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
|
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>networkx.algorithms.cycles — 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/cycles';</script>
<link rel="canonical" href="https://networkx.org/documentation/stable/_modules/networkx/algorithms/cycles.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.cycles</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">========================</span>
<span class="sd">Cycle finding algorithms</span>
<span class="sd">========================</span>
<span class="sd">"""</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</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">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">"cycle_basis"</span><span class="p">,</span>
<span class="s2">"simple_cycles"</span><span class="p">,</span>
<span class="s2">"recursive_simple_cycles"</span><span class="p">,</span>
<span class="s2">"find_cycle"</span><span class="p">,</span>
<span class="s2">"minimum_cycle_basis"</span><span class="p">,</span>
<span class="p">]</span>
<div class="viewcode-block" id="cycle_basis"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.cycles.cycle_basis.html#networkx.algorithms.cycles.cycle_basis">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cycle_basis</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">root</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="sd">"""Returns a list of cycles which form a basis for cycles of G.</span>
<span class="sd"> A basis for cycles of a network is a minimal collection of</span>
<span class="sd"> cycles such that any cycle in the network can be written</span>
<span class="sd"> as a sum of cycles in the basis. Here summation of cycles</span>
<span class="sd"> is defined as "exclusive or" of the edges. Cycle bases are</span>
<span class="sd"> useful, e.g. when deriving equations for electric circuits</span>
<span class="sd"> using Kirchhoff's Laws.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX Graph</span>
<span class="sd"> root : node, optional</span>
<span class="sd"> Specify starting node for basis.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> A list of cycle lists. Each cycle list is a list of nodes</span>
<span class="sd"> which forms a cycle (loop) in G.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.Graph()</span>
<span class="sd"> >>> nx.add_cycle(G, [0, 1, 2, 3])</span>
<span class="sd"> >>> nx.add_cycle(G, [0, 3, 4, 5])</span>
<span class="sd"> >>> print(nx.cycle_basis(G, 0))</span>
<span class="sd"> [[3, 4, 5, 0], [1, 2, 3, 0]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is adapted from algorithm CACM 491 [1]_.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Paton, K. An algorithm for finding a fundamental set of</span>
<span class="sd"> cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> simple_cycles</span>
<span class="sd"> """</span>
<span class="n">gnodes</span> <span class="o">=</span> <span class="nb">set</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">cycles</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="n">gnodes</span><span class="p">:</span> <span class="c1"># loop over connected components</span>
<span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">root</span> <span class="o">=</span> <span class="n">gnodes</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">stack</span> <span class="o">=</span> <span class="p">[</span><span class="n">root</span><span class="p">]</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">root</span><span class="p">:</span> <span class="n">root</span><span class="p">}</span>
<span class="n">used</span> <span class="o">=</span> <span class="p">{</span><span class="n">root</span><span class="p">:</span> <span class="nb">set</span><span class="p">()}</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span> <span class="c1"># walk the spanning tree finding cycles</span>
<span class="n">z</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="c1"># use last-in so cycles easier to find</span>
<span class="n">zused</span> <span class="o">=</span> <span class="n">used</span><span class="p">[</span><span class="n">z</span><span class="p">]</span>
<span class="k">for</span> <span class="n">nbr</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">z</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">nbr</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">used</span><span class="p">:</span> <span class="c1"># new node</span>
<span class="n">pred</span><span class="p">[</span><span class="n">nbr</span><span class="p">]</span> <span class="o">=</span> <span class="n">z</span>
<span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nbr</span><span class="p">)</span>
<span class="n">used</span><span class="p">[</span><span class="n">nbr</span><span class="p">]</span> <span class="o">=</span> <span class="p">{</span><span class="n">z</span><span class="p">}</span>
<span class="k">elif</span> <span class="n">nbr</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span> <span class="c1"># self loops</span>
<span class="n">cycles</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">z</span><span class="p">])</span>
<span class="k">elif</span> <span class="n">nbr</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">zused</span><span class="p">:</span> <span class="c1"># found a cycle</span>
<span class="n">pn</span> <span class="o">=</span> <span class="n">used</span><span class="p">[</span><span class="n">nbr</span><span class="p">]</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="p">[</span><span class="n">nbr</span><span class="p">,</span> <span class="n">z</span><span class="p">]</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">pred</span><span class="p">[</span><span class="n">z</span><span class="p">]</span>
<span class="k">while</span> <span class="n">p</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pn</span><span class="p">:</span>
<span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">pred</span><span class="p">[</span><span class="n">p</span><span class="p">]</span>
<span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">cycles</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span>
<span class="n">used</span><span class="p">[</span><span class="n">nbr</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
<span class="n">gnodes</span> <span class="o">-=</span> <span class="nb">set</span><span class="p">(</span><span class="n">pred</span><span class="p">)</span>
<span class="n">root</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">return</span> <span class="n">cycles</span></div>
<div class="viewcode-block" id="simple_cycles"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.cycles.simple_cycles.html#networkx.algorithms.cycles.simple_cycles">[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">simple_cycles</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">"""Find simple cycles (elementary circuits) of a directed graph.</span>
<span class="sd"> A `simple cycle`, or `elementary circuit`, is a closed path where</span>
<span class="sd"> no node appears twice. Two elementary circuits are distinct if they</span>
<span class="sd"> are not cyclic permutations of each other.</span>
<span class="sd"> This is a nonrecursive, iterator/generator version of Johnson's</span>
<span class="sd"> algorithm [1]_. There may be better algorithms for some cases [2]_ [3]_.</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"> list of nodes</span>
<span class="sd"> Each cycle is represented by a list of nodes along the cycle.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]</span>
<span class="sd"> >>> G = nx.DiGraph(edges)</span>
<span class="sd"> >>> sorted(nx.simple_cycles(G))</span>
<span class="sd"> [[0], [0, 1, 2], [0, 2], [1, 2], [2]]</span>
<span class="sd"> To filter the cycles so that they don't include certain nodes or edges,</span>
<span class="sd"> copy your graph and eliminate those nodes or edges before calling.</span>
<span class="sd"> For example, to exclude self-loops from the above example:</span>
<span class="sd"> >>> H = G.copy()</span>
<span class="sd"> >>> H.remove_edges_from(nx.selfloop_edges(G))</span>
<span class="sd"> >>> sorted(nx.simple_cycles(H))</span>
<span class="sd"> [[0, 1, 2], [0, 2], [1, 2]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation follows pp. 79-80 in [1]_.</span>
<span class="sd"> The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$</span>
<span class="sd"> elementary circuits.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Finding all the elementary circuits of a directed graph.</span>
<span class="sd"> D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.</span>
<span class="sd"> https://doi.org/10.1137/0204007</span>
<span class="sd"> .. [2] Enumerating the cycles of a digraph: a new preprocessing strategy.</span>
<span class="sd"> G. Loizou and P. Thanish, Information Sciences, v. 27, 163-182, 1982.</span>
<span class="sd"> .. [3] A search strategy for the elementary cycles of a directed graph.</span>
<span class="sd"> J.L. Szwarcfiter and P.E. Lauer, BIT NUMERICAL MATHEMATICS,</span>
<span class="sd"> v. 16, no. 2, 192-204, 1976.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> cycle_basis</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">_unblock</span><span class="p">(</span><span class="n">thisnode</span><span class="p">,</span> <span class="n">blocked</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
<span class="n">stack</span> <span class="o">=</span> <span class="p">{</span><span class="n">thisnode</span><span class="p">}</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
<span class="n">node</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="k">if</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">blocked</span><span class="p">:</span>
<span class="n">blocked</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<span class="n">stack</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">node</span><span class="p">])</span>
<span class="n">B</span><span class="p">[</span><span class="n">node</span><span class="p">]</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
<span class="c1"># Johnson's algorithm requires some ordering of the nodes.</span>
<span class="c1"># We assign the arbitrary ordering given by the strongly connected comps</span>
<span class="c1"># There is no need to track the ordering as each node removed as processed.</span>
<span class="c1"># Also we save the actual graph so we can mutate it. We only take the</span>
<span class="c1"># edges because we do not want to copy edge and node attributes here.</span>
<span class="n">subG</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">G</span><span class="p">)(</span><span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">())</span>
<span class="n">sccs</span> <span class="o">=</span> <span class="p">[</span><span class="n">scc</span> <span class="k">for</span> <span class="n">scc</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">strongly_connected_components</span><span class="p">(</span><span class="n">subG</span><span class="p">)</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">scc</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">]</span>
<span class="c1"># Johnson's algorithm exclude self cycle edges like (v, v)</span>
<span class="c1"># To be backward compatible, we record those cycles in advance</span>
<span class="c1"># and then remove from subG</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">subG</span><span class="p">:</span>
<span class="k">if</span> <span class="n">subG</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
<span class="k">yield</span> <span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="n">subG</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">while</span> <span class="n">sccs</span><span class="p">:</span>
<span class="n">scc</span> <span class="o">=</span> <span class="n">sccs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">sccG</span> <span class="o">=</span> <span class="n">subG</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">scc</span><span class="p">)</span>
<span class="c1"># order of scc determines ordering of nodes</span>
<span class="n">startnode</span> <span class="o">=</span> <span class="n">scc</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># Processing node runs "circuit" routine from recursive version</span>
<span class="n">path</span> <span class="o">=</span> <span class="p">[</span><span class="n">startnode</span><span class="p">]</span>
<span class="n">blocked</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span> <span class="c1"># vertex: blocked from search?</span>
<span class="n">closed</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span> <span class="c1"># nodes involved in a cycle</span>
<span class="n">blocked</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">startnode</span><span class="p">)</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span> <span class="c1"># graph portions that yield no elementary circuit</span>
<span class="n">stack</span> <span class="o">=</span> <span class="p">[(</span><span class="n">startnode</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">sccG</span><span class="p">[</span><span class="n">startnode</span><span class="p">]))]</span> <span class="c1"># sccG gives comp nbrs</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
<span class="n">thisnode</span><span class="p">,</span> <span class="n">nbrs</span> <span class="o">=</span> <span class="n">stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">if</span> <span class="n">nbrs</span><span class="p">:</span>
<span class="n">nextnode</span> <span class="o">=</span> <span class="n">nbrs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">nextnode</span> <span class="o">==</span> <span class="n">startnode</span><span class="p">:</span>
<span class="k">yield</span> <span class="n">path</span><span class="p">[:]</span>
<span class="n">closed</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
<span class="c1"># print "Found a cycle", path, closed</span>
<span class="k">elif</span> <span class="n">nextnode</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">blocked</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">nextnode</span><span class="p">)</span>
<span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nextnode</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">sccG</span><span class="p">[</span><span class="n">nextnode</span><span class="p">])))</span>
<span class="n">closed</span><span class="o">.</span><span class="n">discard</span><span class="p">(</span><span class="n">nextnode</span><span class="p">)</span>
<span class="n">blocked</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">nextnode</span><span class="p">)</span>
<span class="k">continue</span>
<span class="c1"># done with nextnode... look for more neighbors</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nbrs</span><span class="p">:</span> <span class="c1"># no more nbrs</span>
<span class="k">if</span> <span class="n">thisnode</span> <span class="ow">in</span> <span class="n">closed</span><span class="p">:</span>
<span class="n">_unblock</span><span class="p">(</span><span class="n">thisnode</span><span class="p">,</span> <span class="n">blocked</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">for</span> <span class="n">nbr</span> <span class="ow">in</span> <span class="n">sccG</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">thisnode</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">B</span><span class="p">[</span><span class="n">nbr</span><span class="p">]:</span>
<span class="n">B</span><span class="p">[</span><span class="n">nbr</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">thisnode</span><span class="p">)</span>
<span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># assert path[-1] == thisnode</span>
<span class="n">path</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="c1"># done processing this node</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">subG</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">scc</span><span class="p">)</span> <span class="c1"># make smaller to avoid work in SCC routine</span>
<span class="n">sccs</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">scc</span> <span class="k">for</span> <span class="n">scc</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">strongly_connected_components</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">scc</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">)</span></div>
<div class="viewcode-block" id="recursive_simple_cycles"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.cycles.recursive_simple_cycles.html#networkx.algorithms.cycles.recursive_simple_cycles">[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">recursive_simple_cycles</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="sd">"""Find simple cycles (elementary circuits) of a directed graph.</span>
<span class="sd"> A `simple cycle`, or `elementary circuit`, is a closed path where</span>
<span class="sd"> no node appears twice. Two elementary circuits are distinct if they</span>
<span class="sd"> are not cyclic permutations of each other.</span>
<span class="sd"> This version uses a recursive algorithm to build a list of cycles.</span>
<span class="sd"> You should probably use the iterator version called simple_cycles().</span>
<span class="sd"> Warning: This recursive version uses lots of RAM!</span>
<span class="sd"> It appears in NetworkX for pedagogical value.</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"> A list of cycles, where each cycle is represented by a list of nodes</span>
<span class="sd"> along the cycle.</span>
<span class="sd"> Example:</span>
<span class="sd"> >>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]</span>
<span class="sd"> >>> G = nx.DiGraph(edges)</span>
<span class="sd"> >>> nx.recursive_simple_cycles(G)</span>
<span class="sd"> [[0], [2], [0, 1, 2], [0, 2], [1, 2]]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The implementation follows pp. 79-80 in [1]_.</span>
<span class="sd"> The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$</span>
<span class="sd"> elementary circuits.</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Finding all the elementary circuits of a directed graph.</span>
<span class="sd"> D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.</span>
<span class="sd"> https://doi.org/10.1137/0204007</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> simple_cycles, cycle_basis</span>
<span class="sd"> """</span>
<span class="c1"># Jon Olav Vik, 2010-08-09</span>
<span class="k">def</span> <span class="nf">_unblock</span><span class="p">(</span><span class="n">thisnode</span><span class="p">):</span>
<span class="sd">"""Recursively unblock and remove nodes from B[thisnode]."""</span>
<span class="k">if</span> <span class="n">blocked</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]:</span>
<span class="n">blocked</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
<span class="k">while</span> <span class="n">B</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]:</span>
<span class="n">_unblock</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]</span><span class="o">.</span><span class="n">pop</span><span class="p">())</span>
<span class="k">def</span> <span class="nf">circuit</span><span class="p">(</span><span class="n">thisnode</span><span class="p">,</span> <span class="n">startnode</span><span class="p">,</span> <span class="n">component</span><span class="p">):</span>
<span class="n">closed</span> <span class="o">=</span> <span class="kc">False</span> <span class="c1"># set to True if elementary path is closed</span>
<span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">thisnode</span><span class="p">)</span>
<span class="n">blocked</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">for</span> <span class="n">nextnode</span> <span class="ow">in</span> <span class="n">component</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]:</span> <span class="c1"># direct successors of thisnode</span>
<span class="k">if</span> <span class="n">nextnode</span> <span class="o">==</span> <span class="n">startnode</span><span class="p">:</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">path</span><span class="p">[:])</span>
<span class="n">closed</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">elif</span> <span class="ow">not</span> <span class="n">blocked</span><span class="p">[</span><span class="n">nextnode</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">circuit</span><span class="p">(</span><span class="n">nextnode</span><span class="p">,</span> <span class="n">startnode</span><span class="p">,</span> <span class="n">component</span><span class="p">):</span>
<span class="n">closed</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">if</span> <span class="n">closed</span><span class="p">:</span>
<span class="n">_unblock</span><span class="p">(</span><span class="n">thisnode</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">for</span> <span class="n">nextnode</span> <span class="ow">in</span> <span class="n">component</span><span class="p">[</span><span class="n">thisnode</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">thisnode</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">B</span><span class="p">[</span><span class="n">nextnode</span><span class="p">]:</span> <span class="c1"># TODO: use set for speedup?</span>
<span class="n">B</span><span class="p">[</span><span class="n">nextnode</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">thisnode</span><span class="p">)</span>
<span class="n">path</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span> <span class="c1"># remove thisnode from path</span>
<span class="k">return</span> <span class="n">closed</span>
<span class="n">path</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># stack of nodes in current path</span>
<span class="n">blocked</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span> <span class="c1"># vertex: blocked from search?</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span> <span class="c1"># graph portions that yield no elementary circuit</span>
<span class="n">result</span> <span class="o">=</span> <span class="p">[]</span> <span class="c1"># list to accumulate the circuits found</span>
<span class="c1"># Johnson's algorithm exclude self cycle edges like (v, v)</span>
<span class="c1"># To be backward compatible, we record those cycles in advance</span>
<span class="c1"># and then remove from subG</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">G</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
<span class="n">result</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">G</span><span class="o">.</span><span class="n">remove_edge</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="c1"># Johnson's algorithm requires some ordering of the nodes.</span>
<span class="c1"># They might not be sortable so we assign an arbitrary ordering.</span>
<span class="n">ordering</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">))))</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">ordering</span><span class="p">:</span>
<span class="c1"># Build the subgraph induced by s and following nodes in the ordering</span>
<span class="n">subgraph</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">node</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="n">ordering</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">>=</span> <span class="n">ordering</span><span class="p">[</span><span class="n">s</span><span class="p">])</span>
<span class="c1"># Find the strongly connected component in the subgraph</span>
<span class="c1"># that contains the least node according to the ordering</span>
<span class="n">strongcomp</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">strongly_connected_components</span><span class="p">(</span><span class="n">subgraph</span><span class="p">)</span>
<span class="n">mincomp</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">strongcomp</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">ns</span><span class="p">:</span> <span class="nb">min</span><span class="p">(</span><span class="n">ordering</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">ns</span><span class="p">))</span>
<span class="n">component</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">mincomp</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">component</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
<span class="c1"># smallest node in the component according to the ordering</span>
<span class="n">startnode</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">component</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">ordering</span><span class="o">.</span><span class="fm">__getitem__</span><span class="p">)</span>
<span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">component</span><span class="p">:</span>
<span class="n">blocked</span><span class="p">[</span><span class="n">node</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
<span class="n">B</span><span class="p">[</span><span class="n">node</span><span class="p">][:]</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">dummy</span> <span class="o">=</span> <span class="n">circuit</span><span class="p">(</span><span class="n">startnode</span><span class="p">,</span> <span class="n">startnode</span><span class="p">,</span> <span class="n">component</span><span class="p">)</span>
<span class="k">return</span> <span class="n">result</span></div>
<div class="viewcode-block" id="find_cycle"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.cycles.find_cycle.html#networkx.algorithms.cycles.find_cycle">[docs]</a><span class="k">def</span> <span class="nf">find_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">orientation</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="sd">"""Returns a cycle found via depth-first traversal.</span>
<span class="sd"> The cycle is a list of edges indicating the cyclic path.</span>
<span class="sd"> Orientation of directed edges is controlled by `orientation`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : graph</span>
<span class="sd"> A directed/undirected graph/multigraph.</span>
<span class="sd"> source : node, list of nodes</span>
<span class="sd"> The node from which the traversal begins. If None, then a source</span>
<span class="sd"> is chosen arbitrarily and repeatedly until all edges from each node in</span>
<span class="sd"> the graph are searched.</span>
<span class="sd"> orientation : None | 'original' | 'reverse' | 'ignore' (default: None)</span>
<span class="sd"> For directed graphs and directed multigraphs, edge traversals need not</span>
<span class="sd"> respect the original orientation of the edges.</span>
<span class="sd"> When set to 'reverse' every edge is traversed in the reverse direction.</span>
<span class="sd"> When set to 'ignore', every edge is treated as undirected.</span>
<span class="sd"> When set to 'original', every edge is treated as directed.</span>
<span class="sd"> In all three cases, the yielded edge tuples add a last entry to</span>
<span class="sd"> indicate the direction in which that edge was traversed.</span>
<span class="sd"> If orientation is None, the yielded edge has no direction indicated.</span>
<span class="sd"> The direction is respected, but not reported.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> edges : directed edges</span>
<span class="sd"> A list of directed edges indicating the path taken for the loop.</span>
<span class="sd"> If no cycle is found, then an exception is raised.</span>
<span class="sd"> For graphs, an edge is of the form `(u, v)` where `u` and `v`</span>
<span class="sd"> are the tail and head of the edge as determined by the traversal.</span>
<span class="sd"> For multigraphs, an edge is of the form `(u, v, key)`, where `key` is</span>
<span class="sd"> the key of the edge. When the graph is directed, then `u` and `v`</span>
<span class="sd"> are always in the order of the actual directed edge.</span>
<span class="sd"> If orientation is not None then the edge tuple is extended to include</span>
<span class="sd"> the direction of traversal ('forward' or 'reverse') on that edge.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXNoCycle</span>
<span class="sd"> If no cycle was found.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> In this example, we construct a DAG and find, in the first call, that there</span>
<span class="sd"> are no directed cycles, and so an exception is raised. In the second call,</span>
<span class="sd"> we ignore edge orientations and find that there is an undirected cycle.</span>
<span class="sd"> Note that the second call finds a directed cycle while effectively</span>
<span class="sd"> traversing an undirected graph, and so, we found an "undirected cycle".</span>
<span class="sd"> This means that this DAG structure does not form a directed tree (which</span>
<span class="sd"> is also known as a polytree).</span>
<span class="sd"> >>> G = nx.DiGraph([(0, 1), (0, 2), (1, 2)])</span>
<span class="sd"> >>> nx.find_cycle(G, orientation="original")</span>
<span class="sd"> Traceback (most recent call last):</span>
<span class="sd"> ...</span>
<span class="sd"> networkx.exception.NetworkXNoCycle: No cycle found.</span>
<span class="sd"> >>> list(nx.find_cycle(G, orientation="ignore"))</span>
<span class="sd"> [(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> simple_cycles</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="ow">or</span> <span class="n">orientation</span> <span class="ow">in</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="s2">"original"</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">tailhead</span><span class="p">(</span><span class="n">edge</span><span class="p">):</span>
<span class="k">return</span> <span class="n">edge</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">orientation</span> <span class="o">==</span> <span class="s2">"reverse"</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">tailhead</span><span class="p">(</span><span class="n">edge</span><span class="p">):</span>
<span class="k">return</span> <span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">orientation</span> <span class="o">==</span> <span class="s2">"ignore"</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">tailhead</span><span class="p">(</span><span class="n">edge</span><span class="p">):</span>
<span class="k">if</span> <span class="n">edge</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"reverse"</span><span class="p">:</span>
<span class="k">return</span> <span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">return</span> <span class="n">edge</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
<span class="n">explored</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="n">cycle</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">final_node</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">start_node</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">nbunch_iter</span><span class="p">(</span><span class="n">source</span><span class="p">):</span>
<span class="k">if</span> <span class="n">start_node</span> <span class="ow">in</span> <span class="n">explored</span><span class="p">:</span>
<span class="c1"># No loop is possible.</span>
<span class="k">continue</span>
<span class="n">edges</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># All nodes seen in this iteration of edge_dfs</span>
<span class="n">seen</span> <span class="o">=</span> <span class="p">{</span><span class="n">start_node</span><span class="p">}</span>
<span class="c1"># Nodes in active path.</span>
<span class="n">active_nodes</span> <span class="o">=</span> <span class="p">{</span><span class="n">start_node</span><span class="p">}</span>
<span class="n">previous_head</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">edge_dfs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">start_node</span><span class="p">,</span> <span class="n">orientation</span><span class="p">):</span>
<span class="c1"># Determine if this edge is a continuation of the active path.</span>
<span class="n">tail</span><span class="p">,</span> <span class="n">head</span> <span class="o">=</span> <span class="n">tailhead</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
<span class="k">if</span> <span class="n">head</span> <span class="ow">in</span> <span class="n">explored</span><span class="p">:</span>
<span class="c1"># Then we've already explored it. No loop is possible.</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="n">previous_head</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">tail</span> <span class="o">!=</span> <span class="n">previous_head</span><span class="p">:</span>
<span class="c1"># This edge results from backtracking.</span>
<span class="c1"># Pop until we get a node whose head equals the current tail.</span>
<span class="c1"># So for example, we might have:</span>
<span class="c1"># (0, 1), (1, 2), (2, 3), (1, 4)</span>
<span class="c1"># which must become:</span>
<span class="c1"># (0, 1), (1, 4)</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">popped_edge</span> <span class="o">=</span> <span class="n">edges</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
<span class="n">edges</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">active_nodes</span> <span class="o">=</span> <span class="p">{</span><span class="n">tail</span><span class="p">}</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">popped_head</span> <span class="o">=</span> <span class="n">tailhead</span><span class="p">(</span><span class="n">popped_edge</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
<span class="n">active_nodes</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">popped_head</span><span class="p">)</span>
<span class="k">if</span> <span class="n">edges</span><span class="p">:</span>
<span class="n">last_head</span> <span class="o">=</span> <span class="n">tailhead</span><span class="p">(</span><span class="n">edges</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])[</span><span class="mi">1</span><span class="p">]</span>
<span class="k">if</span> <span class="n">tail</span> <span class="o">==</span> <span class="n">last_head</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">edges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
<span class="k">if</span> <span class="n">head</span> <span class="ow">in</span> <span class="n">active_nodes</span><span class="p">:</span>
<span class="c1"># We have a loop!</span>
<span class="n">cycle</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span>
<span class="n">final_node</span> <span class="o">=</span> <span class="n">head</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">head</span><span class="p">)</span>
<span class="n">active_nodes</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">head</span><span class="p">)</span>
<span class="n">previous_head</span> <span class="o">=</span> <span class="n">head</span>
<span class="k">if</span> <span class="n">cycle</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">explored</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">seen</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">exception</span><span class="o">.</span><span class="n">NetworkXNoCycle</span><span class="p">(</span><span class="s2">"No cycle found."</span><span class="p">)</span>
<span class="c1"># We now have a list of edges which ends on a cycle.</span>
<span class="c1"># So we need to remove from the beginning edges that are not relevant.</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">edge</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cycle</span><span class="p">):</span>
<span class="n">tail</span><span class="p">,</span> <span class="n">head</span> <span class="o">=</span> <span class="n">tailhead</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
<span class="k">if</span> <span class="n">tail</span> <span class="o">==</span> <span class="n">final_node</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">cycle</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span></div>
<div class="viewcode-block" id="minimum_cycle_basis"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.cycles.minimum_cycle_basis.html#networkx.algorithms.cycles.minimum_cycle_basis">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"directed"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">minimum_cycle_basis</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="kc">None</span><span class="p">):</span>
<span class="sd">"""Returns a minimum weight cycle basis for G</span>
<span class="sd"> Minimum weight means a cycle basis for which the total weight</span>
<span class="sd"> (length for unweighted graphs) of all the cycles is minimum.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX Graph</span>
<span class="sd"> weight: string</span>
<span class="sd"> name of the edge attribute to use for edge weights</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> A list of cycle lists. Each cycle list is a list of nodes</span>
<span class="sd"> which forms a cycle (loop) in G. Note that the nodes are not</span>
<span class="sd"> necessarily returned in a order by which they appear in the cycle</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.Graph()</span>
<span class="sd"> >>> nx.add_cycle(G, [0, 1, 2, 3])</span>
<span class="sd"> >>> nx.add_cycle(G, [0, 3, 4, 5])</span>
<span class="sd"> >>> print([sorted(c) for c in nx.minimum_cycle_basis(G)])</span>
<span class="sd"> [[0, 1, 2, 3], [0, 3, 4, 5]]</span>
<span class="sd"> References:</span>
<span class="sd"> [1] Kavitha, Telikepalli, et al. "An O(m^2n) Algorithm for</span>
<span class="sd"> Minimum Cycle Basis of Graphs."</span>
<span class="sd"> http://link.springer.com/article/10.1007/s00453-007-9064-z</span>
<span class="sd"> [2] de Pina, J. 1995. Applications of shortest path methods.</span>
<span class="sd"> Ph.D. thesis, University of Amsterdam, Netherlands</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> simple_cycles, cycle_basis</span>
<span class="sd"> """</span>
<span class="c1"># We first split the graph in commected subgraphs</span>
<span class="k">return</span> <span class="nb">sum</span><span class="p">(</span>
<span class="p">(</span><span class="n">_min_cycle_basis</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="n">c</span><span class="p">),</span> <span class="n">weight</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">connected_components</span><span class="p">(</span><span class="n">G</span><span class="p">)),</span>
<span class="p">[],</span>
<span class="p">)</span></div>
<span class="k">def</span> <span class="nf">_min_cycle_basis</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">weight</span><span class="p">):</span>
<span class="n">cb</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># We extract the edges not in a spanning tree. We do not really need a</span>
<span class="c1"># *minimum* spanning tree. That is why we call the next function with</span>
<span class="c1"># weight=None. Depending on implementation, it may be faster as well</span>
<span class="n">spanning_tree_edges</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">minimum_spanning_edges</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">edges_excl</span> <span class="o">=</span> <span class="p">[</span><span class="nb">frozenset</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">comp</span><span class="o">.</span><span class="n">edges</span><span class="p">()</span> <span class="k">if</span> <span class="n">e</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">spanning_tree_edges</span><span class="p">]</span>
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">edges_excl</span><span class="p">)</span>
<span class="c1"># We maintain a set of vectors orthogonal to sofar found cycles</span>
<span class="n">set_orth</span> <span class="o">=</span> <span class="p">[{</span><span class="n">edge</span><span class="p">}</span> <span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">edges_excl</span><span class="p">]</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="c1"># kth cycle is "parallel" to kth vector in set_orth</span>
<span class="n">new_cycle</span> <span class="o">=</span> <span class="n">_min_cycle</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">set_orth</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="n">cb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">()</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="o">*</span><span class="n">new_cycle</span><span class="p">)))</span>
<span class="c1"># now update set_orth so that k+1,k+2... th elements are</span>
<span class="c1"># orthogonal to the newly found cycle, as per [p. 336, 1]</span>
<span class="n">base</span> <span class="o">=</span> <span class="n">set_orth</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
<span class="n">set_orth</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">:]</span> <span class="o">=</span> <span class="p">[</span>
<span class="n">orth</span> <span class="o">^</span> <span class="n">base</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">orth</span> <span class="o">&</span> <span class="n">new_cycle</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="k">else</span> <span class="n">orth</span>
<span class="k">for</span> <span class="n">orth</span> <span class="ow">in</span> <span class="n">set_orth</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">:]</span>
<span class="p">]</span>
<span class="k">return</span> <span class="n">cb</span>
<span class="k">def</span> <span class="nf">_min_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">orth</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Computes the minimum weight cycle in G,</span>
<span class="sd"> orthogonal to the vector orth as per [p. 338, 1]</span>
<span class="sd"> """</span>
<span class="n">T</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">Graph</span><span class="p">()</span>
<span class="n">nodes_idx</span> <span class="o">=</span> <span class="p">{</span><span class="n">node</span><span class="p">:</span> <span class="n">idx</span> <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">node</span> <span class="ow">in</span> <span class="nb">enumerate</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">idx_nodes</span> <span class="o">=</span> <span class="p">{</span><span class="n">idx</span><span class="p">:</span> <span class="n">node</span> <span class="k">for</span> <span class="n">node</span><span class="p">,</span> <span class="n">idx</span> <span class="ow">in</span> <span class="n">nodes_idx</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span>
<span class="n">nnodes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodes_idx</span><span class="p">)</span>
<span class="c1"># Add 2 copies of each edge in G to T. If edge is in orth, add cross edge;</span>
<span class="c1"># otherwise in-plane edge</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</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">edges</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="n">uidx</span><span class="p">,</span> <span class="n">vidx</span> <span class="o">=</span> <span class="n">nodes_idx</span><span class="p">[</span><span class="n">u</span><span class="p">],</span> <span class="n">nodes_idx</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="n">edge_w</span> <span class="o">=</span> <span class="n">data</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="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">frozenset</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">orth</span><span class="p">:</span>
<span class="n">T</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span>
<span class="p">[(</span><span class="n">uidx</span><span class="p">,</span> <span class="n">nnodes</span> <span class="o">+</span> <span class="n">vidx</span><span class="p">),</span> <span class="p">(</span><span class="n">nnodes</span> <span class="o">+</span> <span class="n">uidx</span><span class="p">,</span> <span class="n">vidx</span><span class="p">)],</span> <span class="n">weight</span><span class="o">=</span><span class="n">edge_w</span>
<span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">T</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span>
<span class="p">[(</span><span class="n">uidx</span><span class="p">,</span> <span class="n">vidx</span><span class="p">),</span> <span class="p">(</span><span class="n">nnodes</span> <span class="o">+</span> <span class="n">uidx</span><span class="p">,</span> <span class="n">nnodes</span> <span class="o">+</span> <span class="n">vidx</span><span class="p">)],</span> <span class="n">weight</span><span class="o">=</span><span class="n">edge_w</span>
<span class="p">)</span>
<span class="n">all_shortest_pathlens</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">shortest_path_length</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span>
<span class="n">cross_paths_w_lens</span> <span class="o">=</span> <span class="p">{</span>
<span class="n">n</span><span class="p">:</span> <span class="n">all_shortest_pathlens</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">nnodes</span> <span class="o">+</span> <span class="n">n</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nnodes</span><span class="p">)</span>
<span class="p">}</span>
<span class="c1"># Now compute shortest paths in T, which translates to cyles in G</span>
<span class="n">start</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cross_paths_w_lens</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">cross_paths_w_lens</span><span class="o">.</span><span class="n">get</span><span class="p">)</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">nnodes</span> <span class="o">+</span> <span class="n">start</span>
<span class="n">min_path</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">shortest_path</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">source</span><span class="o">=</span><span class="n">start</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">end</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="c1"># Now we obtain the actual path, re-map nodes in T to those in G</span>
<span class="n">min_path_nodes</span> <span class="o">=</span> <span class="p">[</span><span class="n">node</span> <span class="k">if</span> <span class="n">node</span> <span class="o"><</span> <span class="n">nnodes</span> <span class="k">else</span> <span class="n">node</span> <span class="o">-</span> <span class="n">nnodes</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">min_path</span><span class="p">]</span>
<span class="c1"># Now remove the edges that occur two times</span>
<span class="n">mcycle_pruned</span> <span class="o">=</span> <span class="n">_path_to_cycle</span><span class="p">(</span><span class="n">min_path_nodes</span><span class="p">)</span>
<span class="k">return</span> <span class="p">{</span><span class="nb">frozenset</span><span class="p">((</span><span class="n">idx_nodes</span><span class="p">[</span><span class="n">u</span><span class="p">],</span> <span class="n">idx_nodes</span><span class="p">[</span><span class="n">v</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">mcycle_pruned</span><span class="p">}</span>
<span class="k">def</span> <span class="nf">_path_to_cycle</span><span class="p">(</span><span class="n">path</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Removes the edges from path that occur even number of times.</span>
<span class="sd"> Returns a set of edges</span>
<span class="sd"> """</span>
<span class="n">edges</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="k">for</span> <span class="n">edge</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="c1"># Toggle whether to keep the current edge.</span>
<span class="n">edges</span> <span class="o">^=</span> <span class="p">{</span><span class="n">edge</span><span class="p">}</span>
<span class="k">return</span> <span class="n">edges</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 2004-2022, 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>
|
