|
\pagebreak
|
||||||
\end{document}
|
|
||||||
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
|
||||||
|
}
|
||||||
|
}
|
||||||
Reference in New Issue
Block a user