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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
|
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>networkx.algorithms.dag — NetworkX 3.0rc2.dev0 documentation</title>
<script data-cfasync="false">
document.documentElement.dataset.mode = localStorage.getItem("mode") || "light";
document.documentElement.dataset.theme = localStorage.getItem("theme") || "light";
</script>
<!-- Loaded before other Sphinx assets -->
<link href="../../../_static/styles/theme.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../_static/styles/bootstrap.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../_static/styles/pydata-sphinx-theme.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../_static/vendor/fontawesome/6.1.2/css/all.min.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-solid-900.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-brands-400.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-regular-400.woff2">
<link rel="stylesheet" type="text/css" href="../../../_static/pygments.css" />
<link rel="stylesheet" type="text/css" href="../../../_static/custom.css" />
<link rel="stylesheet" type="text/css" href="../../../_static/sg_gallery.css" />
<link rel="stylesheet" type="text/css" href="../../../_static/sg_gallery-binder.css" />
<link rel="stylesheet" type="text/css" href="../../../_static/sg_gallery-dataframe.css" />
<link rel="stylesheet" type="text/css" href="../../../_static/sg_gallery-rendered-html.css" />
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../../../_static/scripts/bootstrap.js?digest=796348d33e8b1d947c94">
<link rel="preload" as="script" href="../../../_static/scripts/pydata-sphinx-theme.js?digest=796348d33e8b1d947c94">
<script data-url_root="../../../" id="documentation_options" src="../../../_static/documentation_options.js"></script>
<script src="../../../_static/jquery.js"></script>
<script src="../../../_static/underscore.js"></script>
<script src="../../../_static/_sphinx_javascript_frameworks_compat.js"></script>
<script src="../../../_static/doctools.js"></script>
<script src="../../../_static/sphinx_highlight.js"></script>
<script src="../../../_static/copybutton.js"></script>
<script>DOCUMENTATION_OPTIONS.pagename = '_modules/networkx/algorithms/dag';</script>
<link rel="canonical" href="https://networkx.org/documentation/stable/_modules/networkx/algorithms/dag.html" />
<link rel="search" type="application/opensearchdescription+xml"
title="Search within NetworkX 3.0rc2.dev0 documentation"
href="../../../_static/opensearch.xml"/>
<link rel="index" title="Index" href="../../../genindex.html" />
<link rel="search" title="Search" href="../../../search.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="en">
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="180" data-default-mode="light">
<a class="skip-link" href="#main-content">Skip to main content</a>
<div class="container-fluid version-alert devbar">
<div class="row no-gutters">
<div class="col-12 text-center">
This page is documentation for a DEVELOPMENT / PRE-RELEASE version.
<a
class="btn version-stable font-weight-bold ml-3 my-3 align-baseline"
href="https://networkx.org/documentation/stable/"
>Switch to stable version</a
>
</div>
</div>
</div>
<input type="checkbox" class="sidebar-toggle" name="__primary" id="__primary">
<label class="overlay overlay-primary" for="__primary"></label>
<input type="checkbox" class="sidebar-toggle" name="__secondary" id="__secondary">
<label class="overlay overlay-secondary" for="__secondary"></label>
<div class="search-button__wrapper">
<div class="search-button__overlay"></div>
<div class="search-button__search-container">
<form class="bd-search d-flex align-items-center" action="../../../search.html" method="get">
<i class="fa-solid fa-magnifying-glass"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search the docs ..." aria-label="Search the docs ..." autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false">
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
</form>
</div>
</div>
<nav class="bd-header navbar navbar-expand-lg bd-navbar" id="navbar-main"><div class="bd-header__inner bd-page-width">
<label class="sidebar-toggle primary-toggle" for="__primary">
<span class="fa-solid fa-bars"></span>
</label>
<div id="navbar-start">
<a class="navbar-brand logo" href="../../../index.html">
<img src="../../../_static/networkx_banner.svg" class="logo__image only-light" alt="Logo image">
<img src="../../../_static/networkx_banner.svg" class="logo__image only-dark" alt="Logo image">
</a>
</div>
<div class="col-lg-9 navbar-header-items">
<div id="navbar-center" class="mr-auto">
<div class="navbar-center-item">
<nav class="navbar-nav">
<p class="sidebar-header-items__title" role="heading" aria-level="1" aria-label="Site Navigation">
Site Navigation
</p>
<ul id="navbar-main-elements" class="navbar-nav">
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../install.html">
Install
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../tutorial.html">
Tutorial
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../reference/index.html">
Reference
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../auto_examples/index.html">
Gallery
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../developer/index.html">
Developer
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../release/index.html">
Releases
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-external" href="https://networkx.org/nx-guides/">
Guides
</a>
</li>
</ul>
</nav>
</div>
</div>
<div id="navbar-end">
<div class="navbar-end-item navbar-persistent--container">
<button class="btn btn-sm navbar-btn search-button search-button__button" title="Search" aria-label="Search" data-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
</button>
</div>
<div class="navbar-end-item">
<button class="theme-switch-button btn btn-sm btn-outline-primary navbar-btn rounded-circle" title="light/dark" aria-label="light/dark" data-toggle="tooltip">
<span class="theme-switch" data-mode="light"><i class="fa-solid fa-sun"></i></span>
<span class="theme-switch" data-mode="dark"><i class="fa-solid fa-moon"></i></span>
<span class="theme-switch" data-mode="auto"><i class="fa-solid fa-circle-half-stroke"></i></span>
</button>
</div>
<div class="navbar-end-item">
<ul id="navbar-icon-links" class="navbar-nav" aria-label="Icon Links">
<li class="nav-item">
<a href="https://networkx.org" title="Home Page" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fas fa-home"></i></span>
<label class="sr-only">Home Page</label></a>
</li>
<li class="nav-item">
<a href="https://github.com/networkx/networkx" title="GitHub" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fab fa-github-square"></i></span>
<label class="sr-only">GitHub</label></a>
</li>
</ul>
</div>
<div class="navbar-end-item">
<ul class="navbar-nav">
<li class="mr-2 dropdown">
<button
type="button"
class="btn btn-version btn-sm navbar-btn dropdown-toggle"
id="dLabelMore"
data-toggle="dropdown"
>
v3.0rc2.dev0
<span class="caret"></span>
</button>
<ul class="dropdown-menu" aria-labelledby="dLabelMore">
<li>
<a href="https://networkx.org/documentation/latest/index.html"
>devel (latest)</a
>
</li>
<li>
<a href="https://networkx.org/documentation/stable/index.html"
>current (stable)</a
>
</li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div class="navbar-persistent--mobile">
<button class="btn btn-sm navbar-btn search-button search-button__button" title="Search" aria-label="Search" data-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
</button>
</div>
<label class="sidebar-toggle secondary-toggle" for="__secondary">
<span class="fa-solid fa-outdent"></span>
</label>
</div>
</nav>
<div class="bd-container">
<div class="bd-container__inner bd-page-width">
<div class="bd-sidebar-primary bd-sidebar">
<div class="sidebar-header-items sidebar-primary__section">
<div class="sidebar-header-items__center">
<div class="navbar-center-item">
<nav class="navbar-nav">
<p class="sidebar-header-items__title" role="heading" aria-level="1" aria-label="Site Navigation">
Site Navigation
</p>
<ul id="navbar-main-elements" class="navbar-nav">
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../install.html">
Install
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../tutorial.html">
Tutorial
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../reference/index.html">
Reference
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../auto_examples/index.html">
Gallery
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../developer/index.html">
Developer
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../release/index.html">
Releases
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-external" href="https://networkx.org/nx-guides/">
Guides
</a>
</li>
</ul>
</nav>
</div>
</div>
<div class="sidebar-header-items__end">
<div class="navbar-end-item">
<button class="theme-switch-button btn btn-sm btn-outline-primary navbar-btn rounded-circle" title="light/dark" aria-label="light/dark" data-toggle="tooltip">
<span class="theme-switch" data-mode="light"><i class="fa-solid fa-sun"></i></span>
<span class="theme-switch" data-mode="dark"><i class="fa-solid fa-moon"></i></span>
<span class="theme-switch" data-mode="auto"><i class="fa-solid fa-circle-half-stroke"></i></span>
</button>
</div>
<div class="navbar-end-item">
<ul id="navbar-icon-links" class="navbar-nav" aria-label="Icon Links">
<li class="nav-item">
<a href="https://networkx.org" title="Home Page" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fas fa-home"></i></span>
<label class="sr-only">Home Page</label></a>
</li>
<li class="nav-item">
<a href="https://github.com/networkx/networkx" title="GitHub" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fab fa-github-square"></i></span>
<label class="sr-only">GitHub</label></a>
</li>
</ul>
</div>
<div class="navbar-end-item">
<ul class="navbar-nav">
<li class="mr-2 dropdown">
<button
type="button"
class="btn btn-version btn-sm navbar-btn dropdown-toggle"
id="dLabelMore"
data-toggle="dropdown"
>
v3.0rc2.dev0
<span class="caret"></span>
</button>
<ul class="dropdown-menu" aria-labelledby="dLabelMore">
<li>
<a href="https://networkx.org/documentation/latest/index.html"
>devel (latest)</a
>
</li>
<li>
<a href="https://networkx.org/documentation/stable/index.html"
>current (stable)</a
>
</li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div class="sidebar-start-items sidebar-primary__section">
<div class="sidebar-start-items__item">
</div>
</div>
<div class="sidebar-end-items sidebar-primary__section">
<div class="sidebar-end-items__item">
</div>
</div>
<div id="rtd-footer-container"></div>
</div>
<main id="main-content" class="bd-main">
<div class="bd-content">
<div class="bd-article-container">
<div class="bd-header-article">
</div>
<article class="bd-article" role="main">
<h1>Source code for networkx.algorithms.dag</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Algorithms for directed acyclic graphs (DAGs).</span>
<span class="sd">Note that most of these functions are only guaranteed to work for DAGs.</span>
<span class="sd">In general, these functions do not check for acyclic-ness, so it is up</span>
<span class="sd">to the user to check for that.</span>
<span class="sd">"""</span>
<span class="kn">import</span> <span class="nn">heapq</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">deque</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span><span class="p">,</span> <span class="n">combinations</span><span class="p">,</span> <span class="n">product</span><span class="p">,</span> <span class="n">starmap</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">gcd</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">arbitrary_element</span><span class="p">,</span> <span class="n">not_implemented_for</span><span class="p">,</span> <span class="n">pairwise</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"descendants"</span><span class="p">,</span>
<span class="s2">"ancestors"</span><span class="p">,</span>
<span class="s2">"topological_sort"</span><span class="p">,</span>
<span class="s2">"lexicographical_topological_sort"</span><span class="p">,</span>
<span class="s2">"all_topological_sorts"</span><span class="p">,</span>
<span class="s2">"topological_generations"</span><span class="p">,</span>
<span class="s2">"is_directed_acyclic_graph"</span><span class="p">,</span>
<span class="s2">"is_aperiodic"</span><span class="p">,</span>
<span class="s2">"transitive_closure"</span><span class="p">,</span>
<span class="s2">"transitive_closure_dag"</span><span class="p">,</span>
<span class="s2">"transitive_reduction"</span><span class="p">,</span>
<span class="s2">"antichains"</span><span class="p">,</span>
<span class="s2">"dag_longest_path"</span><span class="p">,</span>
<span class="s2">"dag_longest_path_length"</span><span class="p">,</span>
<span class="s2">"dag_to_branching"</span><span class="p">,</span>
<span class="s2">"compute_v_structures"</span><span class="p">,</span>
<span class="p">]</span>
<span class="n">chaini</span> <span class="o">=</span> <span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span>
<div class="viewcode-block" id="descendants"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.descendants.html#networkx.algorithms.dag.descendants">[docs]</a><span class="nd">@nx</span><span class="o">.</span><span class="n">_dispatch</span>
<span class="k">def</span> <span class="nf">descendants</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns all nodes reachable from `source` in `G`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX Graph</span>
<span class="sd"> source : node in `G`</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> set()</span>
<span class="sd"> The descendants of `source` in `G`</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If node `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.path_graph(5, create_using=nx.DiGraph)</span>
<span class="sd"> >>> sorted(nx.descendants(DG, 2))</span>
<span class="sd"> [3, 4]</span>
<span class="sd"> The `source` node is not a descendant of itself, but can be included manually:</span>
<span class="sd"> >>> sorted(nx.descendants(DG, 2) | {2})</span>
<span class="sd"> [2, 3, 4]</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> ancestors</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="p">{</span><span class="n">child</span> <span class="k">for</span> <span class="n">parent</span><span class="p">,</span> <span class="n">child</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">bfs_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">)}</span></div>
<div class="viewcode-block" id="ancestors"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.ancestors.html#networkx.algorithms.dag.ancestors">[docs]</a><span class="nd">@nx</span><span class="o">.</span><span class="n">_dispatch</span>
<span class="k">def</span> <span class="nf">ancestors</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns all nodes having a path to `source` in `G`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX Graph</span>
<span class="sd"> source : node in `G`</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> set()</span>
<span class="sd"> The ancestors of `source` in `G`</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If node `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.path_graph(5, create_using=nx.DiGraph)</span>
<span class="sd"> >>> sorted(nx.ancestors(DG, 2))</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> The `source` node is not an ancestor of itself, but can be included manually:</span>
<span class="sd"> >>> sorted(nx.ancestors(DG, 2) | {2})</span>
<span class="sd"> [0, 1, 2]</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> descendants</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="p">{</span><span class="n">child</span> <span class="k">for</span> <span class="n">parent</span><span class="p">,</span> <span class="n">child</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">bfs_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)}</span></div>
<span class="k">def</span> <span class="nf">has_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Decides whether the directed graph has a cycle."""</span>
<span class="k">try</span><span class="p">:</span>
<span class="c1"># Feed the entire iterator into a zero-length deque.</span>
<span class="n">deque</span><span class="p">(</span><span class="n">topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">),</span> <span class="n">maxlen</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">except</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnfeasible</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<div class="viewcode-block" id="is_directed_acyclic_graph"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.is_directed_acyclic_graph.html#networkx.algorithms.dag.is_directed_acyclic_graph">[docs]</a><span class="k">def</span> <span class="nf">is_directed_acyclic_graph</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns True if the graph `G` is a directed acyclic graph (DAG) or</span>
<span class="sd"> False if not.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> True if `G` is a DAG, False otherwise</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> Undirected graph::</span>
<span class="sd"> >>> G = nx.Graph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> nx.is_directed_acyclic_graph(G)</span>
<span class="sd"> False</span>
<span class="sd"> Directed graph with cycle::</span>
<span class="sd"> >>> G = nx.DiGraph([(1, 2), (2, 3), (3, 1)])</span>
<span class="sd"> >>> nx.is_directed_acyclic_graph(G)</span>
<span class="sd"> False</span>
<span class="sd"> Directed acyclic graph::</span>
<span class="sd"> >>> G = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> nx.is_directed_acyclic_graph(G)</span>
<span class="sd"> True</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> topological_sort</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">has_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">)</span></div>
<div class="viewcode-block" id="topological_generations"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.topological_generations.html#networkx.algorithms.dag.topological_generations">[docs]</a><span class="k">def</span> <span class="nf">topological_generations</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Stratifies a DAG into generations.</span>
<span class="sd"> A topological generation is node collection in which ancestors of a node in each</span>
<span class="sd"> generation are guaranteed to be in a previous generation, and any descendants of</span>
<span class="sd"> a node are guaranteed to be in a following generation. Nodes are guaranteed to</span>
<span class="sd"> be in the earliest possible generation that they can belong to.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX digraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> sets of nodes</span>
<span class="sd"> Yields sets of nodes representing each generation.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> Generations are defined for directed graphs only. If the graph</span>
<span class="sd"> `G` is undirected, a :exc:`NetworkXError` is raised.</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` is not a directed acyclic graph (DAG) no topological generations</span>
<span class="sd"> exist and a :exc:`NetworkXUnfeasible` exception is raised. This can also</span>
<span class="sd"> be raised if `G` is changed while the returned iterator is being processed</span>
<span class="sd"> RuntimeError</span>
<span class="sd"> If `G` is changed while the returned iterator is being processed.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(2, 1), (3, 1)])</span>
<span class="sd"> >>> [sorted(generation) for generation in nx.topological_generations(DG)]</span>
<span class="sd"> [[2, 3], [1]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The generation in which a node resides can also be determined by taking the</span>
<span class="sd"> max-path-distance from the node to the farthest leaf node. That value can</span>
<span class="sd"> be obtained with this function using `enumerate(topological_generations(G))`.</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> topological_sort</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Topological sort not defined on undirected graphs."</span><span class="p">)</span>
<span class="n">multigraph</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">()</span>
<span class="n">indegree_map</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">d</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">></span> <span class="mi">0</span><span class="p">}</span>
<span class="n">zero_indegree</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="k">while</span> <span class="n">zero_indegree</span><span class="p">:</span>
<span class="n">this_generation</span> <span class="o">=</span> <span class="n">zero_indegree</span>
<span class="n">zero_indegree</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">this_generation</span><span class="p">:</span>
<span class="k">if</span> <span class="n">node</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">"Graph changed during iteration"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">child</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">neighbors</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span> <span class="o">-=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">node</span><span class="p">][</span><span class="n">child</span><span class="p">])</span> <span class="k">if</span> <span class="n">multigraph</span> <span class="k">else</span> <span class="mi">1</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">"Graph changed during iteration"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span>
<span class="k">if</span> <span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">zero_indegree</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
<span class="k">del</span> <span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span>
<span class="k">yield</span> <span class="n">this_generation</span>
<span class="k">if</span> <span class="n">indegree_map</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnfeasible</span><span class="p">(</span>
<span class="s2">"Graph contains a cycle or graph changed during iteration"</span>
<span class="p">)</span></div>
<div class="viewcode-block" id="topological_sort"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.topological_sort.html#networkx.algorithms.dag.topological_sort">[docs]</a><span class="k">def</span> <span class="nf">topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns a generator of nodes in topologically sorted order.</span>
<span class="sd"> A topological sort is a nonunique permutation of the nodes of a</span>
<span class="sd"> directed graph such that an edge from u to v implies that u</span>
<span class="sd"> appears before v in the topological sort order. This ordering is</span>
<span class="sd"> valid only if the graph has no directed cycles.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX digraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> nodes</span>
<span class="sd"> Yields the nodes in topological sorted order.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> Topological sort is defined for directed graphs only. If the graph `G`</span>
<span class="sd"> is undirected, a :exc:`NetworkXError` is raised.</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` is not a directed acyclic graph (DAG) no topological sort exists</span>
<span class="sd"> and a :exc:`NetworkXUnfeasible` exception is raised. This can also be</span>
<span class="sd"> raised if `G` is changed while the returned iterator is being processed</span>
<span class="sd"> RuntimeError</span>
<span class="sd"> If `G` is changed while the returned iterator is being processed.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> To get the reverse order of the topological sort:</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> list(reversed(list(nx.topological_sort(DG))))</span>
<span class="sd"> [3, 2, 1]</span>
<span class="sd"> If your DiGraph naturally has the edges representing tasks/inputs</span>
<span class="sd"> and nodes representing people/processes that initiate tasks, then</span>
<span class="sd"> topological_sort is not quite what you need. You will have to change</span>
<span class="sd"> the tasks to nodes with dependence reflected by edges. The result is</span>
<span class="sd"> a kind of topological sort of the edges. This can be done</span>
<span class="sd"> with :func:`networkx.line_graph` as follows:</span>
<span class="sd"> >>> list(nx.topological_sort(nx.line_graph(DG)))</span>
<span class="sd"> [(1, 2), (2, 3)]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This algorithm is based on a description and proof in</span>
<span class="sd"> "Introduction to Algorithms: A Creative Approach" [1]_ .</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> is_directed_acyclic_graph, lexicographical_topological_sort</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Manber, U. (1989).</span>
<span class="sd"> *Introduction to Algorithms - A Creative Approach.* Addison-Wesley.</span>
<span class="sd"> """</span>
<span class="k">for</span> <span class="n">generation</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">topological_generations</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">yield from</span> <span class="n">generation</span></div>
<div class="viewcode-block" id="lexicographical_topological_sort"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.lexicographical_topological_sort.html#networkx.algorithms.dag.lexicographical_topological_sort">[docs]</a><span class="k">def</span> <span class="nf">lexicographical_topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Generate the nodes in the unique lexicographical topological sort order.</span>
<span class="sd"> Generates a unique ordering of nodes by first sorting topologically (for which there are often</span>
<span class="sd"> multiple valid orderings) and then additionally by sorting lexicographically.</span>
<span class="sd"> A topological sort arranges the nodes of a directed graph so that the</span>
<span class="sd"> upstream node of each directed edge precedes the downstream node.</span>
<span class="sd"> It is always possible to find a solution for directed graphs that have no cycles.</span>
<span class="sd"> There may be more than one valid solution.</span>
<span class="sd"> Lexicographical sorting is just sorting alphabetically. It is used here to break ties in the</span>
<span class="sd"> topological sort and to determine a single, unique ordering. This can be useful in comparing</span>
<span class="sd"> sort results.</span>
<span class="sd"> The lexicographical order can be customized by providing a function to the `key=` parameter.</span>
<span class="sd"> The definition of the key function is the same as used in python's built-in `sort()`.</span>
<span class="sd"> The function takes a single argument and returns a key to use for sorting purposes.</span>
<span class="sd"> Lexicographical sorting can fail if the node names are un-sortable. See the example below.</span>
<span class="sd"> The solution is to provide a function to the `key=` argument that returns sortable keys.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX digraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> key : function, optional</span>
<span class="sd"> A function of one argument that converts a node name to a comparison key.</span>
<span class="sd"> It defines and resolves ambiguities in the sort order. Defaults to the identity function.</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> nodes</span>
<span class="sd"> Yields the nodes of G in lexicographical topological sort order.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> Topological sort is defined for directed graphs only. If the graph `G`</span>
<span class="sd"> is undirected, a :exc:`NetworkXError` is raised.</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` is not a directed acyclic graph (DAG) no topological sort exists</span>
<span class="sd"> and a :exc:`NetworkXUnfeasible` exception is raised. This can also be</span>
<span class="sd"> raised if `G` is changed while the returned iterator is being processed</span>
<span class="sd"> RuntimeError</span>
<span class="sd"> If `G` is changed while the returned iterator is being processed.</span>
<span class="sd"> TypeError</span>
<span class="sd"> Results from un-sortable node names.</span>
<span class="sd"> Consider using `key=` parameter to resolve ambiguities in the sort order.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(2, 1), (2, 5), (1, 3), (1, 4), (5, 4)])</span>
<span class="sd"> >>> list(nx.lexicographical_topological_sort(DG))</span>
<span class="sd"> [2, 1, 3, 5, 4]</span>
<span class="sd"> >>> list(nx.lexicographical_topological_sort(DG, key=lambda x: -x))</span>
<span class="sd"> [2, 5, 1, 4, 3]</span>
<span class="sd"> The sort will fail for any graph with integer and string nodes. Comparison of integer to strings</span>
<span class="sd"> is not defined in python. Is 3 greater or less than 'red'?</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 'red'), (3, 'red'), (1, 'green'), (2, 'blue')])</span>
<span class="sd"> >>> list(nx.lexicographical_topological_sort(DG))</span>
<span class="sd"> Traceback (most recent call last):</span>
<span class="sd"> ...</span>
<span class="sd"> TypeError: '<' not supported between instances of 'str' and 'int'</span>
<span class="sd"> ...</span>
<span class="sd"> Incomparable nodes can be resolved using a `key` function. This example function</span>
<span class="sd"> allows comparison of integers and strings by returning a tuple where the first</span>
<span class="sd"> element is True for `str`, False otherwise. The second element is the node name.</span>
<span class="sd"> This groups the strings and integers separately so they can be compared only among themselves.</span>
<span class="sd"> >>> key = lambda node: (isinstance(node, str), node)</span>
<span class="sd"> >>> list(nx.lexicographical_topological_sort(DG, key=key))</span>
<span class="sd"> [1, 2, 3, 'blue', 'green', 'red']</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This algorithm is based on a description and proof in</span>
<span class="sd"> "Introduction to Algorithms: A Creative Approach" [1]_ .</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> topological_sort</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Manber, U. (1989).</span>
<span class="sd"> *Introduction to Algorithms - A Creative Approach.* Addison-Wesley.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"Topological sort not defined on undirected graphs."</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
<span class="k">if</span> <span class="n">key</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
<span class="k">return</span> <span class="n">node</span>
<span class="n">nodeid_map</span> <span class="o">=</span> <span class="p">{</span><span class="n">n</span><span class="p">:</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">)}</span>
<span class="k">def</span> <span class="nf">create_tuple</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
<span class="k">return</span> <span class="n">key</span><span class="p">(</span><span class="n">node</span><span class="p">),</span> <span class="n">nodeid_map</span><span class="p">[</span><span class="n">node</span><span class="p">],</span> <span class="n">node</span>
<span class="n">indegree_map</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">d</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">></span> <span class="mi">0</span><span class="p">}</span>
<span class="c1"># These nodes have zero indegree and ready to be returned.</span>
<span class="n">zero_indegree</span> <span class="o">=</span> <span class="p">[</span><span class="n">create_tuple</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">heapq</span><span class="o">.</span><span class="n">heapify</span><span class="p">(</span><span class="n">zero_indegree</span><span class="p">)</span>
<span class="k">while</span> <span class="n">zero_indegree</span><span class="p">:</span>
<span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">node</span> <span class="o">=</span> <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">zero_indegree</span><span class="p">)</span>
<span class="k">if</span> <span class="n">node</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">"Graph changed during iteration"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">child</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s2">"Graph changed during iteration"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span>
<span class="k">if</span> <span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">zero_indegree</span><span class="p">,</span> <span class="n">create_tuple</span><span class="p">(</span><span class="n">child</span><span class="p">))</span>
<span class="k">except</span> <span class="ne">TypeError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
<span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">err</span><span class="si">}</span><span class="se">\n</span><span class="s2">Consider using `key=` parameter to resolve ambiguities in the sort order."</span>
<span class="p">)</span>
<span class="k">del</span> <span class="n">indegree_map</span><span class="p">[</span><span class="n">child</span><span class="p">]</span>
<span class="k">yield</span> <span class="n">node</span>
<span class="k">if</span> <span class="n">indegree_map</span><span class="p">:</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"Graph contains a cycle or graph changed during iteration"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnfeasible</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span></div>
<div class="viewcode-block" id="all_topological_sorts"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.all_topological_sorts.html#networkx.algorithms.dag.all_topological_sorts">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">all_topological_sorts</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns a generator of _all_ topological sorts of the directed graph G.</span>
<span class="sd"> A topological sort is a nonunique permutation of the nodes such that an</span>
<span class="sd"> edge from u to v implies that u appears before v in the topological sort</span>
<span class="sd"> order.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed graph</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> topological_sort_order : list</span>
<span class="sd"> a list of nodes in `G`, representing one of the topological sort orders</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` is not acyclic</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> To enumerate all topological sorts of directed graph:</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3), (2, 4)])</span>
<span class="sd"> >>> list(nx.all_topological_sorts(DG))</span>
<span class="sd"> [[1, 2, 4, 3], [1, 2, 3, 4]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Implements an iterative version of the algorithm given in [1].</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Knuth, Donald E., Szwarcfiter, Jayme L. (1974).</span>
<span class="sd"> "A Structured Program to Generate All Topological Sorting Arrangements"</span>
<span class="sd"> Information Processing Letters, Volume 2, Issue 6, 1974, Pages 153-157,</span>
<span class="sd"> ISSN 0020-0190,</span>
<span class="sd"> https://doi.org/10.1016/0020-0190(74)90001-5.</span>
<span class="sd"> Elsevier (North-Holland), Amsterdam</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Topological sort not defined on undirected graphs."</span><span class="p">)</span>
<span class="c1"># the names of count and D are chosen to match the global variables in [1]</span>
<span class="c1"># number of edges originating in a vertex v</span>
<span class="n">count</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">())</span>
<span class="c1"># vertices with indegree 0</span>
<span class="n">D</span> <span class="o">=</span> <span class="n">deque</span><span class="p">([</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">])</span>
<span class="c1"># stack of first value chosen at a position k in the topological sort</span>
<span class="n">bases</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">current_sort</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># do-while construct</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">assert</span> <span class="nb">all</span><span class="p">([</span><span class="n">count</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">D</span><span class="p">])</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">current_sort</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="k">yield</span> <span class="nb">list</span><span class="p">(</span><span class="n">current_sort</span><span class="p">)</span>
<span class="c1"># clean-up stack</span>
<span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">current_sort</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">current_sort</span><span class="p">)</span>
<span class="n">q</span> <span class="o">=</span> <span class="n">current_sort</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># "restores" all edges (q, x)</span>
<span class="c1"># NOTE: it is important to iterate over edges instead</span>
<span class="c1"># of successors, so count is updated correctly in multigraphs</span>
<span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">out_edges</span><span class="p">(</span><span class="n">q</span><span class="p">):</span>
<span class="n">count</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">assert</span> <span class="n">count</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">>=</span> <span class="mi">0</span>
<span class="c1"># remove entries from D</span>
<span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">D</span><span class="p">)</span> <span class="o">></span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">count</span><span class="p">[</span><span class="n">D</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># corresponds to a circular shift of the values in D</span>
<span class="c1"># if the first value chosen (the base) is in the first</span>
<span class="c1"># position of D again, we are done and need to consider the</span>
<span class="c1"># previous condition</span>
<span class="n">D</span><span class="o">.</span><span class="n">appendleft</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="k">if</span> <span class="n">D</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">bases</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
<span class="c1"># all possible values have been chosen at current position</span>
<span class="c1"># remove corresponding marker</span>
<span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># there are still elements that have not been fixed</span>
<span class="c1"># at the current position in the topological sort</span>
<span class="c1"># stop removing elements, escape inner loop</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">D</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnfeasible</span><span class="p">(</span><span class="s2">"Graph contains a cycle."</span><span class="p">)</span>
<span class="c1"># choose next node</span>
<span class="n">q</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># "erase" all edges (q, x)</span>
<span class="c1"># NOTE: it is important to iterate over edges instead</span>
<span class="c1"># of successors, so count is updated correctly in multigraphs</span>
<span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">out_edges</span><span class="p">(</span><span class="n">q</span><span class="p">):</span>
<span class="n">count</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">assert</span> <span class="n">count</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">>=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="n">count</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
<span class="n">current_sort</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="c1"># base for current position might _not_ be fixed yet</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o"><</span> <span class="nb">len</span><span class="p">(</span><span class="n">current_sort</span><span class="p">):</span>
<span class="n">bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">break</span></div>
<div class="viewcode-block" id="is_aperiodic"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.is_aperiodic.html#networkx.algorithms.dag.is_aperiodic">[docs]</a><span class="k">def</span> <span class="nf">is_aperiodic</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns True if `G` is aperiodic.</span>
<span class="sd"> A directed graph is aperiodic if there is no integer k > 1 that</span>
<span class="sd"> divides the length of every cycle in the graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed graph</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> True if the graph is aperiodic False otherwise</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> A graph consisting of one cycle, the length of which is 2. Therefore ``k = 2``</span>
<span class="sd"> divides the length of every cycle in the graph and thus the graph</span>
<span class="sd"> is *not aperiodic*::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 1)])</span>
<span class="sd"> >>> nx.is_aperiodic(DG)</span>
<span class="sd"> False</span>
<span class="sd"> A graph consisting of two cycles: one of length 2 and the other of length 3.</span>
<span class="sd"> The cycle lengths are coprime, so there is no single value of k where ``k > 1``</span>
<span class="sd"> that divides each cycle length and therefore the graph is *aperiodic*::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1), (1, 4), (4, 1)])</span>
<span class="sd"> >>> nx.is_aperiodic(DG)</span>
<span class="sd"> True</span>
<span class="sd"> A graph consisting of two cycles: one of length 2 and the other of length 4.</span>
<span class="sd"> The lengths of the cycles share a common factor ``k = 2``, and therefore</span>
<span class="sd"> the graph is *not aperiodic*::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 1), (3, 4), (4, 5), (5, 6), (6, 3)])</span>
<span class="sd"> >>> nx.is_aperiodic(DG)</span>
<span class="sd"> False</span>
<span class="sd"> An acyclic graph, therefore the graph is *not aperiodic*::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> nx.is_aperiodic(DG)</span>
<span class="sd"> False</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This uses the method outlined in [1]_, which runs in $O(m)$ time</span>
<span class="sd"> given $m$ edges in `G`. Note that a graph is not aperiodic if it is</span>
<span class="sd"> acyclic as every integer trivial divides length 0 cycles.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Jarvis, J. P.; Shier, D. R. (1996),</span>
<span class="sd"> "Graph-theoretic analysis of finite Markov chains,"</span>
<span class="sd"> in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling:</span>
<span class="sd"> A Multidisciplinary Approach, CRC Press.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"is_aperiodic not defined for undirected graphs"</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">arbitrary_element</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">levels</span> <span class="o">=</span> <span class="p">{</span><span class="n">s</span><span class="p">:</span> <span class="mi">0</span><span class="p">}</span>
<span class="n">this_level</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">]</span>
<span class="n">g</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">lev</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">this_level</span><span class="p">:</span>
<span class="n">next_level</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">this_level</span><span class="p">:</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">levels</span><span class="p">:</span> <span class="c1"># Non-Tree Edge</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">levels</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">-</span> <span class="n">levels</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span> <span class="c1"># Tree Edge</span>
<span class="n">next_level</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">levels</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">lev</span>
<span class="n">this_level</span> <span class="o">=</span> <span class="n">next_level</span>
<span class="n">lev</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">levels</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">):</span> <span class="c1"># All nodes in tree</span>
<span class="k">return</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_aperiodic</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">levels</span><span class="p">)))</span></div>
<div class="viewcode-block" id="transitive_closure"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.transitive_closure.html#networkx.algorithms.dag.transitive_closure">[docs]</a><span class="k">def</span> <span class="nf">transitive_closure</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">reflexive</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns transitive closure of a graph</span>
<span class="sd"> The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that</span>
<span class="sd"> for all v, w in V there is an edge (v, w) in E+ if and only if there</span>
<span class="sd"> is a path from v to w in G.</span>
<span class="sd"> Handling of paths from v to v has some flexibility within this definition.</span>
<span class="sd"> A reflexive transitive closure creates a self-loop for the path</span>
<span class="sd"> from v to v of length 0. The usual transitive closure creates a</span>
<span class="sd"> self-loop only if a cycle exists (a path from v to v with length > 0).</span>
<span class="sd"> We also allow an option for no self-loops.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX Graph</span>
<span class="sd"> A directed/undirected graph/multigraph.</span>
<span class="sd"> reflexive : Bool or None, optional (default: False)</span>
<span class="sd"> Determines when cycles create self-loops in the Transitive Closure.</span>
<span class="sd"> If True, trivial cycles (length 0) create self-loops. The result</span>
<span class="sd"> is a reflexive transitive closure of G.</span>
<span class="sd"> If False (the default) non-trivial cycles create self-loops.</span>
<span class="sd"> If None, self-loops are not created.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX graph</span>
<span class="sd"> The transitive closure of `G`</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `reflexive` not in `{None, True, False}`</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> The treatment of trivial (i.e. length 0) cycles is controlled by the</span>
<span class="sd"> `reflexive` parameter.</span>
<span class="sd"> Trivial (i.e. length 0) cycles do not create self-loops when</span>
<span class="sd"> ``reflexive=False`` (the default)::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> TC = nx.transitive_closure(DG, reflexive=False)</span>
<span class="sd"> >>> TC.edges()</span>
<span class="sd"> OutEdgeView([(1, 2), (1, 3), (2, 3)])</span>
<span class="sd"> However, nontrivial (i.e. length greater then 0) cycles create self-loops</span>
<span class="sd"> when ``reflexive=False`` (the default)::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1)])</span>
<span class="sd"> >>> TC = nx.transitive_closure(DG, reflexive=False)</span>
<span class="sd"> >>> TC.edges()</span>
<span class="sd"> OutEdgeView([(1, 2), (1, 3), (1, 1), (2, 3), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3)])</span>
<span class="sd"> Trivial cycles (length 0) create self-loops when ``reflexive=True``::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> TC = nx.transitive_closure(DG, reflexive=True)</span>
<span class="sd"> >>> TC.edges()</span>
<span class="sd"> OutEdgeView([(1, 2), (1, 1), (1, 3), (2, 3), (2, 2), (3, 3)])</span>
<span class="sd"> And the third option is not to create self-loops at all when ``reflexive=None``::</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1)])</span>
<span class="sd"> >>> TC = nx.transitive_closure(DG, reflexive=None)</span>
<span class="sd"> >>> TC.edges()</span>
<span class="sd"> OutEdgeView([(1, 2), (1, 3), (2, 3), (2, 1), (3, 1), (3, 2)])</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] https://www.ics.uci.edu/~eppstein/PADS/PartialOrder.py</span>
<span class="sd"> """</span>
<span class="n">TC</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="k">if</span> <span class="n">reflexive</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">{</span><span class="kc">None</span><span class="p">,</span> <span class="kc">True</span><span class="p">,</span> <span class="kc">False</span><span class="p">}:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Incorrect value for the parameter `reflexive`"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">if</span> <span class="n">reflexive</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">TC</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">descendants</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">if</span> <span class="n">u</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">TC</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="k">elif</span> <span class="n">reflexive</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">TC</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span>
<span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">descendants</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">|</span> <span class="p">{</span><span class="n">v</span><span class="p">}</span> <span class="k">if</span> <span class="n">u</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">TC</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">elif</span> <span class="n">reflexive</span> <span class="ow">is</span> <span class="kc">False</span><span class="p">:</span>
<span class="n">TC</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">edge_bfs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">if</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">TC</span><span class="p">[</span><span class="n">v</span><span class="p">])</span>
<span class="k">return</span> <span class="n">TC</span></div>
<div class="viewcode-block" id="transitive_closure_dag"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.transitive_closure_dag.html#networkx.algorithms.dag.transitive_closure_dag">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">transitive_closure_dag</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">topo_order</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the transitive closure of a directed acyclic graph.</span>
<span class="sd"> This function is faster than the function `transitive_closure`, but fails</span>
<span class="sd"> if the graph has a cycle.</span>
<span class="sd"> The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that</span>
<span class="sd"> for all v, w in V there is an edge (v, w) in E+ if and only if there</span>
<span class="sd"> is a non-null path from v to w in G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> topo_order: list or tuple, optional</span>
<span class="sd"> A topological order for G (if None, the function will compute one)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX DiGraph</span>
<span class="sd"> The transitive closure of `G`</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` has a cycle</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3)])</span>
<span class="sd"> >>> TC = nx.transitive_closure_dag(DG)</span>
<span class="sd"> >>> TC.edges()</span>
<span class="sd"> OutEdgeView([(1, 2), (1, 3), (2, 3)])</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This algorithm is probably simple enough to be well-known but I didn't find</span>
<span class="sd"> a mention in the literature.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">topo_order</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">topo_order</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="n">TC</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="c1"># idea: traverse vertices following a reverse topological order, connecting</span>
<span class="c1"># each vertex to its descendants at distance 2 as we go</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">topo_order</span><span class="p">):</span>
<span class="n">TC</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">descendants_at_distance</span><span class="p">(</span><span class="n">TC</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="k">return</span> <span class="n">TC</span></div>
<div class="viewcode-block" id="transitive_reduction"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.transitive_reduction.html#networkx.algorithms.dag.transitive_reduction">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">transitive_reduction</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns transitive reduction of a directed graph</span>
<span class="sd"> The transitive reduction of G = (V,E) is a graph G- = (V,E-) such that</span>
<span class="sd"> for all v,w in V there is an edge (v,w) in E- if and only if (v,w) is</span>
<span class="sd"> in E and there is no path from v to w in G with length greater than 1.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX DiGraph</span>
<span class="sd"> The transitive reduction of `G`</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If `G` is not a directed acyclic graph (DAG) transitive reduction is</span>
<span class="sd"> not uniquely defined and a :exc:`NetworkXError` exception is raised.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> To perform transitive reduction on a DiGraph:</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (2, 3), (1, 3)])</span>
<span class="sd"> >>> TR = nx.transitive_reduction(DG)</span>
<span class="sd"> >>> list(TR.edges)</span>
<span class="sd"> [(1, 2), (2, 3)]</span>
<span class="sd"> To avoid unnecessary data copies, this implementation does not return a</span>
<span class="sd"> DiGraph with node/edge data.</span>
<span class="sd"> To perform transitive reduction on a DiGraph and transfer node/edge data:</span>
<span class="sd"> >>> DG = nx.DiGraph()</span>
<span class="sd"> >>> DG.add_edges_from([(1, 2), (2, 3), (1, 3)], color='red')</span>
<span class="sd"> >>> TR = nx.transitive_reduction(DG)</span>
<span class="sd"> >>> TR.add_nodes_from(DG.nodes(data=True))</span>
<span class="sd"> >>> TR.add_edges_from((u, v, DG.edges[u, v]) for u, v in TR.edges)</span>
<span class="sd"> >>> list(TR.edges(data=True))</span>
<span class="sd"> [(1, 2, {'color': 'red'}), (2, 3, {'color': 'red'})]</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> https://en.wikipedia.org/wiki/Transitive_reduction</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">is_directed_acyclic_graph</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"Directed Acyclic Graph required for transitive_reduction"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
<span class="n">TR</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="n">TR</span><span class="o">.</span><span class="n">add_nodes_from</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nodes</span><span class="p">())</span>
<span class="n">descendants</span> <span class="o">=</span> <span class="p">{}</span>
<span class="c1"># count before removing set stored in descendants</span>
<span class="n">check_count</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">)</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">u_nbrs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">])</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">u_nbrs</span><span class="p">:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">descendants</span><span class="p">:</span>
<span class="n">descendants</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="p">{</span><span class="n">y</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">dfs_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)}</span>
<span class="n">u_nbrs</span> <span class="o">-=</span> <span class="n">descendants</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="n">check_count</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">check_count</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">del</span> <span class="n">descendants</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="n">TR</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">u_nbrs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">TR</span></div>
<div class="viewcode-block" id="antichains"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.antichains.html#networkx.algorithms.dag.antichains">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">antichains</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">topo_order</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Generates antichains from a directed acyclic graph (DAG).</span>
<span class="sd"> An antichain is a subset of a partially ordered set such that any</span>
<span class="sd"> two elements in the subset are incomparable.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> topo_order: list or tuple, optional</span>
<span class="sd"> A topological order for G (if None, the function will compute one)</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> antichain : list</span>
<span class="sd"> a list of nodes in `G` representing an antichain</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> NetworkXUnfeasible</span>
<span class="sd"> If `G` contains a cycle</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(1, 2), (1, 3)])</span>
<span class="sd"> >>> list(nx.antichains(DG))</span>
<span class="sd"> [[], [3], [2], [2, 3], [1]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function was originally developed by Peter Jipsen and Franco Saliola</span>
<span class="sd"> for the SAGE project. It's included in NetworkX with permission from the</span>
<span class="sd"> authors. Original SAGE code at:</span>
<span class="sd"> https://github.com/sagemath/sage/blob/master/src/sage/combinat/posets/hasse_diagram.py</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Free Lattices, by R. Freese, J. Jezek and J. B. Nation,</span>
<span class="sd"> AMS, Vol 42, 1995, p. 226.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">topo_order</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">topo_order</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
<span class="n">TC</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">transitive_closure_dag</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">topo_order</span><span class="p">)</span>
<span class="n">antichains_stacks</span> <span class="o">=</span> <span class="p">[([],</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">topo_order</span><span class="p">)))]</span>
<span class="k">while</span> <span class="n">antichains_stacks</span><span class="p">:</span>
<span class="p">(</span><span class="n">antichain</span><span class="p">,</span> <span class="n">stack</span><span class="p">)</span> <span class="o">=</span> <span class="n">antichains_stacks</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># Invariant:</span>
<span class="c1"># - the elements of antichain are independent</span>
<span class="c1"># - the elements of stack are independent from those of antichain</span>
<span class="k">yield</span> <span class="n">antichain</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">new_antichain</span> <span class="o">=</span> <span class="n">antichain</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span>
<span class="n">new_stack</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">stack</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="n">t</span> <span class="ow">in</span> <span class="n">TC</span><span class="p">[</span><span class="n">x</span><span class="p">])</span> <span class="ow">or</span> <span class="p">(</span><span class="n">x</span> <span class="ow">in</span> <span class="n">TC</span><span class="p">[</span><span class="n">t</span><span class="p">]))]</span>
<span class="n">antichains_stacks</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">new_antichain</span><span class="p">,</span> <span class="n">new_stack</span><span class="p">))</span></div>
<div class="viewcode-block" id="dag_longest_path"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.dag_longest_path.html#networkx.algorithms.dag.dag_longest_path">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">dag_longest_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">default_weight</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">topo_order</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the longest path in a directed acyclic graph (DAG).</span>
<span class="sd"> If `G` has edges with `weight` attribute the edge data are used as</span>
<span class="sd"> weight values.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> weight : str, optional</span>
<span class="sd"> Edge data key to use for weight</span>
<span class="sd"> default_weight : int, optional</span>
<span class="sd"> The weight of edges that do not have a weight attribute</span>
<span class="sd"> topo_order: list or tuple, optional</span>
<span class="sd"> A topological order for `G` (if None, the function will compute one)</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> Longest path</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(0, 1, {'cost':1}), (1, 2, {'cost':1}), (0, 2, {'cost':42})])</span>
<span class="sd"> >>> list(nx.all_simple_paths(DG, 0, 2))</span>
<span class="sd"> [[0, 1, 2], [0, 2]]</span>
<span class="sd"> >>> nx.dag_longest_path(DG)</span>
<span class="sd"> [0, 1, 2]</span>
<span class="sd"> >>> nx.dag_longest_path(DG, weight="cost")</span>
<span class="sd"> [0, 2]</span>
<span class="sd"> In the case where multiple valid topological orderings exist, `topo_order`</span>
<span class="sd"> can be used to specify a specific ordering:</span>
<span class="sd"> >>> DG = nx.DiGraph([(0, 1), (0, 2)])</span>
<span class="sd"> >>> sorted(nx.all_topological_sorts(DG)) # Valid topological orderings</span>
<span class="sd"> [[0, 1, 2], [0, 2, 1]]</span>
<span class="sd"> >>> nx.dag_longest_path(DG, topo_order=[0, 1, 2])</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> >>> nx.dag_longest_path(DG, topo_order=[0, 2, 1])</span>
<span class="sd"> [0, 2]</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> dag_longest_path_length</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[]</span>
<span class="k">if</span> <span class="n">topo_order</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">topo_order</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">topological_sort</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{}</span> <span class="c1"># stores {v : (length, u)}</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">topo_order</span><span class="p">:</span>
<span class="n">us</span> <span class="o">=</span> <span class="p">[</span>
<span class="p">(</span>
<span class="n">dist</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="o">+</span> <span class="p">(</span>
<span class="nb">max</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">values</span><span class="p">(),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">))</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">()</span>
<span class="k">else</span> <span class="n">data</span>
<span class="p">)</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">),</span>
<span class="n">u</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">data</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()</span>
<span class="p">]</span>
<span class="c1"># Use the best predecessor if there is one and its distance is</span>
<span class="c1"># non-negative, otherwise terminate.</span>
<span class="n">maxu</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">us</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="k">if</span> <span class="n">us</span> <span class="k">else</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">dist</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">maxu</span> <span class="k">if</span> <span class="n">maxu</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">>=</span> <span class="mi">0</span> <span class="k">else</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">v</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">dist</span><span class="p">[</span><span class="n">x</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="n">path</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="n">u</span> <span class="o">!=</span> <span class="n">v</span><span class="p">:</span>
<span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">v</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">dist</span><span class="p">[</span><span class="n">v</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
<span class="n">path</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="k">return</span> <span class="n">path</span></div>
<div class="viewcode-block" id="dag_longest_path_length"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.dag_longest_path_length.html#networkx.algorithms.dag.dag_longest_path_length">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">dag_longest_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">default_weight</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the longest path length in a DAG</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX DiGraph</span>
<span class="sd"> A directed acyclic graph (DAG)</span>
<span class="sd"> weight : string, optional</span>
<span class="sd"> Edge data key to use for weight</span>
<span class="sd"> default_weight : int, optional</span>
<span class="sd"> The weight of edges that do not have a weight attribute</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> int</span>
<span class="sd"> Longest path length</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> DG = nx.DiGraph([(0, 1, {'cost':1}), (1, 2, {'cost':1}), (0, 2, {'cost':42})])</span>
<span class="sd"> >>> list(nx.all_simple_paths(DG, 0, 2))</span>
<span class="sd"> [[0, 1, 2], [0, 2]]</span>
<span class="sd"> >>> nx.dag_longest_path_length(DG)</span>
<span class="sd"> 2</span>
<span class="sd"> >>> nx.dag_longest_path_length(DG, weight="cost")</span>
<span class="sd"> 42</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> dag_longest_path</span>
<span class="sd"> """</span>
<span class="n">path</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">dag_longest_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">)</span>
<span class="n">path_length</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">path</span><span class="p">):</span>
<span class="n">i</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">))</span>
<span class="n">path_length</span> <span class="o">+=</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">for</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">pairwise</span><span class="p">(</span><span class="n">path</span><span class="p">):</span>
<span class="n">path_length</span> <span class="o">+=</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="n">default_weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">path_length</span></div>
<span class="k">def</span> <span class="nf">root_to_leaf_paths</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Yields root-to-leaf paths in a directed acyclic graph.</span>
<span class="sd"> `G` must be a directed acyclic graph. If not, the behavior of this</span>
<span class="sd"> function is undefined. A "root" in this graph is a node of in-degree</span>
<span class="sd"> zero and a "leaf" a node of out-degree zero.</span>
<span class="sd"> When invoked, this function iterates over each path from any root to</span>
<span class="sd"> any leaf. A path is a list of nodes.</span>
<span class="sd"> """</span>
<span class="n">roots</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">in_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">leaves</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">()</span> <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">all_paths</span> <span class="o">=</span> <span class="n">partial</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">all_simple_paths</span><span class="p">,</span> <span class="n">G</span><span class="p">)</span>
<span class="c1"># TODO In Python 3, this would be better as `yield from ...`.</span>
<span class="k">return</span> <span class="n">chaini</span><span class="p">(</span><span class="n">starmap</span><span class="p">(</span><span class="n">all_paths</span><span class="p">,</span> <span class="n">product</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">leaves</span><span class="p">)))</span>
<div class="viewcode-block" id="dag_to_branching"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.dag.dag_to_branching.html#networkx.algorithms.dag.dag_to_branching">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">dag_to_branching</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns a branching representing all (overlapping) paths from</span>
<span class="sd"> root nodes to leaf nodes in the given directed acyclic graph.</span>
<span class="sd"> As described in :mod:`networkx.algorithms.tree.recognition`, a</span>
<span class="sd"> *branching* is a directed forest in which each node has at most one</span>
<span class="sd"> parent. In other words, a branching is a disjoint union of</span>
<span class="sd"> *arborescences*. For this function, each node of in-degree zero in</span>
<span class="sd"> `G` becomes a root of one of the arborescences, and there will be</span>
<span class="sd"> one leaf node for each distinct path from that root to a leaf node</span>
<span class="sd"> in `G`.</span>
<span class="sd"> Each node `v` in `G` with *k* parents becomes *k* distinct nodes in</span>
<span class="sd"> the returned branching, one for each parent, and the sub-DAG rooted</span>
<span class="sd"> at `v` is duplicated for each copy. The algorithm then recurses on</span>
<span class="sd"> the children of each copy of `v`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed acyclic graph.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> DiGraph</span>
<span class="sd"> The branching in which there is a bijection between root-to-leaf</span>
<span class="sd"> paths in `G` (in which multiple paths may share the same leaf)</span>
<span class="sd"> and root-to-leaf paths in the branching (in which there is a</span>
<span class="sd"> unique path from a root to a leaf).</span>
<span class="sd"> Each node has an attribute 'source' whose value is the original</span>
<span class="sd"> node to which this node corresponds. No other graph, node, or</span>
<span class="sd"> edge attributes are copied into this new graph.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNotImplemented</span>
<span class="sd"> If `G` is not directed, or if `G` is a multigraph.</span>
<span class="sd"> HasACycle</span>
<span class="sd"> If `G` is not acyclic.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> To examine which nodes in the returned branching were produced by</span>
<span class="sd"> which original node in the directed acyclic graph, we can collect</span>
<span class="sd"> the mapping from source node to new nodes into a dictionary. For</span>
<span class="sd"> example, consider the directed diamond graph::</span>
<span class="sd"> >>> from collections import defaultdict</span>
<span class="sd"> >>> from operator import itemgetter</span>
<span class="sd"> >>></span>
<span class="sd"> >>> G = nx.DiGraph(nx.utils.pairwise("abd"))</span>
<span class="sd"> >>> G.add_edges_from(nx.utils.pairwise("acd"))</span>
<span class="sd"> >>> B = nx.dag_to_branching(G)</span>
<span class="sd"> >>></span>
<span class="sd"> >>> sources = defaultdict(set)</span>
<span class="sd"> >>> for v, source in B.nodes(data="source"):</span>
<span class="sd"> ... sources[source].add(v)</span>
<span class="sd"> >>> len(sources["a"])</span>
<span class="sd"> 1</span>
<span class="sd"> >>> len(sources["d"])</span>
<span class="sd"> 2</span>
<span class="sd"> To copy node attributes from the original graph to the new graph,</span>
<span class="sd"> you can use a dictionary like the one constructed in the above</span>
<span class="sd"> example::</span>
<span class="sd"> >>> for source, nodes in sources.items():</span>
<span class="sd"> ... for v in nodes:</span>
<span class="sd"> ... B.nodes[v].update(G.nodes[source])</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function is not idempotent in the sense that the node labels in</span>
<span class="sd"> the returned branching may be uniquely generated each time the</span>
<span class="sd"> function is invoked. In fact, the node labels may not be integers;</span>
<span class="sd"> in order to relabel the nodes to be more readable, you can use the</span>
<span class="sd"> :func:`networkx.convert_node_labels_to_integers` function.</span>
<span class="sd"> The current implementation of this function uses</span>
<span class="sd"> :func:`networkx.prefix_tree`, so it is subject to the limitations of</span>
<span class="sd"> that function.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">has_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"dag_to_branching is only defined for acyclic graphs"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">HasACycle</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
<span class="n">paths</span> <span class="o">=</span> <span class="n">root_to_leaf_paths</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">prefix_tree</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span>
<span class="c1"># Remove the synthetic `root`(0) and `NIL`(-1) nodes from the tree</span>
<span class="n">B</span><span class="o">.</span><span class="n">remove_node</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">B</span><span class="o">.</span><span class="n">remove_node</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">B</span></div>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">compute_v_structures</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Iterate through the graph to compute all v-structures.</span>
<span class="sd"> V-structures are triples in the directed graph where</span>
<span class="sd"> two parent nodes point to the same child and the two parent nodes</span>
<span class="sd"> are not adjacent.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A networkx DiGraph.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> vstructs : iterator of tuples</span>
<span class="sd"> The v structures within the graph. Each v structure is a 3-tuple with the</span>
<span class="sd"> parent, collider, and other parent.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> https://en.wikipedia.org/wiki/Collider_(statistics)</span>
<span class="sd"> """</span>
<span class="k">for</span> <span class="n">collider</span><span class="p">,</span> <span class="n">preds</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">pred</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="k">for</span> <span class="n">common_parents</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="n">preds</span><span class="p">,</span> <span class="n">r</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
<span class="c1"># ensure that the colliders are the same</span>
<span class="n">common_parents</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">common_parents</span><span class="p">)</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">common_parents</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">collider</span><span class="p">,</span> <span class="n">common_parents</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</article>
</div>
<div class="bd-sidebar-secondary bd-toc">
<div class="toc-item">
<div id="searchbox"></div>
</div>
<div class="toc-item">
</div>
<div class="toc-item">
</div>
</div>
</div>
<footer class="bd-footer-content">
<div class="bd-footer-content__inner">
</div>
</footer>
</main>
</div>
</div>
<!-- Scripts loaded after <body> so the DOM is not blocked -->
<script src="../../../_static/scripts/bootstrap.js?digest=796348d33e8b1d947c94"></script>
<script src="../../../_static/scripts/pydata-sphinx-theme.js?digest=796348d33e8b1d947c94"></script>
<footer class="bd-footer"><div class="bd-footer__inner container">
<div class="footer-item">
<p class="copyright">
© Copyright 2004-2023, NetworkX Developers.<br>
</p>
</div>
<div class="footer-item">
<p class="theme-version">
Built with the
<a href="https://pydata-sphinx-theme.readthedocs.io/en/stable/index.html">
PyData Sphinx Theme
</a>
0.12.0.
</p>
</div>
<div class="footer-item">
<p class="sphinx-version">
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 5.2.3.<br>
</p>
</div>
</div>
</footer>
</body>
</html>
|
