Added "applications" section
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@ -6,59 +6,8 @@
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solutions
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solutions
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]{ormc_handout}
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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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\input{tikxset}
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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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\begin{document}
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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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{Prepared by Mark on \today}
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\subfile{parts/00 review}
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\input{parts/00 review}
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\subfile{parts/01 flow}
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\input{parts/01 flow}
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\subfile{parts/02 residual}
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\input{parts/02 residual}
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\subfile{parts/03 fulkerson}
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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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\end{document}
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@ -1,8 +1,3 @@
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Review}
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\section{Review}
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\definition{}
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\definition{}
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@ -87,5 +82,3 @@
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\vfill
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\vfill
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\pagebreak
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\pagebreak
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\end{document}
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@ -1,8 +1,3 @@
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Network Flow}
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\section{Network Flow}
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\generic{Networks}
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\generic{Networks}
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@ -359,5 +354,3 @@
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\vfill
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\vfill
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\pagebreak
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\pagebreak
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\end{document}
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@ -1,8 +1,3 @@
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Residual Graphs}
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\section{Residual Graphs}
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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.
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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.
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@ -214,6 +209,4 @@
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\footnotetext{Integral = \say{integer} as an adjective.}
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\footnotetext{Integral = \say{integer} as an adjective.}
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\vfill
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\vfill
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\pagebreak
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\end{document}
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@ -1,7 +1,3 @@
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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{The Ford-Fulkerson Algorithm}
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\section{The Ford-Fulkerson Algorithm}
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We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
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We now have all the tools we need to construct an algorithm that finds a maximal flow. \\
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@ -125,4 +121,3 @@
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\vfill
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\vfill
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\pagebreak
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\pagebreak
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\end{document}
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187
Advanced/Graph Algorithms/parts/04 applications.tex
Executable file
187
Advanced/Graph Algorithms/parts/04 applications.tex
Executable file
@ -0,0 +1,187 @@
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\section{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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\vfill
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\pagebreak
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\problem{Circulations with Demand}
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Say we have a network of cities and power stations. Stations produce power; cities consume it.
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Each station produces a limited amount of power, and each city has limited demand.
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\vspace{2ex}
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We can represent this power grid as a graph, with cities and stations as nodes and transmission lines as edges.
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\vspace{2ex}
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A simple example is below. There are two cities ($2$ and $4$) and two stations (both $-3$).
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We'll represent station capacity with a negative number, since they \textit{consume} a negative amount of energy.
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\begin{center}
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\begin{tikzpicture}[
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node distance = 25mm,
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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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minimum size = 8mm
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},
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]
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% Nodes
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\begin{scope}[layer = nodes]
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\node[main] (S1) {$-3$};
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\node[main] (S2) [below left of = S1] {$-3$};
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\node[main] (C1) [below right of = S1] {$2$};
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\node[main] (C2) [below right of = S2] {$4$};
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\end{scope}
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% Edges
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\draw[->]
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(S1) edge node[label] {$3$} (S2)
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(S1) edge node[label] {$3$} (C1)
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(S2) edge node[label] {$2$} (C1)
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(S2) edge node[label] {$2$} (C2)
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(C1) edge node[label] {$2$} (C2)
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;
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\end{tikzpicture}
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\end{center}
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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?
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\vspace{2ex}
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\textbf{Your job:} Devise an algorithm that solve this problem.
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\vspace{2ex}
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\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.}
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\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