-
Notifications
You must be signed in to change notification settings - Fork 487
/
10-heaps-huffman.html
1069 lines (937 loc) · 38.2 KB
/
10-heaps-huffman.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
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>CS 2150: 10-heaps-huffman slide set</title>
<meta name="description" content="A set of slides for a course on Program and Data Representation">
<meta name="author" content="Aaron Bloomfield">
<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/css/reveal.css">
<link rel="stylesheet" href="../slides/reveal.js/css/theme/black.css" id="theme">
<link rel="stylesheet" href="../slides/css/pdr.css">
<!-- Code syntax highlighting -->
<link rel="stylesheet" href="../slides/reveal.js/lib/css/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.css' : '../slides/reveal.js/css/print/paper.css';
document.getElementsByTagName( 'head' )[0].appendChild( link );
</script>
<!--[if lt IE 9]>
<script src="../slides/reveal.js/lib/js/html5shiv.js"></script>
<![endif]-->
<script type="text/javascript" src="../slides/js/dhtmlwindow.js"></script>
<script type="text/javascript" src="../slides/js/canvas.js"></script>
<link rel="stylesheet" href="../slides/css/dhtmlwindow.css" type="text/css">
</head>
<body onload="canvasinit()">
<div id="dhtmlwindowholder"><span style="display:none"></span></div>
<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
<center><small>[Aaron Bloomfield](http://www.cs.virginia.edu/~asb) / [aaron@virginia.edu](mailto:aaron@virginia.edu) / [@bloomfieldaaron](http://twitter.com/bloomfieldaaron)</small></center>
<center><small>Repository: [github.com/aaronbloomfield/pdr](http://github.com/aaronbloomfield/pdr) / [↑](index.html) / <a href="10-heaps-huffman.html?print-pdf"><img class="print" width="20" src="images/print-icon.png"></a></small></center>
<center><small>Themes: [default](?#/cover), [sky](?theme=sky#/cover), [beige](?theme=beige#/cover), [simple](?theme=simple#/cover), [serif](?theme=serif#/cover), [night](?theme=night#/cover), [moon](?theme=moon#/cover), [solarized](?theme=solarized#/cover)</small></center>
## Heaps (Priority Queues) & Huffman Coding
</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
[Priority Queues](#priorityqueues)
[Binary Heaps](#heaps)
[Heap Operations](#heapops)
[File Compression](#filecomp)
[Huffman Coding](#huffman)
[Priority Queue Example](#priorityqueueex)
</script></section>
<section>
<section id="priorityqueues" data-markdown><script type="text/template">
# Priority Queues
</script></section>
<section data-markdown><script type="text/template">
## Motivation
- Multiuser environment
- Operating system must choose which process to run on CPU
- Management of limited resources
- Bandwidth on network router
- Limited bandwidth, but want to give best possible performance
- Send traffic from highest priority queue first
- Example: VoIP
</script></section>
<section data-markdown><script type="text/template">
## Priority Queue ADT - Model
- operations:
- insert
- inserts with a *priority*
- findMin
- finds the minimum element
- deleteMin
- finds, returns, and removes minimum element
</script></section>
<section data-markdown><script type="text/template">
## Implementations of Priority Queue ADT
| Data structure | insert | findMin | deleteMin |
|-|-|-|-|
| Unsorted array | Θ(1) ammortized | Θ(*n*) | Θ(*n*) |
| Unsorted linked list | Θ(1) | Θ(*n*) | Θ(*n*) |
| Sorted array | Θ(*n*) | Θ(1) | Θ(1) |
| Sorted linked list | Θ(*n*) | Θ(1) | Θ(1) |
| Binary search tree | Θ(*n*) or Θ(1) | Θ(*n*) | Θ(*n*) |
| AVL or red-black tree | Θ(log *n*) | Θ(log *n*) | Θ(log *n*) |
| Hash table | ideally constant | Θ(*n*) | Θ(*n*) |
- We would like:
- findMin: constant
- insert: worst case Θ(log *n*), typical case constant
- deleteMin: worst and average case Θ(log *n*)
</script></section>
</section>
<section>
<section id="heaps" data-markdown><script type="text/template">
# Binary Heaps
</script></section>
<section data-markdown><script type="text/template">
## Binary Heap
- A binary heap is a data structure that is one possible implementation of a priority queue
- It's a binary tree (*not* a BST), with a different:
- structure property
- ordering property
</script></section>
<section data-markdown><script type="text/template">
## Definitions
A *perfect* (or *complete*) binary tree is a binary tree with all leaf nodes at the same depth; all internal nodes have 2 children.
- This has height *h*, 2<sup>*h*+1</sup>-1 nodes, 2<sup>*h*</sup>-1 non-leaves, and 2<sup>*h*</sup> leaves
- Note that about half of the nodes are leaves
- Note that this example is a complete binary tree, not a heap
</script></section>
<section data-markdown><script type="text/template">
## Full Binary Tree
- A binary tree in which each node has exactly zero or two children.
- Also known as a proper binary tree
- We will use this later for Huffman trees
</script></section>
<section>
<h2>Heap Structure Property</h2>
<p>A binary heap is an <i>almost complete</i> binary tree, which is a binary tree that is completely filled, with the possible exception of the bottom level, which is filled left to right.</p>
<p> </p>
<p>Examples:</p>
<table class="transparent"><tr><td><img alt="heap 4" src="graphs/heap-4.svg"></td><td style="width:50px"></td><td><img alt="heap 3" src="graphs/heap-3.svg"></td></tr></table>
</section>
<section>
<h2>Almost complete binary tree of height <i>h</i></h2>
<table class="transparent">
<tr><td class="middle" style="text-align:left"><ul><li>For <i>h</i> = 0, just a single node</li></ul></td><td><img alt="heap 5" src="graphs/heap-5.svg"></td><td></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>For <i>h</i> = 1, left child or two children</li></ul></td><td><img alt="heap 6" src="graphs/heap-6.svg"></td><td><img alt="heap 7" src="graphs/heap-7.svg"></td></tr>
<tr><td colspan="3">
<ul>
<li>For <i>h</i> ≥ 2, either:<ul>
<li>the left subtree of the root is <i>complete</i> with height <i>h</i>-1 and the right is <i>almost complete</i> with height <i>h</i>-1, OR</li>
<li>the left is <i>almost complete</i> with height <i>h</i>-1 and the right is <i>complete</i> with height <i>h</i>-2</li></ul></li>
</ul>
</td></tr></table>
</section>
<section>
<h2>Representing Complete Binary Trees in Arrays</h2>
<table class="transparent">
<tr><td class="top"><img alt="heap 8" src="graphs/heap-8.svg"></td>
<td>
<p> </p>
<p>From node <i>i</i>:</p>
<p> </p>
<p>left child: 2*<i>i</i></p>
<p>right child: (2*<i>i</i>)+1</p>
<p>parent: floor(<i>i</i>/2)</p>
</td></tr></table>
<p>Implicit (array) representation:</p>
<table class="transparent" style="border-collapse:collapse"><tr class="bordercfifty" style="border-bottom:medium solid;"><td> </td><td>A</td><td>B</td><td>C</td><td>D</td><td>E</td><td>F</td><td>G</td><td>H</td><td>I</td><td>J</td><td>K</td><td>L</td><td> </td></tr>
<tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td></tr>
</table>
</section>
<section data-markdown><script type="text/template">
## Why better than pointers?
- Saves space
- No need to store parent/child pointers
- Arrays are more compact than dynamic memory
- Saves time
- Arrays work better with cache (we'll see why later this semester)
- \*2, /2, + operations are faster than dereferencing pointer
- Dynamic memory allocation is slow
- Parent easy to locate
</script></section>
<section>
<h2>Heap Ordering Property</h2>
<p>Heap ordering property: For every non-root node <i>X</i>, the key in the parent of <i>X</i> is less than (or equal to) the key in <i>X</i>. Thus, the tree is <i>partially</i> ordered.</p>
<table class="transparent"><tr>
<td class="top"><img alt="heap 9" src="graphs/heap-9.svg"> not a heap</td><td style="width:50px"></td><td><img alt="heap 10" src="graphs/heap-10.svg"> min-heap</td></tr>
</table>
</section>
</section>
<section>
<section id="heapops" data-markdown><script type="text/template">
# Heap Operations
</script></section>
<section data-markdown><script type="text/template">
## Heap Operations
- findMin: just look at the root node
- insert(val): percolate up
- deleteMin: percolate down
See [here](http://people.ksp.sk/~kuko/bak/index.html) for a good heap animation site
</script></section>
<section data-markdown><script type="text/template">
## Heap: Insert(val)
Basic Idea:
1. Put val at "next" leaf position
2. Repeatedly exchange node with its parent if needed
</script></section>
<section data-markdown><script type="text/template">
## insert(int x)
Note that `heap` is an int vector, and `heap_size` is the number of *heap elements* inserted into that vector
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
void binary_heap::insert(int x) {
// a vector push_back will resize as necessary
heap.push_back(x);
// move it up into the right position
percolateUp(++heap_size);
}
void binary_heap::percolateUp(int hole) {
// get the value just inserted
int x = heap[hole];
// while we haven't run off the top and while the
// value is less than the parent...
for ( ; (hole > 1) && (x < heap[hole/2]); hole /= 2 )
heap[hole] = heap[hole/2]; // move the parent down
// correct position found! insert at that spot
heap[hole] = x;
}
```
</script></section>
<section>
<h2>Insert: percolate up</h2>
<table class="transparent"><tr><td><img alt="heap 12" src="graphs/heap-12.svg"></td><td class="middle">→</td><td><img alt="heap 13" src="graphs/heap-13.svg"></td></tr></table>
</section>
<section>
<h2>Insert expected running time</h2>
<ul>
<li>How far to move up?<ul>
<li>Half of the nodes are leaves, so half of the inserts will only move up one level</li>
<li>A quarter of the nodes are one level above the leaves, so one quarter of the inserts will move up two levels</li>
<li>One eighth will require moving up 3 levels</li>
<li>One sixteenth will require moving up 4 levels</li>
<li>Etc.</li></ul></li>
<li>Expected running time:<br> </li>
<ul>
\( \frac{1}{2}*1 + \frac{1}{4}*2 + \frac{1}{8}*3 + \ldots = \sum_{i=1}^{n} \frac{1}{2^i}*i = 2 \)
</section>
<section data-markdown><script type="text/template">
## Heap: DeleteMin
Basic Idea:
1. Remove root (that is always the min!)
2. Put "last" leaf node at root
3. Find smallest child (why?)
4. Swap node with smallest child if needed.
5. Repeat steps 3 & 4 until no swaps needed.
</script></section>
<section>
<h2>Which child to swap with</h2>
<table class="transparent">
<tr><td class="middle" style="text-align:left"><ul><li>Consider this min-heap:<ul>
<li>25 needs percolating!</li>
<li>But which way?</li></ul></li>
</ul></td><td><img alt="heap 14" src="graphs/heap-14.svg"></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>If we swap 25 with the smallest child:<ul>
<li>All's good!</li></ul></li>
</ul></td><td><img alt="heap 15" src="graphs/heap-15.svg"></td></tr>
<tr><td class="middle" style="text-align:left"><ul><li>If we swap 25 with the largest child:<ul>
<li>No longer a min-heap!</li></ul></li>
</ul></td><td><img alt="heap 16" src="graphs/heap-16.svg"></td></tr>
</table>
</section>
<section>
<h2>DeleteMin: percolate down</h2>
<table class="transparent"><tr><td><img alt="heap 17" src="graphs/heap-17.svg"></td><td class="middle">→</td><td><img alt="heap 18" src="graphs/heap-18.svg"></td></tr></table>
</section>
<section data-markdown><script type="text/template">
## deleteMin()
Note that `heap` is an int vector, and `heap_size` is the number of *heap elements* inserted into that vector
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
int binary_heap::deleteMin() {
// make sure the heap is not empty
if ( heap_size == 0 )
throw "deleteMin() called on empty heap";
// save the value to be returned
int ret = heap[1];
// move the last inserted position into the root
heap[1] = heap[heap_size--];
// make sure the vector knows that there is one less
// element
heap.pop_back();
// percolate the root down to the proper position
percolateDown(1);
// return the old root node
return ret;
}
```
</script></section>
<section data-markdown><script type="text/template">
## percolateDown()
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
void binary_heap::percolateDown(int hole) {
// get the value to percolate down
int x = heap[hole];
// while there is a left child...
while ( hole*2 <= heap_size ) {
int child = hole*2; // the left child
// is there a right child? if so, is it lesser?
if ( (child+1 <= heap_size) &&
(heap[child+1] < heap[child]) )
child++;
// if the child is greater than the node...
if ( x > heap[child] ) {
heap[hole] = heap[child]; // move child up
hole = child; // move hole down
} else
break;
}
// correct position found! insert at that spot
heap[hole] = x;
}
```
</script></section>
<section data-markdown><script type="text/template">
## Other Possible Heap Operations
- `decreaseKey(processID, amount)`: "raise" the priority of a process, percolate up
- `increaseKey(processID, amount)`: "lower" the priority of a process, percolate down
- `remove(processID)`: remove a process, move to top, then delete.
1. `decreaseKey(processID, -infinity)`
2. `deleteMin()`
- Worst case running time for all of these: Θ(*n*), because of the find() required; without the find(), it's Θ(log *n*)
- What about FindMax?
- ExpandHeap: when heap fills, copy into new space.
</script></section>
<section data-markdown><script type="text/template">
## Heaps (Summary)
- insert: percolate up; Θ(log *n*) time worst case, but constant expected time
- deleteMin: percolate down; Θ(log *n*) time worst case; also logarithmic expected time
- findMin: Θ(1) time
</script></section>
<section data-markdown><script type="text/template">
## Heapsort
- Insert *n* elements, then remove n elements
- Each one has an insertion time of log *n*
- And then a removal time of log *n*
- Hence Θ(*n* log *n*)
- But it's not a *stable* sort, so it's not used as often as mergesort
</script></section>
<section>
<h2>An xkcd about heaps...</h2>
<img class="stretch" src="images/10-heaps-huffman/tree.png" title="Not only is that terrible in general, but you just KNOW Billy's going to open the root present first, and then everyone will have to wait while the heap is rebuilt." alt="Tree" />
<p class="center"><a href="http://xkcd.com/835/">xkcd # 835</a></p>
</section>
</section>
<section>
<section id="filecomp" data-markdown><script type="text/template">
# File Compression
</script></section>
<section data-markdown><script type="text/template">
## Why compress files?
- Disk space is limited
- File transfer
- Bigger files take longer to transfer
- Smaller file might fit in memory more easily
</script></section>
<section data-markdown><script type="text/template">
## What is a file?
- Named collection of information
- C++ program
- Application executables
- Word documents
- Email
- Web pages
- Pictures, audio, video
- Which of these needs to be exactly the same when we use them again?
</script></section>
<section data-markdown><script type="text/template">
## Data Compression
- Lossless compression: X = X'
- Lossy compression: X != X'
- Information is lost (irreversible)
- Compression ratio: |X|/|Y|
- Where |X| is the number of bits (i.e., file size) of X
</script></section>
<section data-markdown><script type="text/template">
## Lossy Compression
- Some data is lost, but not "noticable" data
- Standards:
- JPEG (Joint Photographic Experts Group)
- Still images (pictures)
- MP3 (MPEG-1, Layer 3), Ogg vorbis, AAC
- Audio
- MPEG (Motion Picture Experts Group), as well as most video codecs (standards)
- Audio and video
- Compression ratios of 10:1 are possible
</script></section>
<section>
<table class="transparent">
<tr><td colspan="2" class="top"><h2> <br>JPEG image quality<br>comparison</h2></td><td style="width:30%"><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="1" alt="jpeg @ 25%" src="images/10-heaps-huffman/jpeg-025.jpg"></a></td></tr>
<tr>
<td class="top">
<ul>
<li style="text-align:left" class="fragment" data-fragment-index="1">Quality = 100; image size: 83,261 (100%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="2">Quality = 50; image size: 15,138 (18%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="3">Quality = 25; image size: 9,553 (11%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="4">Quality = 10; image size: 4,787 (6%)</li>
<li style="text-align:left" class="fragment" data-fragment-index="5">Quality = 1; image size: 1,523 (2%)</li>
</ul>
</td>
<td style="width:30%"><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="2" alt="jpeg @ 100%" src="images/10-heaps-huffman/jpeg-100.jpg"></a>
<a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="3" alt="jpeg @ 50%" src="images/10-heaps-huffman/jpeg-050.jpg"></a></td>
<td><a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="4" alt="jpeg @ 10%" src="images/10-heaps-huffman/jpeg-010.jpg"></a>
<a href="http://en.wikipedia.org/wiki/Jpg#Sample_photographs"><img class="fragment" data-fragment-index="5" alt="jpeg @ 1%" src="images/10-heaps-huffman/jpeg-001.jpg"></a></td></tr>
</table>
</section>
<section data-markdown><script type="text/template">
## Lossless Compression
- No data is lost.
- Standards:
- Gzip, Unix compress, zip, Morse code
- PNG image file formats
- Run-length encoding (RLE)
- Can get compression ratios of 4:1
</script></section>
<section data-markdown><script type="text/template">
## Lossless Compression of Text
- ASCII = fixed 8 bits per character
- Example: "hello there"
- 11 characters * 8 bits = 88 bits
- Can we encode this message using fewer bits?
</script></section>
</section>
<section>
<section id="huffman" data-markdown><script type="text/template">
# Huffman Coding
</script></section>
<section>
<h2>Huffman Coding</h2>
<table>
<tr style="background-color: transparent"><td style="width:60%">
<ul><li>Uses <i>frequencies</i> of symbols in a string to build a <i>prefix code</i></li>
<li>The more frequent a character is, the fewer bits we'll use to represent it</li>
<li>Prefix code: no code in our encoding is a prefix of another code</li>
</ul></td><td class="top">
<table>
<tr><th>Letter</th><th>Code</th></tr>
<tr><td>a</td><td>0</td></tr>
<tr><td>b</td><td>100</td></tr>
<tr><td>c</td><td>101</td></tr>
<tr><td>d</td><td>11</td></tr>
</table>
</td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Decoding a Prefix Code
- Create the Huffman coding tree
- Loop
- start at root of tree
- loop
- if bit read = 1 then go right
- else, go left
- until node is a leaf
- Report character found!
- Until end of the message
</script></section>
<section id="lab10tree">
<h2>Decode: 1110001010011</h2>
<table>
<tr style="background-color: transparent"><td class="top">
<table>
<tr><th>Letter</th><th>Code</th></tr>
<tr><td>a</td><td>0</td></tr>
<tr><td>b</td><td>100</td></tr>
<tr><td>c</td><td>101</td></tr>
<tr><td>d</td><td>11</td></tr>
</table>
</td><td style="width:50px"></td><td class="top"><img alt="huffman 13" src="graphs/huffman-13.svg"></td><td style="width:50px"></td><td class="middle">This is a full<br>binary tree!</td></tr></table>
<p> </p>
<p class="center">11 100 0 101 0 0 11 = dbacaad</p>
</section>
<section>
<h2>Huffman Trees</h2>
<p>Cost of a file encoded via a Huffman Tree containing <i>n</i> symbols:</p>
<p> </p>
\( C(T) = p_1 * r_1 + p_2 * r_2 + p_3 * r_3 + \ldots + p_n * r_n \)
<p> </p>
<p>Where:</p>
<ul>
<li><i>p<sub>i</sub></i> = the frequency (or probability) that a symbol occurs</li>
<li><i>r<sub>i</sub></i> = the length of the path from the root to the node</li>
</ul>
</section>
<section>
<h2>Huffman encoding costs</h2>
<table>
<tr style="background-color: transparent"><td class="top">
<table><tr style="background-color: transparent"><td>This is the example<br> from 2 slides ago</td></tr>
<tr style="background-color: transparent"><td> </td></tr>
<tr style="background-color: transparent"><td> </td></tr>
<tr style="background-color: transparent"><td>
<table>
<tr><th>Letter</th><th>Frequency</th><th>Code</th></tr>
<tr><td>a</td><td>3/7</td><td>0</td></tr>
<tr><td>b</td><td>1/7</td><td>100</td></tr>
<tr><td>c</td><td>1/7</td><td>101</td></tr>
<tr><td>d</td><td>2/7</td><td>11</td></tr>
</table>
</td></tr></table>
</td><td style="width:50px"></td><td style="text-align:left">
<ul>
<li>a: 3/7 * 1 = 3/7</li>
<li>b: 1/7 * 3 = 3/7</li>
<li>c: 1/7 * 3 = 3/7</li>
<li>d: 2/7 * 2 = 4/7</li>
</ul>
<p> </p>
<ul>
<li>Cost is 3/7 + 3/7 + 3/7 + 4/7 = 13/7 = 1.85 bits per character</li>
<li>ASCII is 8 bits per char</li>
<li>"Straight" encoding is 2 bits per char</li>
</ul></td></tr></table>
</section>
<section data-markdown><script type="text/template">
## Compression Phase
1. Determine the frequencies of the characters stored in the source file
- Read the source file
- Then store the character frequencies in a *min-heap*
2. Build a *tree* of prefix codes (a Huffman code) that determines the unique bit codes for each character
3. Write the prefix codes or code tree to the output file
4. Re-read the source file and for each character read, write its prefix code into the output file
- You must WRITE the prefix code/tree and the encoded file to the SAME output file
</script></section>
<section data-markdown><script type="text/template">
## Huffman coding example
- For the quote "if it is to be, it is up to me"
- Total of 12 different characters; string length is 30
- Normal ASCII encoding is 8 bits per character
- 30\*8 = 240 bits
- Which is 30 bytes, obviously
- "Straight" encoding is 4 bits per character
- 30\*4 = 120 bits
- Which is 15 bytes
</script></section>
<section>
<h2>Compression step 1 (a)</h2>
<p>Determine frequencies of letters</p>
<table style="font-size:80%">
<tr><th>Character</th><th>Frequency</th></tr>
<tr><td>b</td><td>1</td></tr>
<tr><td>e</td><td>2</td></tr>
<tr><td>f</td><td>1</td></tr>
<tr><td>i</td><td>5</td></tr>
<tr><td>m</td><td>1</td></tr>
<tr><td>o</td><td>2</td></tr>
<tr><td>p</td><td>1</td></tr>
<tr><td>s</td><td>2</td></tr>
<tr><td>t</td><td>4</td></tr>
<tr><td>u</td><td>1</td></tr>
<tr><td>, (comma)</td><td>1</td></tr>
<tr><td>(space)</td><td>9</td></tr>
</table>
</section>
<section>
<h2>Compression step 1 (b)</h2>
<table><tr style="background-color: transparent"><td class="top"><p>Build a min-heap, sorted by frequency</p><img alt="huffman-14" src="graphs/huffman-14.svg"></td><td class="top">
<table>
<tr><th>Character</th><th>Frequency</th></tr>
<tr><td>b</td><td>1</td></tr>
<tr><td>e</td><td>2</td></tr>
<tr><td>f</td><td>1</td></tr>
<tr><td>i</td><td>5</td></tr>
<tr><td>m</td><td>1</td></tr>
<tr><td>o</td><td>2</td></tr>
<tr><td>p</td><td>1</td></tr>
<tr><td>s</td><td>2</td></tr>
<tr><td>t</td><td>4</td></tr>
<tr><td>u</td><td>1</td></tr>
<tr><td>, (comma)</td><td>1</td></tr>
<tr><td>(space)</td><td>9</td></tr>
</table>
</td></table>
</section>
<section data-markdown><script type="text/template">
## Compression step 2
- Build the tree, starting with a "forest" of trees:
- Repeat
- Take the two trees that have the lowest frequency
- The next two removals from the heap
- Make them children of a new node
- Keep track of the total frequency of that node
- And stick that tree back into the heap
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## Building the Huffman Tree
</script></section>
<section data-markdown><script type="text/template">
## The final Huffman coding tree
</script></section>
<section>
<h2>The Prefix codes</h2>
<table><tr style="background-color: transparent"><td class="top"><img alt="huffman-12" src="graphs/huffman-12.svg"></td><td class="top">
<table>
<tr><th>Character</th><th>Prefix code</th></tr>
<tr><td>b</td><td>00000</td></tr>
<tr><td>e</td><td>0011</td></tr>
<tr><td>f</td><td>00001</td></tr>
<tr><td>i</td><td>11</td></tr>
<tr><td>m</td><td>00010</td></tr>
<tr><td>o</td><td>1000</td></tr>
<tr><td>p</td><td>00011</td></tr>
<tr><td>s</td><td>1001</td></tr>
<tr><td>t</td><td>101</td></tr>
<tr><td>u</td><td>00100</td></tr>
<tr><td>, (comma)</td><td>00101</td></tr>
<tr><td>(space)</td><td>01</td></tr>
</table>
</td></table>
</section>
<section>
<h2>Resulting Encoding Table</h2>
<table style="font-size:90%">
<tr><th>Character</th><th>Frequency</th><th>Prefix code</th><th>Total bits</th></tr>
<tr><td>b</td><td>1</td><td>00000</td><td>5</td></tr>
<tr><td>e</td><td>2</td><td>0011</td><td>8</td></tr>
<tr><td>f</td><td>1</td><td>00001</td><td>5</td></tr>
<tr><td>i</td><td>5</td><td>11</td><td>10</td></tr>
<tr><td>m</td><td>1</td><td>00010</td><td>5</td></tr>
<tr><td>o</td><td>2</td><td>1000</td><td>8</td></tr>
<tr><td>p</td><td>1</td><td>00011</td><td>5</td></tr>
<tr><td>s</td><td>2</td><td>1001</td><td>8</td></tr>
<tr><td>t</td><td>4</td><td>101</td><td>12</td></tr>
<tr><td>u</td><td>1</td><td>00100</td><td>5</td></tr>
<tr><td>, (comma)</td><td>1</td><td>00101</td><td>5</td></tr>
<tr><td>(space)</td><td>9</td><td>01</td><td>18</td></tr>
</table>
<p>Total is 94 bits</p>
</section>
<section data-markdown><script type="text/template">
## Huffman Encoding Statistics
- Total encoding is 94 bits for 30 characeters
- Or an average of 3.13 bits per character
- ASCII was 240 bits: compression ratio of 2.6
- "Straight" encoding was 120 bits: compression ratio of 1.3
</script></section>
<section data-markdown><script type="text/template">
## Compression step 3
- Write the prefix codes to a file
- We'll use regular ASCII characters
- Does this actually do any compression, then?
```
b 00000
e 0011
f 00001
i 11
m 00010
o 1000
p 00011
s 1001
t 101
u 00100
, 00101
01
```
(note the double-space to indicate a space!)
</script></section>
<section>
<h2>The Prefix codes</h2>
<table><tr style="background-color: transparent">
<td class="top">
<ul>
<li>Write the text to the file using the Huffman encoding<br> </li>
<li>"be" becomes 00000 0011</li>
<li>"set" becomes 1001 0011 101</li>
<li>"stumps" becomes 1001 101 00100 00010 00011 101</li>
</ul>
</td><td style="width:50px"></td><td class="top">
<table>
<tr><th>Character</th><th>Prefix code</th></tr>
<tr><td>b</td><td>00000</td></tr>
<tr><td>e</td><td>0011</td></tr>
<tr><td>f</td><td>00001</td></tr>
<tr><td>i</td><td>11</td></tr>
<tr><td>m</td><td>00010</td></tr>
<tr><td>o</td><td>1000</td></tr>
<tr><td>p</td><td>00011</td></tr>
<tr><td>s</td><td>1001</td></tr>
<tr><td>t</td><td>101</td></tr>
<tr><td>u</td><td>00100</td></tr>
<tr><td>, (comma)</td><td>00101</td></tr>
<tr><td>(space)</td><td>01</td></tr>
</table>
</td></table>
</section>
<section data-markdown><script type="text/template">
## Compression Phase Review
1. Determine the frequencies of the characters stored in the source file
- Read the source file
- Then store the character frequencies in a *min-heap*
2. Build a *tree* of prefix codes (a Huffman code) that determines the unique bit codes for each character
3. Write the prefix codes or code tree to the output file
4. Re-read the source file and for each character read, write its prefix code into the output file
- You must WRITE the prefix code/tree and the encoded file to the SAME output file
</script></section>
<section data-markdown><script type="text/template">
## Decompression Phase Review
1. Read in the prefix code structure (tree or array) from the compressed file.
- Build a new Huffman tree for performing decompression
2. Read in one bit at a time from the compressed file and move through the prefix code tree until a leaf node is reached
3. Write the character stored at the leaf node into the decompressed file
4. While there is still input, repeat
</script></section>
<section data-markdown><script type="text/template">
## Huffman Coding Lab
- Compression (pre-lab)
- Decompression (in-lab)
- Report (post-lab)
</script></section>
<section data-markdown><script type="text/template">
## Huffman Encoding Prelab
- Write a program that reads input from a file (see class website for test files)
- Build a binary heap (priority queue) that calculates the letter frequencies
- Build the huffman tree
- Print the letter-encoding mapping to the file
- There is a specific format, as specified in the lab
- Encode the message, printing this also to the file
</script></section>
<section data-markdown><script type="text/template">
## Tips/Hints
- You may use the heap code included here, or create your own heap code, but you may not use it from other sources (including the STL)
- File input/output reference material:
- The [fileio.cpp](../labs/lab10/fileio.cpp.html) ([src](../labs/lab10/fileio.cpp)) file from [lab 10 (Huffman)](../labs/lab10/index.html)
- The [Input/output with files article](http://www.cplusplus.com/doc/tutorial/files/) on [cplusplus.com](http://www.cplusplus.com/)
- The [getWordInTable.cpp](../labs/lab06/code/getWordInTable.cpp.html) ([src](../labs/lab06/code/getWordInTable.cpp)) file from [lab 6 (hashes)](../labs/lab06/index.html)
</script></section>
<section data-markdown><script type="text/template">
## ASCII
- American Standard Code for Information Interchange (ASCII)
- Character encoding standard
- Specifies correspondence between digital bit patterns and English alphabet
- 7-bit encoding with 1 bit for parity (error correction)
- 128 characters
- 95 printable characters
- 33 non-printable characters
- More details in the [Wikipedia ASCII article](http://en.wikipedia.org/wiki/ASCII)
</script></section>
<section id="asciiset">
<h2>ASCII Character Codes in Hexadecimal</h2>
<table class="transparent">
<tr>
<td class="width:45%"><img alt="ascii" src="images/10-heaps-huffman/ascii.png"></td>
<td style="width:10%"></td>
<td class="top" style="width:45%"><p>For the lab, you only need to account for the printable characters (0x20 to 0x7e)</p><p> </p>
<p>Character codes:</p>
<ul>
<li>32<sub>10</sub> (0x20) = space</li>
<li>33<sub>10</sub> (0x21) = !</li>
<li>126<sub>10</sub>(0x7e) = ~</li>
</ul></td></tr></table>
</section>
</section>
<section>
<section id="priorityqueueex" data-markdown><script type="text/template">
# Priority Queue<br>Example
</script></section>
<section data-markdown><script type="text/template">
## Priority Queue Operations
- insert (x)
- deleteMin()
- findMin()
- isEmpty()
- makeEmpty()
- size()
</script></section>
<section data-markdown><script type="text/template">
## Binary Heap Class
Source code: [binary_heap.h](code/10-heaps-huffman/binary_heap.h.html) ([src](code/10-heaps-huffman/binary_heap.h))
```
class binary_heap {
public:
binary_heap();
binary_heap(vector<int> vec);
~binary_heap();
void insert(int x);
int findMin();
int deleteMin();
unsigned int size();
void makeEmpty();
bool isEmpty();
void print();
private:
vector<int> heap;
unsigned int heap_size;
void percolateUp(int hole);
void percolateDown(int hole);
};
```
</script></section>
<section data-markdown><script type="text/template">
## Heap Data Members
- `int heap_size`
- Size of the heap
- This holds the number of *heap elements*, which is always one less than the number of elements in the vector
- `vector<int> heap`
- The heap implemented as an array (vector)
</script></section>
<section data-markdown><script type="text/template">
## Methods seen already
- `insert(int x)`
- `deleteMin()`
- `percolateUp(int hole)`
- `percolateDown(int hole)`
</script></section>
<section data-markdown><script type="text/template">
## Constructors & Destructors
Source code: [binary_heap.cpp](code/10-heaps-huffman/binary_heap.cpp.html) ([src](code/10-heaps-huffman/binary_heap.cpp))
```
// default constructor
binary_heap::binary_heap() : heap_size(0) {
heap.push_back(0);
}
// builds a heap from an unsorted vector
binary_heap::binary_heap(vector<int> vec) :
heap_size(vec.size()) {
heap = vec;
heap.push_back(heap[0]);
heap[0] = 0;
for ( int i = heap_size/2; i > 0; i-- )
percolateDown(i);
}
// the destructor doesn't need to do much
binary_heap::~binary_heap() {
}
```
</script></section>
<section data-markdown><script type="text/template">
## findMin()