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reductions.html
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reductions.html
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<!DOCTYPE html>
<html lang="en"><head>
<script src="quarto_files/clipboard/clipboard.min.js"></script>
<script src="quarto_files/quarto-html/tabby.min.js"></script>
<script src="quarto_files/quarto-html/popper.min.js"></script>
<script src="quarto_files/quarto-html/tippy.umd.min.js"></script>
<link href="quarto_files/quarto-html/tippy.css" rel="stylesheet">
<link href="quarto_files/quarto-html/light-border.css" rel="stylesheet">
<link href="quarto_files/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="quarto_files/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.5.56">
<title>Reductions</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="quarto_files/revealjs/dist/reset.css">
<link rel="stylesheet" href="quarto_files/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
</style>
<link rel="stylesheet" href="quarto_files/revealjs/dist/theme/quarto.css">
<link rel="stylesheet" href="dsa2.css">
<link href="quarto_files/revealjs/plugin/quarto-line-highlight/line-highlight.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-menu/menu.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-menu/quarto-menu.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-chalkboard/font-awesome/css/all.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-chalkboard/style.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
.callout.callout-style-simple {
padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
.callout.callout-style-default {
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
.callout .callout-body-container {
flex-grow: 1;
}
.callout.callout-style-simple .callout-body {
font-size: 1rem;
font-weight: 400;
}
.callout.callout-style-default .callout-body {
font-size: 0.9rem;
font-weight: 400;
}
.callout.callout-titled.callout-style-simple .callout-body {
margin-top: 0.2em;
}
.callout:not(.callout-titled) .callout-body {
display: flex;
}
.callout:not(.no-icon).callout-titled.callout-style-simple .callout-content {
padding-left: 1.6em;
}
.callout.callout-titled .callout-header {
padding-top: 0.2em;
margin-bottom: -0.2em;
}
.callout.callout-titled .callout-title p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.callout.callout-titled.callout-style-simple .callout-content p {
margin-top: 0;
}
.callout.callout-titled.callout-style-default .callout-content p {
margin-top: 0.7em;
}
.callout.callout-style-simple div.callout-title {
border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
.callout.callout-style-default div.callout-title {
border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-default div.callout-content {
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-simple .callout-icon::before {
height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
}
.callout.callout-style-default .callout-icon::before {
height: 0.9rem;
width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
}
.callout-title {
display: flex
}
.callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
.callout.no-icon::before {
display: none !important;
}
.callout.callout-titled .callout-body > .callout-content > :last-child {
padding-bottom: 0.5rem;
margin-bottom: 0;
}
.callout.callout-titled .callout-icon::before {
margin-top: .5rem;
padding-right: .5rem;
}
.callout:not(.callout-titled) .callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
/* Callout Types */
div.callout-note {
border-left-color: #4582ec !important;
}
div.callout-note .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
}
div.callout-important .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-important.callout-style-default .callout-title {
background-color: #f7dddc
}
div.callout-warning {
border-left-color: #f0ad4e !important;
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<body class="quarto-light">
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<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<p class="titlep"> </p>
<div class="orangetitle">
<h1 class="title" style="color:white !important">CS 3100</h1>
<h2 class="subtitle" style="color:white !important">Data Structures and Algorithms 2</h2>
<h2 class="subtitle" style="color:white !important">Reductions</h2>
</div>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<div class="titlesmall"><p>
<a href="https://www.cs.virginia.edu/~asb">Aaron Bloomfield</a> ([email protected])<br>
<a href="https://engineering.virginia.edu/faculty/raymond-pettit">Raymond Pettit</a> ([email protected])<br>
<a href="https://github.com/uva-cs/dsa2">@github</a> | <a href="index.html">↑</a> | <a href="?print-pdf"><img class="print" width="20" src="images/print-icon.svg" style="top:0px;vertical-align:middle;background-color:transparent;display:inline;width:30px;height:30px"></a>
</p></div>
<p class="titlep"> </p>
</section><section id="TOC">
<nav role="doc-toc">
<h2 id="toc-title">Contents</h2>
<ul>
<li><a href="#/network-flow" id="/toc-network-flow">Network Flow <!-- from day20.pptx --></a></li>
<li><a href="#/max-flow-min-cut" id="/toc-max-flow-min-cut">Max flow / Min cut <!-- from day21.pptx --></a></li>
<li><a href="#/bipartite-matching" id="/toc-bipartite-matching">Bipartite Matching <!-- from day21.pptx --></a></li>
<li><a href="#/reductions" id="/toc-reductions">Reductions <!-- from day22.pptx and day23.pptx --></a></li>
<li><a href="#/np-completeness" id="/toc-np-completeness">NP-Completeness <!-- from day23.pptx --></a></li>
</ul>
</nav>
</section>
<section>
<section id="network-flow" class="title-slide slide level1 center">
<h1>Network Flow <!-- from day20.pptx --></h1>
<div class="titlereading">
<p>Readings in CLRS 4th edition: chapter 24</p>
</div>
</section>
<section id="network-flow-1" class="slide level2">
<h2>Network Flow</h2>
<p><img data-src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cc/OpenRailwayMap_of_Northeastern_US_and_Southeastern_Canada.png/1024px-OpenRailwayMap_of_Northeastern_US_and_Southeastern_Canada.png" class="center" style="object-fit:none;object-position:right;height:513px;width:864px"></p>
<p><span class="font-90"><strong>Question:</strong> What is the maximum throughput of the railroad network from Omaha (OMA, far left) to Boston (BBY, far right)?</span></p>
</section>
<section id="flow-networks" class="slide level2">
<h2>Flow Networks</h2>
<!-- day20 slide 13 -->
<table style="width:100%">
<tbody><tr>
<td class="font-90" style="vertical-align:top;padding-left:0;width:40%">
<p> </p>
<div class="no-bullets-left">
<ul>
<li>Graph <span class="math inline">\(G=(V,E)\)</span></li>
<li><span class="purple">Start node</span> <span class="math inline">\({\color{purple}s}\in V\)</span></li>
<li><span class="navy">Sink node</span> <span class="math inline">\({\color{navy}t}\in V\)</span></li>
<li><span class="green">Edge capacities</span> <span class="math inline">\({\color{green}c(e)} \in \mathbb{R}^+\)</span></li>
</ul>
</div>
</td>
<td class="svg-table-right compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-1.svg"></p>
</div>
</td>
</tr>
</tbody></table>
<p><strong>Max flow intuition:</strong> If <span class="math inline">\(\color{purple}s\)</span> is a faucet, <span class="math inline">\(\color{navy}t\)</span> is a drain, and <span class="math inline">\(\color{purple}s\)</span> connects to <span class="math inline">\(\color{navy}t\)</span> through a network of pipes <span class="math inline">\(E\)</span> with capacities <span class="math inline">\(\color{green}c(e)\)</span>, what is the <span class="und">maximum</span> amount of water which can flow from the faucet to the drain?</p>
</section>
<section id="network-flow-2" class="slide level2">
<h2>Network Flow</h2>
<!-- day20 slide 14 -->
<div class="svg-50-right-flow-graph">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
<p class="font-75 center">
<span class="red">flow</span> / <span class="green">capacity</span>
</p>
</div>
<div class="font-75">
<ul>
<li>Assignment of values <span class="math inline">\(\color{red}f(e)\)</span> to edges
<ul>
<li>“Amount of water going through that pipe”</li>
</ul></li>
<li>Capacity constraint
<ul>
<li><span class="math inline">\({\color{red}f(e)} \le {\color{green}c(e)}\)</span></li>
<li>“Flow cannot exceed capacity”</li>
</ul></li>
<li>Flow constraint
<ul>
<li><span class="math inline">\(\forall v \in V - \{ {\color{purple}s},{\color{navy}t}\}\)</span>, <span class="math inline">\(\text{inflow}(v)=\text{outflow}(v)\)</span></li>
<li><span class="math inline">\(\text{inflow}(v)=\sum_{x \in V}f(x,v)\)</span></li>
<li><span class="math inline">\(\text{outflow}(v)=\sum_{x \in V}f(v,x)\)</span></li>
<li>Water going in must match water coming out</li>
</ul></li>
<li>Flow of <span class="math inline">\(G\)</span>: <span class="math inline">\(|f|=\text{outflow}({\color{purple}s})-\text{inflow}({\color{purple}s})\)</span></li>
<li>Net outflow of <span class="math inline">\(\color{purple}s\)</span>
<ul>
<li><span class="red">3 in this example</span></li>
</ul></li>
</ul>
</div>
</section>
<section id="maximum-flow-problem" class="slide level2">
<h2>Maximum Flow Problem</h2>
<!-- day20 slide 15 -->
<p class="font-90 center">
Of all valid flows through the graph, find the one that maximizes:
</p>
<p><span class="math display">\[|f|=\text{outflow}({\color{purple}s})-\text{inflow}({\color{purple}s})\]</span></p>
<table class="width-100">
<tbody><tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="fragment">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3.svg"></p>
</div>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="greedy-approach" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 16 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow.svg" class="center" style="height:50%;width:50%"></p>
</section>
<section id="greedy-approach-1" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 17 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-1.svg" class="center" style="height:50%;width:50%"></p>
</section>
<section id="greedy-approach-2" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 18 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-1-1.svg" class="center" style="height:50%;width:50%"></p>
<p class="red">
Flow: 20
</p>
</section>
<section id="greedy-approach-3" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 19 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-2-2.svg" class="center" style="height:50%;width:50%"></p>
<p class="red">
Maximum flow: 30
</p>
<p><span class="font-90"><strong>Observe:</strong> highest capacity path is not <span class="und">saturated</span> in optimal solution</span></p>
</section>
<section id="residual-graphs" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 20 -->
<table class="width-100">
<tbody><tr>
<td colspan="2" class="font-70 zero-bottom">
<p>Given a flow <span class="math inline">\(f\)</span> in graph <span class="math inline">\(G\)</span>, the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> models <span class="und">additional</span> flow that is possible</p>
<ul>
<li><span class="orange und">Forward edge</span> for each edge in <span class="math inline">\(G\)</span> with weight set to remaining capacity <span class="math inline">\({\color{green}c(e)}-{\color{red}f(e)}\)</span>
<ul>
<li>Models <span class="und">additional</span> flow that can be sent along the edge: <span class="purple">flow to add</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<ul>
<li><span class="magenta und">Backward edge</span> by flipping each edge <span class="math inline">\(e\)</span> in <span class="math inline">\(G\)</span> with weight set to flow <span class="math inline">\({\color{red}f(e)}\)</span>
<ul>
<li>Models amount of flow that can be <span class="und">removed</span> from the edge: <span class="purple">flow to remove</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
<p class="svgcaption center">
Flow <span class="math inline">\(\color{red}f\)</span> in <span class="math inline">\(G\)</span>
</p>
</td>
<td class="compact">
<div class="fragment">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
<p class="svgcaption center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-1" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 21 -->
<table class="width-100">
<tbody><tr>
<td colspan="2" class="font-70 zero-bottom">
<p>Given a flow <span class="math inline">\(f\)</span> in graph <span class="math inline">\(G\)</span>, the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> models <span class="und">additional</span> flow that is possible</p>
<ul>
<li><span class="orange und">Forward edge</span> for each edge in <span class="math inline">\(G\)</span> with weight set to remaining capacity <span class="math inline">\({\color{green}c(e)}-{\color{red}f(e)}\)</span>
<ul>
<li>Models <span class="und">additional</span> flow that can be sent along the edge: <span class="purple">flow to add</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li><span class="magenta und">Backward edge</span> by flipping each edge <span class="math inline">\(e\)</span> in <span class="math inline">\(G\)</span> with weight set to flow <span class="math inline">\({\color{red}f(e)}\)</span>
<ul>
<li>Models amount of flow that can be <span class="und">removed</span> from the edge: <span class="purple">flow to remove</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
<p class="svgcaption center">
Flow <span class="math inline">\(\color{red}f\)</span> in <span class="math inline">\(G\)</span>
</p>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
<p class="svgcaption center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</p>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-example" class="slide level2">
<h2>Residual Graphs Example</h2>
<!-- day20 slide 22 -->
<table style="width:80%">
<tbody><tr>
<td class="und bold" style="text-align:center;width:50%">
Flow Graph
</td>
<td class="und bold" style="text-align:center">
Residual Graph
</td>
</tr>
<tr>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0">
<img src="graphs/reductions/diamond_flow-0-3.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment fade-out" data-fragment-index="1">
<img src="graphs/reductions/diamond_flow-0-4-True.svg" class="center diamond_flow_in_table">
</p>
<p style="margin:0" class="fragment" data-fragment-index="1">
<img src="graphs/reductions/diamond_flow-0-4-True-2.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
</tr>
<tr>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment" data-fragment-index="2">
<img src="graphs/reductions/diamond_flow-0-2.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment" data-fragment-index="3">
<img src="graphs/reductions/diamond_flow-0-5-True.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphsx" class="slide level2">
<h2>Residual Graphsx</h2>
<!-- day20 slide 23 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{navy}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-2" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 24 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{navy}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-False-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True-1.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-3" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 25 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{navy}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3-False-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3-True.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-algorithm" class="slide level2">
<h2>Ford-Fulkerson Algorithm</h2>
<p class="font-75">
Define an augmenting path to be an <span class="math inline">\({\color{purple}s} \rightarrow {\color{navy}t}\)</span> path in the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> (using edges of non-zero weight)
</p>
<table class="width-100 font-75">
<tbody><tr>
<td style="padding-left:0;padding-top:0" colspan="2">
<p>
Ford-Fulkerson max-flow algorithm:
</p>
<ul>
<li class="">
Initialize <span class="math inline">\({\color{red}f(e)}=0\)</span> for all all <span class="math inline">\(e \in E\)</span>
</li>
<li class="">
Construct the residual network <span class="math inline">\(G_{\color{red}f}\)</span>
</li>
<li class="">
While there is an augmenting path <span class="math inline">\(p\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span>:
<ul>
<li>
Let <span class="math inline">\(c=\min_{e \in E}c_f(e)\)</span> where <span class="math inline">\(c_f(e)\)</span> is the weight of edge <span class="math inline">\(e\)</span> in the residual network <span class="math inline">\(G_{\color{red}f}\)</span>
</li>
<li>
Add <span class="math inline">\(c\)</span> units of flow to <span class="math inline">\(G\)</span> based on the augmenting path <span class="math inline">\(p\)</span>
</li>
<li>
Update the residual network <span class="math inline">\(G_{\color{red}f}\)</span> for the updated flow
</li>
</ul>
</li>
</ul>
</td>
</tr>
</tbody></table>
<p> </p>
<div class="fragment">
<table class="navynote font-75">
<tbody><tr>
<td>
<span class="bold">Ford-Fulkerson approach:</span> take <span class="und">any</span> augmenting path (will revisit this later)
</td>
</tr>
</tbody></table>
</div>
</section>
<section id="ford-fulkerson-example" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 29 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:0;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True.svg"></p>
</div>
</td>
</tr>
<tr>
<td class="font-90">
<span class="bold">Initially:</span> <span class="math inline">\({\color{red}f(e)}=0\)</span> for all <span class="math inline">\(e \in E\)</span>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-1" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 30 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<div class="fragment">
<p class="center">
Increase flow by 1 unit
</p>
</div>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True-2.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-2" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 31 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-False-2.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True-2.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-3" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 32 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:0;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-True.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-4" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 33 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<div class="fragment">
<p class="center">
Increase flow by 1 unit
</p>
</div>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-True-3.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-5" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 34 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-6-False-3.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-True-3.svg"></p>
</div>
</td>
</tr>
<tr>
<td>