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<!doctype html>
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<head>
<meta charset="utf-8">
<title>CS 2150: 11-graphs slide set</title>
<meta name="description" content="A set of slides for a course on Program and Data Representation">
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<div class="slides">
<section data-markdown id="cover"><script type="text/template">
# CS 2150
### Program and Data Representation
<center><small><a href="http://www.cs.virginia.edu/~asb">Aaron Bloomfield</a> (aaron@virginia.edu)</small></center>
<center><small><a href="http://@github/uva-cs/pdr">@github</a> | <a href="index.html">↑</a> | <a href="daily-announcements.html?print-pdf"><img class="print" width="20" src="../slides/images/print-icon.png"></a></small></center>
## Graphs
</script></section>
<section>
<h2>CS 2150 Roadmap</h2>
<table class="wide">
<tr><td colspan="3"><p class="center">Data Representation</p></td><td></td><td colspan="3"><p class="center">Program Representation</p></td></tr>
<tr>
<td class="top"><small> <br> <br>string<br> <br> <br> <br>int x[3]<br> <br> <br> <br>char x<br> <br> <br> <br>0x9cd0f0ad<br> <br> <br> <br>01101011</small></td>
<!-- image adapted from http://openclipart.org/detail/3677/arrow-left-right-by-torfnase -->
<td><img class="noborder" src="images/red-double-arrow.png" height="500" alt="vertical red double arrow"></td>
<td class="top"> <br>Objects<br> <br>Arrays<br> <br>Primitive types<br> <br>Addresses<br> <br>bits</td>
<td> </td>
<td class="top"><small> <br> <br>Java code<br> <br> <br>C++ code<br> <br> <br>C code<br> <br> <br>x86 code<br> <br> <br>IBCM<br> <br> <br>hexadecimal</small></td>
<!-- image adapted from http://openclipart.org/detail/3677/arrow-left-right-by-torfnase -->
<td><img class="noborder" src="images/green-double-arrow.png" height="500" alt="vertical green double arrow"></td>
<td class="top"> <br>High-level language<br> <br>Low-level language<br> <br>Assembly language<br> <br>Machine code</td>
</tr>
</table>
</section>
<section data-markdown><script type="text/template">
# Contents
[Introduction](#introduction)
[Topological Sort](#topological)
[Shortest Path Algorithms](#shortestpath)
[Map Routing](#maproutes)
[Travelling Salesperson (TSP)](#tsp)
[Minimum Spanning Tree (MST)](#mst)
</script></section>
<section>
<section id="introduction" data-markdown><script type="text/template">
# Introduction
</script></section>
<section data-markdown><script type="text/template">
## Graph examples
- Google Maps, of course
- Which actually started off using MapQuest data
- But we'll call it Google Maps data
- Others?
</script></section>
<section>
<h2>Airline Routes</h2>
<img class="stretch" src="images/11-graphs/airline-routes-from-openflights.jpg" alt="airline routes">
</section>
<section>
<h2>Flowcharts</h2>
<img class="stretch" src="images/11-graphs/geek-gift-flowchart.gif" alt="geek gift flowchart">
</section>
<section>
<h2>Pre-requisite Diagrams</h2>
<img class="stretch" src="images/11-graphs/bs-cs-chart.svg" alt="bs cs pre-req chart">
</section>
<section data-markdown><script type="text/template">
## Graphs
- *G* = (*V*, *E*)
- *V* are the vertices; *E* are the edges
- Edges are of the form (*v*, *w*), where *v*, *w* ∈ *V*
- ordered pair: directed graph or digraph
- unordered pair: undirected graph
</script></section>
<section data-markdown><script type="text/template">
## How big are these graphs?
- Airline routes: [Openflights](http://openflights.org), as of January 2012, has 59,036 routes between 3,209 airports ([source](http://openflights.org/data.html))
- Google maps: information is hard to find
- So this is a not-so-educated guess
- There are probably 50 million intersections (vertices) in the US
- Assume each one connects to four others
- That's 4 * 50 million = 200 million edges
</script></section>
<section data-markdown><script type="text/template">
## Terminology
- A **weight** or **cost** can be associated with each edge
- *w* is **adjacent** to *v* iff (*v*, *w*) ∈ *E*
- **path**: sequence of vertices *w*<sub>1</sub>, *w*<sub>2</sub>, *w*<sub>3</sub>, ..., *w*<sub>n</sub> such that (*w*<sub>*i*</sub>, *w*<sub>*i*+1</sub>) ∈ *E* for 1 ≤ *i* < *n*
- **length** of a path: number of edges in the path
- **simple path**: all vertices are distinct
</script></section>
<section data-markdown><script type="text/template">
## How to weight a graph?
- For Google maps?
- For airline routes?
</script></section>
<section data-markdown><script type="text/template">
## More terminology
- **cycle**:
- **directed graph**: path of length ≥ 1 such that *w*<sub>1</sub> = *w*<sub>n</sub>
- **undirected graph**: same, except all edges are distinct
- **connected**: there is a path from every vertex to every other vertex
- **loop**: (*v*, *v*) ∈ *E*
- **complete graph**: there is an edge between every pair of vertices
</script></section>
<section data-markdown><script type="text/template">
## Digraph terminology
- **directed acyclic graph**: no cycles; often called a "DAG"
- **strongly connected**: there is a path from every vertex to every other vertex
- **weakly connected**: the underlying undirected graph is connected
</script></section>
<section>
<h2>Representation: Adjacency Matrix</h2>
<p>\( A[u][v] = \left\{
\begin{array}{l l}
weight & \quad \text{if ($u$,$v$) $\in$ $E$}\\
0 & \quad \text{if ($u$,$v$) $\notin$ $E$}\\
\end{array} \right.
\) </p>
<p> </p>
<table><tr style="background-color:transparent"><td class="top">
<table>
<tr><th style="width:50px"></th><th style="width:50px">1</th><th style="width:50px">2</th><th style="width:50px">3</th><th style="width:50px">4</th></tr>
<tr><th>1</th><td></td><td></td><td></td><td></td></tr>
<tr><th>2</th><td></td><td></td><td></td><td></td></tr>
<tr><th>3</th><td></td><td></td><td></td><td></td></tr>
<tr><th>4</th><td></td><td></td><td></td><td></td></tr>
</table>
</td><td style="width:100px"></td><td class="top"><img src="graphs/graph-3.svg" alt="graph"></td></tr></table>
<script type="text/javascript">insertCanvas();</script>
</section>
<section>
<h2>Representation: Adjacency List</h2>
<table><tr style="background-color:transparent"><td class="top">
<td class="middle"><img src="images/11-graphs/adjacency-list.svg" alt="linked list"></td>
<td style="width:50px"></td>
<td class="top"><img src="graphs/graph-3.svg" alt="graph"></td>
</tr></table>
</section>
<section data-markdown><script type="text/template">
## Representation in the real world
- Two types of representation
- Adjacency matrix
- Adjacency list
- How does Google maps probably store it?
- How do airline routes probably store it?
</script></section>
</section>
<section>
<section id="topological" data-markdown><script type="text/template">
# Topological Sort
</script></section>
<section data-markdown><script type="text/template">
## Topological Sort
- Given a *directed acyclic graph*, construct an **ordering** of the vertices such that if there is a path from *v<sub>i</sub>* to *v<sub>j</sub>*, then *v<sub>j</sub>* appears after *v<sub>i</sub>* in the ordering
- The result is a linear list of vertices
- **indegree** of *v*: number of edges (*u*, *v*) -- meaning the number of *incoming* edges
</script></section>
<section>
<h2>Topological Sort</h2>
<img src="graphs/graph-4.svg" alt="graph">
<p class="fragment" style="text-align:center">One valid topological sort is: v1, v6, v8, v3, v2, v7, v4, v5</p>
<script type="text/javascript">insertCanvas();</script>
</section>
<section data-markdown><script type="text/template">
## What is the topological sort?
</script></section>
<section data-markdown><script type="text/template">
## What is the topological sort?
</script></section>
<section>
<h2>This is already topologically sorted!</h2>
<img class="stretch" src="images/11-graphs/bs-cs-chart.svg" alt="bs cs pre-req chart">
</section>
<section data-markdown><script type="text/template">
## Topological sort
```
void Graph::topsort() {
Vertex v, w;
for (int counter=0; counter < NUM_VERTICES;
counter++) {
v = findNewVertexOfInDegreeZero();
if (v == NOT_A_VERTEX)
throw CycleFound();
v.topologicalNum = counter;
for each w adjacent to v
w.indegree--;
}
}
```
- What's the big-Theta running time?
- Observation: The only new (eligible) vertices with indegree 0 are the ones adjacent to the vertex just processed
</script></section>
<section data-markdown><script type="text/template">
## Topological sort
```
void Graph::topsort() {
Queue q(NUM_VERTICES);
int counter = 0;
Vertex v, w;
q.makeEmpty(); // initialize the queue
for each vertex v
if (v.indegree == 0)
q.enqueue(v);
while (!q.isEmpty()) {
v = q.dequeue(); // get vertex of indegree 0
v.topologicalNum = ++counter;
for each w adjacent to v: // insert eligible verts
if (--w.indegree == 0)
q.enqueue(w);
}
if (counter != NUM_VERTICES)
throw CycleFound();
}
```
</script></section>
<section>
<h2>Another Topological Sort Example</h2>
<img src="graphs/graph-7.svg" alt="graph">
<script type="text/javascript">insertCanvas();</script>
</section>
</section>
<section>
<section id="shortestpath" data-markdown><script type="text/template">
# Shortest Path<br>Algorithms
</script></section>
<section data-markdown><script type="text/template">
## Why do we care about shortest paths?
- The obvious answers:
- Map routing (car navigation systems, Google Maps, flights)
- 6 degrees of separation
- But what else?
- Internet routing
- Puzzle answers (Rubik's cube)
</script></section>
<section data-markdown><script type="text/template">
## Three types of algorithms
- Single pair
- Single source
- All pairs
- We won't see this in this course
- And a variant that we'll see later:
- Travelling salesperson
</script></section>
<section>
<h2>Shortest Path Algorithms</h2>
<ul>
<li>This version is called the "single-source" shortest path</li>
<li>Given a graph \( G = (V, E) \) and a single distinguished vertex <i>s</i>, find the shortest weighted path from <i>s</i> to every other vertex in <i>G</i></li>
</ul>
<p> </p>
<p class="center">The <b>weighted path length</b> of \( v_1, v_2, \ldots , v_n \):</p>
<p> </p>
<p class="center">\( \sum_{i=1}^{n-1}c_{i,i+1} \) where \( c_{i,i+1} \) is the cost of edge \( (v_i,v_{i+1}) \)</p>
</section>
<section>
<h2>Unweighted Shortest Path</h2>
<ul>
<li>Special case of the weighted problem: all weights are 1</li>
<li>Solution: breadth-first search; similar to level-order traversal for trees</li>
</ul>
<img src="graphs/graph-8.svg" alt="graph">
<script type="text/javascript">insertCanvas();</script>
</section>
<section data-markdown><script type="text/template">
## Unweighted Shortest Path
```
void Graph::unweighted (Vertex s) {
Queue q(NUM_VERTICES);
Vertex v, w;
q.enqueue(s);
s.dist = 0;
while (!q.isEmpty()) {
v = q.dequeue();
for each w adjacent to v
// each edge examined at most once,
// if adjacency lists are used
if (w.dist == INFINITY) {
w.dist = v.dist + 1;
w.path = v;
q.enqueue(w); // each vertex added at most once
}
}
}
```
What is the big-Theta running time?
</script></section>
<section data-markdown><script type="text/template">
## Weighted Shortest Path
- We assume no negative weight edges
- Dijkstra's algorithm: uses similar ideas as the unweighted case
- Greedy algorithms: do what seems to be best at every decision point
</script></section>
<section data-markdown><script type="text/template">
## Dijkstra's algorithm
- Initialize each vertex's distance as infinity
- Start at a given vertex *s*
- Update *s*'s distance to be 0
- Repeat
- Pick the next unknown vertex with the shortest distance to be the next *v*
- If no more vertices are unknown, terminate loop
- Mark *v* as known
- For each edge from *v* to adjacent unknown vertices *w*
- If the total distance to *w* is less than the current distance to *w*
- Update *w*'s distance and the path to *w*
</script></section>
<section>
<h2>Dijkstra's Algorithm</h2>
<table>
<tr style="background-color:transparent"><td class="top">
<table>
<tr><th>V</th><th>Known?</th><th style="width:150px">Dist</th><th style="width:150px">Path</th></tr>
<tr><th><i>v</i>0</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>1</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>2</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>3</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>4</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>5</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>6</th><td></td><td></td><td></td></tr>
</table>
</td><td style="width:50px"></td><td class="top"><img alt="graph" style="width:400px" src="graphs/graph-9.svg"></td></tr>
</table>
<script type="text/javascript">insertCanvas();</script>
</section>
<section data-markdown><script type="text/template">
## Dijkstra's algorithm
```
void Graph::dijkstra(Vertex s) {
Vertex v,w;
s.dist = 0;
while (there exist unknown vertices, find the
unknown v with the smallest distance) {
v.known = true;
for each w adjacent to v
if (!w.known)
if (v.dist + Cost_VW < w.dist) {
w.dist = v.dist + Cost_VW;
w.path = v;
}
}
}
```
</script></section>
<section>
<h2>Another Dijkstra's Algorithm Example</h2>
<table>
<tr style="background-color:transparent"><td class="top">
<table>
<tr><th>V</th><th>Known?</th><th style="width:150px">Dist</th><th style="width:150px">Path</th></tr>
<tr><th><i>v</i>1</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>2</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>3</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>4</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>5</th><td></td><td></td><td></td></tr>
<tr><th><i>v</i>6</th><td></td><td></td><td></td></tr>
</table>
</td><td style="width:50px"></td><td class="top"><img alt="graph" style="width:400px" src="graphs/graph-10.svg"></td></tr>
</table>
<p> </p>
<p style="font-size:90%">This is the same graph as in the <a href="http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm">Wikipedia article on Dijkstra's algorithm</a></p>
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</section>
<section data-markdown><script type="text/template">
## Analysis
- How long does it take to find the smallest unknown distance?
- Simple scan using an array: Θ(*v*)
- Total running time:
- Using a simple scan: Θ(*v*<sup>2</sup>+*e*) = Θ(*v*<sup>2</sup>)
- Optimizations?
- Use adjacency lists and heaps
- Assuming that the graph is connected (i.e. *e* > *v*-1), then the running time decreases to Θ(*e*+*v* log *v*)
- This can be further simplified to Θ(*e* log *v*)
- Although we won't see how to do that in this course
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<section data-markdown><script type="text/template">
## Negative Cost Edges?
- Perhaps the graph weights are the amount of fuel expended
- Positive means fuel was used
- And passing by a fuel station is a refueling, which is a negative cost edge
- Dijkstra's algorithm does not work for negative cost edges
- Others do (such as the [Bellman-Ford algorithm](http://en.wikipedia.org/wiki/Bellman-ford)), but are much less efficient
- What about negative cost cycles?
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<section>
<h2>Shortest Path Example Problem</h2>
<p class="center">From the ICPC Mid-Atlantic Regionals, 2009</p>
<img src="images/11-graphs/icpc-midatl-2009-shortest-path.png" alt="icpc 2009 problem" class="stretch">
</section>
</section>
<section>
<section id="maproutes" data-markdown><script type="text/template">
# Map Routing
</script></section>
<section data-markdown><script type="text/template">
## More on shortest path
- We studied finding the shortest path from a single vertex to *every* vertex
- But what about just 1 destination?
- Solution: do the same algorithm, but stop when the destination enters the set *S*
- Thus, the running time is the same!
</script></section>
<section>
<h2>How would you drive to Seattle?</h2>
<p class="fragment">And what constitutes a "highway"?</p>
</section>
<section>
<h2>The Eisenhower Interstate System
<a href="http://en.wikipedia.org/wiki/File:Map_of_current_Interstates.svg"><img class="stretch" alt="interstates" src="images/11-graphs/interstates.svg"></a>
</section>
<section>
<h2>A Google Maps Screenshot</h2>
<img class="stretch" alt="google maps" src="images/11-graphs/google-maps.png">
</section>
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## GPS shortest path algorithm
(Google Maps likely uses something much more complicated)
- Assume you are starting on a "side road"
- Transition to a "main road"
- Transition to a "highway"
- Get as close as you can to your destination via the highway system
- Transition to a "main road", and get as close as you can to your destination
- Transition to a "side road", and go to your destination
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</section>
<section>
<section id="tsp" data-markdown><script type="text/template">
# Travelling<br>Salesperson<br>(TSP)
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<section data-markdown><script type="text/template">
## Travelling Salesperson Problem (TSP)
- Given a number of cities and the costs of traveling from any city to any other city, what is the least-cost round-trip route that visits each city exactly once and then returns to the starting city?
- Really important problem for:
- UPS, Federal Express, USPS
- Any transport delivery system
- Cost = distance because more fuel is used
- We'll assume the graph is fully connected
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## Easy
</script></section>
<section>
<h2>Hard</h2>
<img class="stretch" alt="graph" src="graphs/graph-12.svg">
</section>
<section>
<h2>Really Hard</h2>
<img class="stretch" src="images/11-graphs/us-city-map.jpg" alt="graph">
(<a href="http://commons.wikimedia.org/wiki/File:US_map_-_geographic.png">source</a>)
</section>
<section data-markdown><script type="text/template">
## Analysis
- *Hamiltonian path*: a path in a connected graph that visits each vertex exactly once
- *Hamiltonian cycle*: a Hamiltonian path that ends where it started
- The traveling salesperson problem is thus to find the least weight Hamiltonian path (cycle) in a connected, weighted graph
- The size of the solution space is 1/2 (*n*-1)!
- Which means it's a Θ(*n*!) algorithm
- That's exponential (specifically, 2<sup>*n*</sup> < *n*! < *n*<sup>*n*</sup> for large *n*)
- For 10 cities: 181,440 possible cycles
- For 20 cities: 6 * 10<sup>16</sup> possible cycles
- For 21 cities, there are more possibilities than can be stored in a 64-bit `unsigned long`
</script></section>
<section data-markdown><script type="text/template">
## More Analysis
- This problem is ***NP-complete***
- Meaning there is no known efficient solution
- Just to try every possible path
- But there are ways to get a somewhat efficient solution (a *heuristic*)
- It just might not be the most efficient path
- What's the (usually) least expensive way to get between two US cities?
- And is that significantly slower than the "best" algorithm?
</script></section>
<section data-markdown><script type="text/template">
## The Record
- In April 2006, a computer cluster computed a path of 85,900 cities visited in 136 CPU years ([source](http://en.wikipedia.org/wiki/Travelling_salesman_problem#Exact_algorithms))
- About 3-6 months of "wall time"
- 85,900! = 9.61 * 10<sup>386,526</sup>
- Assume you can compute 1 million paths each second
- That would take only 3.04 * 10<sup>386,516</sup> years!
- (the exponent lowered by 10)
- They used acceleration techniques, obviously...
</script></section>
<section data-markdown><script type="text/template">
## Lab 11
- The parts of [lab 11](../labs/lab11/index.html):
- Pre-lab: implement a topological sort
- In-lab: implement a brute-force traveling salesperson problem
- Using locations in Tolkien's Middle Earth
- Post-lab: do a report containing analysis and a study of acceleration techniques
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</section>
<section>
<section id="mst" data-markdown><script type="text/template">
# Minimum Spanning Tree (MST)
</script></section>
<section data-markdown><script type="text/template">
## Spanning Tree
- Suppose you are going to build a transport system:
- Set of cities
- Roads, rail lines, air corridors connecting cities
- Trains, buses or aircraft between cities
- Which links do you actually use?
- Cannot use a complete graph
- Passengers can change at connection points
- Want to minimize number of links used
- Any solution is a *tree*
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<section data-markdown><script type="text/template">
## Spanning Tree (almost...)
[](http://commons.wikimedia.org/wiki/File:London_Underground_Zone_2.svg)
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<section data-markdown><script type="text/template">
## Spanning Tree
- A *spanning tree* of a graph *G* is a subgraph of *G* that contains every vertex of *G* and is a tree
- Any connected graph has a spanning tree
- Any two spanning trees of a graph have the same number of nodes
- Construct a spanning tree:
- Start with the graph
- Remove an edge from each cycle
- What remains has the same set of vertices but is a tree
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<section>
<h2>Spanning Trees</h2>
<p>Original graph:</p>
<img alt="graph" src="graphs/graph-13a.svg">
<p>Possible spanning trees:</p>
<table class="transparent"><tr>
<td><img alt="graph" src="graphs/graph-13b.svg"></td>
<td><img alt="graph" src="graphs/graph-13c.svg"></td>
<td><img alt="graph" src="graphs/graph-13d.svg"></td>
<td><img alt="graph" src="graphs/graph-13e.svg"></td>
</tr></table>
</section>
<section data-markdown><script type="text/template">
## Minimal Spanning Tree
- Spanning trees are simple
- But suppose edges have weights!
- "Cost" associated with the edge
- Miles for a transport link, for example
- Spanning trees each have a different total weight
- Minimal-weight spanning tree: *spanning tree with the minimal total weight*
</script></section>
<section>
<h2>Minimum Spanning Trees</h2>
<ul>
<li>Given a connected and undirected graph <i>G</i> = (<i>V</i>,<i>E</i>), find a graph <i>G'</i> = (<i>V</i>,<i>E'</i>) such that:<ul>
<li><i>E'</i> is a subset of <i>E</i></li>
<li>|<i>E'</i>| = |<i>V</i>| - 1</li>
<li><i>G'</i> is connected</li>
<li>\( \sum_{(u,v) \in E'} c_{uv} \) is minimal</li></ul></li>
<li><i>G'</i> is then a <i>minimal spanning tree</i></li>
<li>Applications: wiring a house, cable TV lines, power grids, Internet connections</li>
</ul>
</section>
<section data-markdown><script type="text/template">
## Generic Minimum Spanning Tree Algorithm
- KnownVertices <- {}
- while KnownVertices does not form a spanning tree, loop:
- find edge (u,v) that is "safe" for KnownVertices
- KnownVertices <- KnownVertices U {(u,v)}
- end loop
But how to find a "safe" edge?
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<section data-markdown><script type="text/template">
## Prim's algorithm
Idea: Grow a tree by adding an edge to the "known" vertices from the "unknown" vertices. Pick the edge with the smallest weight.
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## Prim's Algorithm for MST
- Pick one node as the root,
- Incrementally add edges that connect a "new" vertex to the tree.
- Pick the edge (*u*,*v*) where:
- *u* is in the tree, *v* is not, AND
- where the edge weight is the smallest of all edges (where *u* is in the tree and *v* is not)
</script></section>
<section>
<h2>Prim's MST Algorithm</h2>
<table class="transparent">
<tr><td><img alt="graph" src="graphs/graph-14.svg"></td><td style="width:50px"></td><td><img class="fragment" data-fragment-index="1" alt="graph" src="graphs/graph-15.svg"></td></tr>
</table>
<p> </p>
<p class="fragment" data-fragment-index="1" style="text-align:center">Edges: (v1,v2), (v1,v4), (v3,v4), (v4,v7), (v5,v7), (v6,v7)</p>
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</section>
<section data-markdown><script type="text/template">
## Analysis
- Running time: Same as Dijkstra's: Θ(*e* log *v*)
- Correctness:
- Suppose we have a partially built tree that we know is contained in some minimum spanning tree *T*
- Let (*u*,*v*) ∈ *E*, where *u* is "known" and *v* is "unknown" and has minimal cost
- Then there is a MST *T'* that contains the partially built tree and (*u*,*v*) that has as low a cost as *T*
</script></section>
<section data-markdown><script type="text/template">
## Kruskal's MST Algorithm
Idea: Grow a forest out of edges that do not create a cycle. Pick an edge with the smallest weight.
</script></section>
<section>
<h2>Kruskal's MST Algorithm</h2>
<table class="transparent">
<tr><td><img alt="graph" src="graphs/graph-14.svg"></td><td style="width:50px"></td><td><img class="fragment" data-fragment-index="1" alt="graph" src="graphs/graph-15.svg"></td></tr>
</table>
<p> </p>
<p class="fragment" data-fragment-index="1" style="text-align:center">Edges: (v1,v4), (v6,v7), (v1,v2), (v3,v4), (v4,v7), (v5,v7) </p>
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</section>
<section data-markdown><script type="text/template">
## Kruskal code
```
void Graph::kruskal() {
int edgesAccepted = 0;
DisjSet s(NUM_VERTICES);
while (edgesAccepted < NUM_VERTICES - 1) {
// The next line has |E| heap ops
e = smallest weight edge not deleted yet;
// edge e = (u, v)
uset = s.find(u); // |E| finds
vset = s.find(v); // |E| finds
if (uset != vset) {
edgesAccepted++;
s.unionSets(uset, vset); // |V| unions
}
}
}
```
When optimized, it has the same running time as Prim's and Dijkstra's: Θ(*e* log *v*)
</script></section>
</section>
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