-
Notifications
You must be signed in to change notification settings - Fork 3
/
Copy pathreductions.html
9304 lines (9170 loc) · 288 KB
/
reductions.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>
<script src="quarto_files/clipboard/clipboard.min.js"></script>
<script src="quarto_files/quarto-html/tabby.min.js"></script>
<script src="quarto_files/quarto-html/popper.min.js"></script>
<script src="quarto_files/quarto-html/tippy.umd.min.js"></script>
<link href="quarto_files/quarto-html/tippy.css" rel="stylesheet">
<link href="quarto_files/quarto-html/light-border.css" rel="stylesheet">
<link href="quarto_files/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<link href="quarto_files/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.5.56">
<title>Reductions</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="quarto_files/revealjs/dist/reset.css">
<link rel="stylesheet" href="quarto_files/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { display: inline-block; text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ color: #003b4f; background-color: #f1f3f5; }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span { color: #003b4f; } /* Normal */
code span.al { color: #ad0000; } /* Alert */
code span.an { color: #5e5e5e; } /* Annotation */
code span.at { color: #657422; } /* Attribute */
code span.bn { color: #ad0000; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #003b4f; font-weight: bold; } /* ControlFlow */
code span.ch { color: #20794d; } /* Char */
code span.cn { color: #8f5902; } /* Constant */
code span.co { color: #5e5e5e; } /* Comment */
code span.cv { color: #5e5e5e; font-style: italic; } /* CommentVar */
code span.do { color: #5e5e5e; font-style: italic; } /* Documentation */
code span.dt { color: #ad0000; } /* DataType */
code span.dv { color: #ad0000; } /* DecVal */
code span.er { color: #ad0000; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #ad0000; } /* Float */
code span.fu { color: #4758ab; } /* Function */
code span.im { color: #00769e; } /* Import */
code span.in { color: #5e5e5e; } /* Information */
code span.kw { color: #003b4f; font-weight: bold; } /* Keyword */
code span.op { color: #5e5e5e; } /* Operator */
code span.ot { color: #003b4f; } /* Other */
code span.pp { color: #ad0000; } /* Preprocessor */
code span.sc { color: #5e5e5e; } /* SpecialChar */
code span.ss { color: #20794d; } /* SpecialString */
code span.st { color: #20794d; } /* String */
code span.va { color: #111111; } /* Variable */
code span.vs { color: #20794d; } /* VerbatimString */
code span.wa { color: #5e5e5e; font-style: italic; } /* Warning */
</style>
<link rel="stylesheet" href="quarto_files/revealjs/dist/theme/quarto.css">
<link rel="stylesheet" href="dsa2.css">
<link href="quarto_files/revealjs/plugin/quarto-line-highlight/line-highlight.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-menu/menu.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-menu/quarto-menu.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-chalkboard/font-awesome/css/all.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/reveal-chalkboard/style.css" rel="stylesheet">
<link href="quarto_files/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
.callout.callout-style-simple {
padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
.callout.callout-style-default {
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
.callout .callout-body-container {
flex-grow: 1;
}
.callout.callout-style-simple .callout-body {
font-size: 1rem;
font-weight: 400;
}
.callout.callout-style-default .callout-body {
font-size: 0.9rem;
font-weight: 400;
}
.callout.callout-titled.callout-style-simple .callout-body {
margin-top: 0.2em;
}
.callout:not(.callout-titled) .callout-body {
display: flex;
}
.callout:not(.no-icon).callout-titled.callout-style-simple .callout-content {
padding-left: 1.6em;
}
.callout.callout-titled .callout-header {
padding-top: 0.2em;
margin-bottom: -0.2em;
}
.callout.callout-titled .callout-title p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.callout.callout-titled.callout-style-simple .callout-content p {
margin-top: 0;
}
.callout.callout-titled.callout-style-default .callout-content p {
margin-top: 0.7em;
}
.callout.callout-style-simple div.callout-title {
border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
.callout.callout-style-default div.callout-title {
border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-default div.callout-content {
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-simple .callout-icon::before {
height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
}
.callout.callout-style-default .callout-icon::before {
height: 0.9rem;
width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
}
.callout-title {
display: flex
}
.callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
.callout.no-icon::before {
display: none !important;
}
.callout.callout-titled .callout-body > .callout-content > :last-child {
padding-bottom: 0.5rem;
margin-bottom: 0;
}
.callout.callout-titled .callout-icon::before {
margin-top: .5rem;
padding-right: .5rem;
}
.callout:not(.callout-titled) .callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
/* Callout Types */
div.callout-note {
border-left-color: #4582ec !important;
}
div.callout-note .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
}
div.callout-important .callout-icon::before {
background-image: url('data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAYAAABzenr0AAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAIKADAAQAAAABAAAAIAAAAACshmLzAAAEKklEQVRYCcVXTWhcVRS+575MJym48A+hSRFr00ySRQhURRfd2HYjk2SSTokuBCkU2o0LoSKKraKIBTcuFCoidGFD08nkBzdREbpQ1EDNIv8qSGMFUboImMSZd4/f9zJv8ibJMC8xJQfO3HPPPef7zrvvvnvviIkpC9nsw0UttFunbUhpFzFtarSd6WJkStVMw5xyVqYTvkwfzuf/5FgtkVoB0729j1rjXwThS7Vio+Mo6DNnvLfahoZ+i/o32lULuJ3NNiz7q6+pyAUkJaFF6JwaM2lUJlV0MlnQn5aTRbEu0SEqHUa0A4AdiGuB1kFXRfVyg5d87+Dg4DL6m2TLAub60ilj7A1Ec4odSAc8X95sHh7+ZRPCFo6Fnp7HfU/fBng/hi10CjCnWnJjsxvDNxWw0NfV6Rv5GgP3I3jGWXumdTD/3cbEOP2ZbOZp69yniG3FQ9z1jD7bnBu9Fc2tKGC2q+uAJOQHBDRiZX1x36o7fWBs7J9ownbtO+n0/qWkvW7UPIfc37WgT6ZGR++EOJyeQDSb9UB+DZ1G6DdLDzyS+b/kBCYGsYgJbSQHuThGKRcw5xdeQf8YdNHsc6ePXrlSYMBuSIAFTGAtQo+VuALo4BX83N190NWZWbynBjhOHsmNfFWLeL6v+ynsA58zDvvAC8j5PkbOcXCMg2PZFk3q8MjI7WAG/Dp9AwP7jdGBOOQkAvlFUB+irtm16I1Zw9YBcpGTGXYmk3kQIC/Cds55l+iMI3jqhjAuaoe+am2Jw5GT3Nbz3CkE12NavmzN5+erJW7046n/CH1RO/RVa8lBLozXk9uqykkGAyRXLWlLv5jyp4RFsG5vGVzpDLnIjTWgnRy2Rr+tDKvRc7Y8AyZq10jj8DqXdnIRNtFZb+t/ZRtXcDiVnzpqx8mPcDWxgARUqx0W1QB9MeUZiNrV4qP+Ehc+BpNgATsTX8ozYKL2NtFYAHc84fG7ndxUPr+AR/iQSns7uSUufAymwDOb2+NjK27lEFocm/EE2WpyIy/Hi66MWuMKJn8RvxIcj87IM5Vh9663ziW36kR0HNenXuxmfaD8JC7tfKbrhFr7LiZCrMjrzTeGx+PmkosrkNzW94ObzwocJ7A1HokLolY+AvkTiD/q1H0cN48c5EL8Crkttsa/AXQVDmutfyku0E7jShx49XqV3MFK8IryDhYVbj7Sj2P2eBxwcXoe8T8idsKKPRcnZw1b+slFTubwUwhktrfnAt7J++jwQtLZcm3sr9LQrjRzz6cfMv9aLvgmnAGvpoaGLxM4mAEaLV7iAzQ3oU0IvD5x9ix3yF2RAAuYAOO2f7PEFWCXZ4C9Pb2UsgDeVnFSpbFK7/IWu7TPTvBqzbGdCHOJQSxiEjt6IyZmxQyEJHv6xyQsYk//moVFsN2zP6fRImjfq7/n/wFDguUQFNEwugAAAABJRU5ErkJggg==');
}
div.callout-important.callout-style-default .callout-title {
background-color: #f7dddc
}
div.callout-warning {
border-left-color: #f0ad4e !important;
}
div.callout-warning .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-warning.callout-style-default .callout-title {
background-color: #fcefdc
}
div.callout-tip {
border-left-color: #02b875 !important;
}
div.callout-tip .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-tip.callout-style-default .callout-title {
background-color: #ccf1e3
}
div.callout-caution {
border-left-color: #fd7e14 !important;
}
div.callout-caution .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-caution.callout-style-default .callout-title {
background-color: #ffe5d0
}
</style>
<style type="text/css">
.reveal div.sourceCode {
margin: 0;
overflow: auto;
}
.reveal div.hanging-indent {
margin-left: 1em;
text-indent: -1em;
}
.reveal .slide:not(.center) {
height: 100%;
}
.reveal .slide.scrollable {
overflow-y: auto;
}
.reveal .footnotes {
height: 100%;
overflow-y: auto;
}
.reveal .slide .absolute {
position: absolute;
display: block;
}
.reveal .footnotes ol {
counter-reset: ol;
list-style-type: none;
margin-left: 0;
}
.reveal .footnotes ol li:before {
counter-increment: ol;
content: counter(ol) ". ";
}
.reveal .footnotes ol li > p:first-child {
display: inline-block;
}
.reveal .slide ul,
.reveal .slide ol {
margin-bottom: 0.5em;
}
.reveal .slide ul li,
.reveal .slide ol li {
margin-top: 0.4em;
margin-bottom: 0.2em;
}
.reveal .slide ul[role="tablist"] li {
margin-bottom: 0;
}
.reveal .slide ul li > *:first-child,
.reveal .slide ol li > *:first-child {
margin-block-start: 0;
}
.reveal .slide ul li > *:last-child,
.reveal .slide ol li > *:last-child {
margin-block-end: 0;
}
.reveal .slide .columns:nth-child(3) {
margin-block-start: 0.8em;
}
.reveal blockquote {
box-shadow: none;
}
.reveal .tippy-content>* {
margin-top: 0.2em;
margin-bottom: 0.7em;
}
.reveal .tippy-content>*:last-child {
margin-bottom: 0.2em;
}
.reveal .slide > img.stretch.quarto-figure-center,
.reveal .slide > img.r-stretch.quarto-figure-center {
display: block;
margin-left: auto;
margin-right: auto;
}
.reveal .slide > img.stretch.quarto-figure-left,
.reveal .slide > img.r-stretch.quarto-figure-left {
display: block;
margin-left: 0;
margin-right: auto;
}
.reveal .slide > img.stretch.quarto-figure-right,
.reveal .slide > img.r-stretch.quarto-figure-right {
display: block;
margin-left: auto;
margin-right: 0;
}
</style>
</head>
<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<p class="titlep"> </p>
<div class="orangetitle">
<h1 class="title" style="color:white !important">CS 3100</h1>
<h2 class="subtitle" style="color:white !important">Data Structures and Algorithms 2</h2>
<h2 class="subtitle" style="color:white !important">Reductions</h2>
</div>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<p class="titlep"> </p>
<div class="titlesmall"><p>
<a href="https://www.cs.virginia.edu/~asb">Aaron Bloomfield</a> ([email protected])<br>
<a href="https://engineering.virginia.edu/faculty/raymond-pettit">Raymond Pettit</a> ([email protected])<br>
<a href="https://github.com/uva-cs/dsa2">@github</a> | <a href="index.html">↑</a> | <a href="?print-pdf"><img class="print" width="20" src="images/print-icon.svg" style="top:0px;vertical-align:middle;background-color:transparent;display:inline;width:30px;height:30px"></a>
</p></div>
<p class="titlep"> </p>
</section><section id="TOC">
<nav role="doc-toc">
<h2 id="toc-title">Contents</h2>
<ul>
<li><a href="#/network-flow" id="/toc-network-flow">Network Flow <!-- from day20.pptx --></a></li>
<li><a href="#/max-flow-min-cut" id="/toc-max-flow-min-cut">Max flow / Min cut <!-- from day21.pptx --></a></li>
<li><a href="#/bipartite-matching" id="/toc-bipartite-matching">Bipartite Matching <!-- from day21.pptx --></a></li>
<li><a href="#/reductions" id="/toc-reductions">Reductions <!-- from day22.pptx and day23.pptx --></a></li>
<li><a href="#/np-completeness" id="/toc-np-completeness">NP-Completeness <!-- from day23.pptx --></a></li>
</ul>
</nav>
</section>
<section>
<section id="network-flow" class="title-slide slide level1 center">
<h1>Network Flow <!-- from day20.pptx --></h1>
<div class="titlereading">
<p>Readings in CLRS 4th edition: chapter 24</p>
</div>
</section>
<section id="network-flow-1" class="slide level2">
<h2>Network Flow</h2>
<p><img data-src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cc/OpenRailwayMap_of_Northeastern_US_and_Southeastern_Canada.png/1024px-OpenRailwayMap_of_Northeastern_US_and_Southeastern_Canada.png" class="center" style="object-fit:none;object-position:right;height:513px;width:864px"></p>
<p><span class="font-90"><strong>Question:</strong> What is the maximum throughput of the railroad network from Omaha (OMA, far left) to Boston (BBY, far right)?</span></p>
</section>
<section id="flow-networks" class="slide level2">
<h2>Flow Networks</h2>
<!-- day20 slide 13 -->
<table style="width:100%">
<tbody><tr>
<td class="font-90" style="vertical-align:top;padding-left:0;width:40%">
<p> </p>
<div class="no-bullets-left">
<ul>
<li>Graph <span class="math inline">\(G=(V,E)\)</span></li>
<li><span class="purple">Start node</span> <span class="math inline">\({\color{purple}s}\in V\)</span></li>
<li><span class="navyblue">Sink node</span> <span class="math inline">\({\color{Blue}t}\in V\)</span></li>
<li><span class="green">Edge capacities</span> <span class="math inline">\({\color{green}c(e)} \in \mathbb{R}^+\)</span></li>
</ul>
</div>
</td>
<td class="svg-table-right compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-1.svg"></p>
</div>
</td>
</tr>
</tbody></table>
<p><strong>Max flow intuition:</strong> If <span class="math inline">\(\color{purple}s\)</span> is a faucet, <span class="math inline">\(\color{Blue}t\)</span> is a drain, and <span class="math inline">\(\color{purple}s\)</span> connects to <span class="math inline">\(\color{Blue}t\)</span> through a network of pipes <span class="math inline">\(E\)</span> with capacities <span class="math inline">\(\color{green}c(e)\)</span>, what is the <span class="und">maximum</span> amount of water which can flow from the faucet to the drain?</p>
</section>
<section id="network-flow-2" class="slide level2">
<h2>Network Flow</h2>
<!-- day20 slide 14 -->
<div class="svg-50-right-flow-graph">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
<p class="font-75 center">
<span class="red">flow</span> / <span class="green">capacity</span>
</p>
</div>
<div class="font-75">
<ul>
<li>Assignment of values <span class="math inline">\(\color{red}f(e)\)</span> to edges
<ul>
<li>“Amount of water going through that pipe”</li>
</ul></li>
<li>Capacity constraint
<ul>
<li><span class="math inline">\({\color{red}f(e)} \le {\color{green}c(e)}\)</span></li>
<li>“Flow cannot exceed capacity”</li>
</ul></li>
<li>Flow constraint
<ul>
<li><span class="math inline">\(\forall v \in V - \{ {\color{purple}s},{\color{Blue}t}\}\)</span>, <span class="math inline">\(\text{inflow}(v)=\text{outflow}(v)\)</span></li>
<li><span class="math inline">\(\text{inflow}(v)=\sum_{x \in V}f(x,v)\)</span></li>
<li><span class="math inline">\(\text{outflow}(v)=\sum_{x \in V}f(v,x)\)</span></li>
<li>Water going in must match water coming out</li>
</ul></li>
<li>Flow of <span class="math inline">\(G\)</span>: <span class="math inline">\(|f|=\text{outflow}({\color{purple}s})-\text{inflow}({\color{purple}s})\)</span></li>
<li>Net outflow of <span class="math inline">\(\color{purple}s\)</span>
<ul>
<li><span class="red">3 in this example</span></li>
</ul></li>
</ul>
</div>
</section>
<section id="maximum-flow-problem" class="slide level2">
<h2>Maximum Flow Problem</h2>
<!-- day20 slide 15 -->
<p class="font-90 center">
Of all valid flows through the graph, find the one that maximizes:
</p>
<p><span class="math display">\[|f|=\text{outflow}({\color{purple}s})-\text{inflow}({\color{purple}s})\]</span></p>
<table class="width-100">
<tbody><tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="fragment">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3.svg"></p>
</div>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="greedy-approach" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 16 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow.svg" class="center" style="height:50%;width:50%"></p>
</section>
<section id="greedy-approach-1" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 17 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-1.svg" class="center" style="height:50%;width:50%"></p>
</section>
<section id="greedy-approach-2" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 18 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-1-1.svg" class="center" style="height:50%;width:50%"></p>
<p class="red">
Flow: 20
</p>
</section>
<section id="greedy-approach-3" class="slide level2">
<h2>Greedy Approach</h2>
<!-- day20 slide 19 -->
<p><strong>Greedy choice:</strong> saturate <em>highest</em> capacity path first</p>
<p><img src="graphs/reductions/diamond_flow-2-2.svg" class="center" style="height:50%;width:50%"></p>
<p class="red">
Maximum flow: 30
</p>
<p><span class="font-90"><strong>Observe:</strong> highest capacity path is not <span class="und">saturated</span> in optimal solution</span></p>
</section>
<section id="residual-graphs" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 20 -->
<table class="width-100">
<tbody><tr>
<td colspan="2" class="font-70 zero-bottom">
<p>Given a flow <span class="math inline">\(f\)</span> in graph <span class="math inline">\(G\)</span>, the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> models <span class="und">additional</span> flow that is possible</p>
<ul>
<li><span class="orange und">Forward edge</span> for each edge in <span class="math inline">\(G\)</span> with weight set to remaining capacity <span class="math inline">\({\color{green}c(e)}-{\color{red}f(e)}\)</span>
<ul>
<li>Models <span class="und">additional</span> flow that can be sent along the edge: <span class="purple">flow to add</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<ul>
<li><span class="magenta und">Backward edge</span> by flipping each edge <span class="math inline">\(e\)</span> in <span class="math inline">\(G\)</span> with weight set to flow <span class="math inline">\({\color{red}f(e)}\)</span>
<ul>
<li>Models amount of flow that can be <span class="und">removed</span> from the edge: <span class="purple">flow to remove</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
<p class="svgcaption center">
Flow <span class="math inline">\(\color{red}f\)</span> in <span class="math inline">\(G\)</span>
</p>
</td>
<td class="compact">
<div class="fragment">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
<p class="svgcaption center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-1" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 21 -->
<table class="width-100">
<tbody><tr>
<td colspan="2" class="font-70 zero-bottom">
<p>Given a flow <span class="math inline">\(f\)</span> in graph <span class="math inline">\(G\)</span>, the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> models <span class="und">additional</span> flow that is possible</p>
<ul>
<li><span class="orange und">Forward edge</span> for each edge in <span class="math inline">\(G\)</span> with weight set to remaining capacity <span class="math inline">\({\color{green}c(e)}-{\color{red}f(e)}\)</span>
<ul>
<li>Models <span class="und">additional</span> flow that can be sent along the edge: <span class="purple">flow to add</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li><span class="magenta und">Backward edge</span> by flipping each edge <span class="math inline">\(e\)</span> in <span class="math inline">\(G\)</span> with weight set to flow <span class="math inline">\({\color{red}f(e)}\)</span>
<ul>
<li>Models amount of flow that can be <span class="und">removed</span> from the edge: <span class="purple">flow to remove</span></li>
</ul></li>
</ul>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
<p class="svgcaption center">
Flow <span class="math inline">\(\color{red}f\)</span> in <span class="math inline">\(G\)</span>
</p>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
<p class="svgcaption center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</p>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-example" class="slide level2">
<h2>Residual Graphs Example</h2>
<!-- day20 slide 22 -->
<table style="width:80%">
<tbody><tr>
<td class="und bold" style="text-align:center;width:50%">
Flow Graph
</td>
<td class="und bold" style="text-align:center">
Residual Graph
</td>
</tr>
<tr>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0">
<img src="graphs/reductions/diamond_flow-0-3.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment fade-out" data-fragment-index="1">
<img src="graphs/reductions/diamond_flow-0-4-True.svg" class="center diamond_flow_in_table">
</p>
<p style="margin:0" class="fragment" data-fragment-index="1">
<img src="graphs/reductions/diamond_flow-0-4-True-2.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
</tr>
<tr>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment" data-fragment-index="2">
<img src="graphs/reductions/diamond_flow-0-2.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
<td style="vertical-align:top;padding:0">
<div class="r-stack">
<p style="margin:0" class="fragment" data-fragment-index="3">
<img src="graphs/reductions/diamond_flow-0-5-True.svg" class="center diamond_flow_in_table">
</p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-2" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 23 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{Blue}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-3" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 24 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{Blue}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 opacity-0 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-False-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-2-True-1.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="residual-graphs-4" class="slide level2">
<h2>Residual Graphs</h2>
<!-- day20 slide 25 -->
<table>
<tbody><tr>
<td colspan="2" class="font-75 zero-bottom">
<p>Consider a path from <span class="math inline">\({\color{purple}s} \rightarrow {\color{Blue}t}\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span> using only edges with positive (non-zero) weight Consider the minimum-weight edge along the path: we can increase the flow by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<ul>
<li>Send <span class="math inline">\(w(e)\)</span> flow along all <span class="orange">forward</span> edges (these have at least <span class="math inline">\(w(e)\)</span> capacity)</li>
<li>Remove <span class="math inline">\(w(e)\)</span> flow along all <span class="magenta">backward</span> edges (these contain at least <span class="math inline">\(w(e)\)</span> units of flow)</li>
</ul>
</td>
</tr>
<tr>
<td colspan="2" class="font-75 zero-bottom">
<p><span class="bold">Observe:</span> Flow has <span class="und">increased</span> by <span class="math inline">\(w(e)\)</span></p>
</td>
</tr>
<tr>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3-False-1.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-3-True.svg"></p>
</div>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-algorithm" class="slide level2">
<h2>Ford-Fulkerson Algorithm</h2>
<div class="font-85">
<p>
Define an augmenting path to be an <span class="math inline">\({\color{purple}s} \rightarrow {\color{Blue}t}\)</span> path in the residual graph <span class="math inline">\(G_{\color{red}f}\)</span> (using edges of non-zero weight)
</p>
<p>
Ford-Fulkerson max-flow algorithm:
</p>
<ul>
<li class="">
Initialize <span class="math inline">\({\color{red}f(e)}=0\)</span> for all all <span class="math inline">\(e \in E\)</span>
</li>
<li class="">
Construct the residual network <span class="math inline">\(G_{\color{red}f}\)</span>
</li>
<li class="">
While there is an augmenting path <span class="math inline">\(p\)</span> in <span class="math inline">\(G_{\color{red}f}\)</span>:
<ul>
<li>
Let <span class="math inline">\(c=\min_{e \in E}c_f(e)\)</span> where <span class="math inline">\(c_f(e)\)</span> is the weight of edge <span class="math inline">\(e\)</span> in the residual network <span class="math inline">\(G_{\color{red}f}\)</span>
</li>
<li>
Add <span class="math inline">\(c\)</span> units of flow to <span class="math inline">\(G\)</span> based on the augmenting path <span class="math inline">\(p\)</span>
</li>
<li>
Update the residual network <span class="math inline">\(G_{\color{red}f}\)</span> for the updated flow
</li>
</ul>
</li>
</ul>
<p> </p>
<div class="fragment">
<table class="navynote font-75">
<tbody><tr>
<td>
<span class="bold">Ford-Fulkerson approach:</span> take <span class="und">any</span> augmenting path (will revisit this later)
</td>
</tr>
</tbody></table>
</div>
</div>
</section>
<section id="ford-fulkerson-example" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 29 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:0;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True.svg"></p>
</div>
</td>
</tr>
<tr>
<td class="font-90">
<span class="bold">Initially:</span> <span class="math inline">\({\color{red}f(e)}=0\)</span> for all <span class="math inline">\(e \in E\)</span>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-1" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 30 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<div class="fragment">
<p class="center">
Increase flow by 1 unit
</p>
</div>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True-2.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-2" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 31 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:1;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-False-2.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-4-True-2.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-3" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 32 -->
<p> </p>
<table class="width-100">
<tbody><tr>
<td>
</td>
<td style="opacity:0;text-align:center" class="zero-bottom">
<p class="center">
Increase flow by 1 unit
</p>
</td>
</tr>
<tr>
<td class="compact" style="width:50%">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5.svg"></p>
</div>
</td>
<td class="compact">
<div class="svg-100">
<p><img src="graphs/reductions/flow_graph-5-True.svg"></p>
</div>
</td>
</tr>
<tr>
<td>
</td>
<td style="text-align:center">
Residual graph <span class="math inline">\(G_{\color{red}f}\)</span>
</td>
</tr>
</tbody></table>
</section>
<section id="ford-fulkerson-example-4" class="slide level2">
<h2>Ford-Fulkerson Example</h2>
<!-- day20 slide 33 -->
<p> </p>
<table class="width-100">
<tbody><tr>