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@kekeandzeyu
kekeandzeyu committed Nov 4, 2024
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121 changes: 29 additions & 92 deletions Writerside/topics/Data-Structures-and-Algorithms-3.topic
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<secondary-label ref="wip"></secondary-label>
<chapter title="16 Minimum Spanning Trees" id="16-minimum-spanning-trees">
<chapter title="16.1 Introduction to MSTs" id="16-1-introduction-to-msts">
<p>
<format color="DarkOrange">Spanning tree:</format>
A
<format
color="OrangeRed">spanning tree
</format>
is a subgraph
<math>T
</math>
that is both a
<format color="OrangeRed">tree</format>
(connected and acyclic) and
<format color="OrangeRed">spanning
</format>
(includes all of the vertices).
</p>
<p><format color="DarkOrange">Spanning tree:</format> A <format color="OrangeRed">spanning tree
</format> is a subgraph <math>T</math> that is both a <format color="OrangeRed">tree</format> (connected
and acyclic) and <format color="OrangeRed">spanning</format> (includes all of the vertices).</p>
<img src="../images_data/d16-1-1.png" alt="Spanning Tree"/>
<p>
<format color="BlueViolet">Application:</format>
</p>
<p><format color="BlueViolet">Application:</format></p>
<list type="bullet">
<li>
<p>Dithering</p>
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</p>
</li>
<li>
<p>Autoconfig protocol for Ethernet bridging to avoid cycles in a
network</p>
<p>Autoconfig protocol for Ethernet bridging to avoid cycles in a network</p>
</li>
<li>
<p>Approximation algorithms for NP-hard problems (e.g., TSP, Steiner
tree)</p>
<p>Approximation algorithms for NP-hard problems (e.g., TSP, Steiner tree)</p>
</li>
<li>
<p>Network design (communication, electrical, hydraulic, computer,
road).</p>
<p>Network design (communication, electrical, hydraulic, computer, road)</p>
</li>
</list>
</chapter>
<chapter title="16.2 Greedy Algorithm" id="16-2-greedy-algorithm">
<p>
<format color="BlueViolet">Definitions</format>
</p>
<p><format color="BlueViolet">Definitions</format></p>
<list type="bullet">
<li>
<p>
<format color="DarkOrange">Cut:</format>
A cut in a graph is a
partition of its vertices into two (nonempty) sets.
</p>
<p><format color="DarkOrange">Cut:</format> A cut in a graph is a partition of its vertices into two
(nonempty) sets.</p>
</li>
<li>
<p>
<format color="DarkOrange">Crossing edge:</format>
A crossing
edge connects a vertex in one set with a vertex in the other.
</p>
<p><format color="DarkOrange">Crossing edge:</format> A crossing edge connects a vertex in one set
with a vertex in the other.</p>
</li>
</list>
<img src="../images_data/d16-2-1.png" alt="Greedy Algorithm"/>
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<p>Start with all edges colored gray.</p>
</step>
<step>
<p>Find cut with no black crossing edges; color its
min-weight edge black.</p>
<p>Find cut with no black crossing edges; color its min-weight edge black.</p>
</step>
<step>
<p>Repeat until
<math>V - 1</math>
edges are colored black.
</p>
<p>Repeat until <math>V - 1</math> edges are colored black.</p>
</step>
</procedure>
<p>
<format color="BlueViolet">Correctness Proof</format>
</p>
<p><format color="LawnGreen">Correctness Proof</format></p>
<list type="decimal">
<li>
<p>Given any cut, the crossing edge of min weight is in MST.</p>
<p>
<format color="LawnGreen">Proof</format>
</p>
<p>Suppose min-weight crossing edge
<math>e</math>
is not in the
MST.
</p>
<p><format color="LawnGreen">Proof</format></p>
<p>Suppose min-weight crossing edge <math>e</math> is not in the MST.</p>
<list type="bullet">
<li>
<p>Adding
<math>e</math>
to the MST creates a cycle.
</p>
<p>Adding <math>e</math> to the MST creates a cycle.</p>
</li>
<li>
<p>Some other edge
<math>f</math>
in cycle must be a crossing
edge.
</p>
<p>Some other edge <math>f</math> in cycle must be a crossing edge.</p>
</li>
<li>
<p>Removing
<math>f</math>
and adding
<math>e</math>
is also a
spanning edge.
</p>
<p>Removing <math>f</math> and adding <math>e</math> is also a spanning edge.</p>
</li>
<li>
<p>Since weight of
<math>e</math>
is less than the weight of
<math>f</math>
, that spanning tree is lower height.
</p>
<p>Since weight of <math>e</math> is less than the weight of <math>f</math>, that spanning
tree is lower height.</p>
</li>
<li>
<p>Contradiction.</p>
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</li>
<li>
<p>The greedy algorithm computes the MST.</p>
<p>
<format color="MediumVioletRed">Proof</format>
</p>
<p><format color="LawnGreen">Proof</format></p>
<list type="bullet">
<li>
<p>Any edge colored black is in the MST (via cut property).</p>
</li>
<li>
<p>Fewer than
<math>V - 1</math>
black edges => cut with no
black crossing edges. (consider cut whose vertices are one
connected component)
</p>
<p>Fewer than <math>V - 1</math> black edges =&gt; cut with no black crossing edges
(consider cut whose vertices are one connected component).</p>
</li>
</list>
</li>
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</warning>
</chapter>
<chapter title="16.3 Edge-weighted Graph Implementation" id="16-3-edge-weighted-graph-implementation">
<p>
<format color="BlueViolet">Edge</format>
</p>
<p><format color="BlueViolet">Edge</format></p>
<tabs>
<tab title="Java">
<code-block lang="java" collapsible="true">
public class Edge implements Comparable&lt;Edge&gt; {
private final int source;
private final int destination;
private final double weight;
private final int source;
private final int destination;
private final double weight;

public Edge(int source, int destination, double weight) {
this.source = source;
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