forked from aaronbloomfield/pdr
-
Notifications
You must be signed in to change notification settings - Fork 228
/
11-graphs.html
779 lines (663 loc) · 28 KB
/
11-graphs.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
<!doctype html>
<html lang="en">
<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">
<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="../slides/reveal.js/dist/reset.css">
<link rel="stylesheet" href="../slides/reveal.js/dist/reveal.css">
<link rel="stylesheet" href="../slides/reveal.js/dist/theme/black.css" id="theme">
<link rel="stylesheet" href="../slides/css/pdr.css">
<!-- Code syntax highlighting -->
<link rel="stylesheet" href="../slides/reveal.js/plugin/highlight/zenburn.css">
<!-- Printing and PDF exports -->
<script>
var link = document.createElement( 'link' );
link.rel = 'stylesheet';
link.type = 'text/css';
link.href = window.location.search.match( /print-pdf/gi ) ? '../slides/reveal.js/css/print/pdf.scss' : '../slides/reveal.js/css/print/paper.scss';
document.getElementsByTagName( 'head' )[0].appendChild( link );
</script>
<!--[if lt IE 9]>
<script src="../slides/reveal.js/lib/js/html5shiv.js"></script>
<![endif]-->
<style>.reveal li { font-size:93%; line-height:120%; }</style>
</head>
<body>
<div class="reveal">
<!-- Any section element inside of this container is displayed as a slide -->
<div class="slides">
<section data-markdown id="cover"><script type="text/template">
# CS 2150
### Program and Data Representation
<p class='titlep'> </p>
<div class="titlesmall"><p>
<a href="http://www.cs.virginia.edu/~mrf8t">Mark Floryan</a> ([email protected])<br>
<a href="http://www.cs.virginia.edu/~asb">Aaron Bloomfield</a> ([email protected])<br>
<a href="http://github.com/uva-cs/pdr">@github</a> | <a href="index.html">↑</a> | <a href="./11-graphs.html?print-pdf"><img class="print" width="20" src="../slides/images/print-icon.png" style="top:0px;vertical-align:middle"></a>
</p></div>
<p class='titlep'> </p>
## 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 class="center"><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>
</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 class="center"><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>
</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">
</section>
</section>
<section>
<section id="shortestpath" data-markdown class="center"><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">
</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>
</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>
</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* log *v*)
- Although we won't see how to do that in this course
</script></section>
<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?
</script></section>
<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 class="center"><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>
<section data-markdown><script type="text/template">
## 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
</script></section>
</section>
<section>
<section id="tsp" data-markdown class="center"><script type="text/template">
# Travelling<br>Salesperson<br>(TSP)
</script></section>
<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
</script></section>
<section data-markdown><script type="text/template">
## 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: shortest path solution to a puzzle problem
</script></section>
</section>
<section>
<section id="mst" data-markdown class="center"><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*
</script></section>
<section data-markdown><script type="text/template">
## Spanning Tree (almost...)
[](http://commons.wikimedia.org/wiki/File:London_Underground_Zone_2.svg)
</script></section>
<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
</script></section>
<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?
</script></section>
<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.
</script></section>
<section data-markdown><script type="text/template">
## 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>
</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>
</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>
</div>
</div>
<script src='../slides/reveal.js/dist/reveal.js'></script><script src='../slides/reveal.js/plugin/zoom/zoom.js'></script><script src='../slides/reveal.js/plugin/notes/notes.js'></script><script src='../slides/reveal.js/plugin/search/search.js'></script><script src='../slides/reveal.js/plugin/markdown/markdown.js'></script><script src='../slides/reveal.js/plugin/highlight/highlight.js'></script><script src='../slides/reveal.js/plugin/math/math.js'></script>
<script src="js/settings.js"></script>
</body>
</html>