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
|
# -*- coding: utf-8 -*-
"""
Strongly connected components.
"""
__authors__ = "\n".join(['Eben Kenah',
'Aric Hagberg (hagberg@lanl.gov)'
'Christopher Ellison'])
# Copyright (C) 2004-2010 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__all__ = ['number_strongly_connected_components',
'strongly_connected_components',
'strongly_connected_component_subgraphs',
'is_strongly_connected',
'strongly_connected_components_recursive',
'kosaraju_strongly_connected_components',
'condensation',
]
import networkx as nx
def strongly_connected_components(G):
"""Return nodes in strongly connected components of graph.
Parameters
----------
G : NetworkX Graph
An directed graph.
Returns
-------
comp : list of lists
A list of nodes for each component of G.
The list is ordered from largest connected component to smallest.
See Also
--------
connected_components
Notes
-----
Uses Tarjan's algorithm with Nuutila's modifications.
Nonrecursive version of algorithm.
References
----------
.. [1] Depth-first search and linear graph algorithms, R. Tarjan
SIAM Journal of Computing 1(2):146-160, (1972).
.. [2] On finding the strongly connected components in a directed graph.
E. Nuutila and E. Soisalon-Soinen
Information Processing Letters 49(1): 9-14, (1994)..
"""
preorder={}
lowlink={}
scc_found={}
scc_queue = []
scc_list=[]
i=0 # Preorder counter
for source in G:
if source not in scc_found:
queue=[source]
while queue:
v=queue[-1]
if v not in preorder:
i=i+1
preorder[v]=i
done=1
v_nbrs=G[v]
for w in v_nbrs:
if w not in preorder:
queue.append(w)
done=0
break
if done==1:
lowlink[v]=preorder[v]
for w in v_nbrs:
if w not in scc_found:
if preorder[w]>preorder[v]:
lowlink[v]=min([lowlink[v],lowlink[w]])
else:
lowlink[v]=min([lowlink[v],preorder[w]])
queue.pop()
if lowlink[v]==preorder[v]:
scc_found[v]=True
scc=[v]
while scc_queue and preorder[scc_queue[-1]]>preorder[v]:
k=scc_queue.pop()
scc_found[k]=True
scc.append(k)
scc_list.append(scc)
else:
scc_queue.append(v)
scc_list.sort(key=len,reverse=True)
return scc_list
def kosaraju_strongly_connected_components(G,source=None):
"""Return nodes in strongly connected components of graph.
Parameters
----------
G : NetworkX Graph
An directed graph.
Returns
-------
comp : list of lists
A list of nodes for each component of G.
The list is ordered from largest connected component to smallest.
See Also
--------
connected_components
Notes
-----
Uses Kosaraju's algorithm.
"""
components=[]
G=G.reverse(copy=False)
post=list(nx.dfs_postorder_nodes(G,source=source))
G=G.reverse(copy=False)
seen={}
while post:
r=post.pop()
if r in seen:
continue
c=nx.dfs_preorder_nodes(G,r)
new=[v for v in c if v not in seen]
seen.update([(u,True) for u in new])
components.append(new)
components.sort(key=len,reverse=True)
return components
def strongly_connected_components_recursive(G):
"""Return nodes in strongly connected components of graph.
Recursive version of algorithm.
Parameters
----------
G : NetworkX Graph
An directed graph.
Returns
-------
comp : list of lists
A list of nodes for each component of G.
The list is ordered from largest connected component to smallest.
See Also
--------
connected_components
Notes
-----
Uses Tarjan's algorithm with Nuutila's modifications.
References
----------
.. [1] Depth-first search and linear graph algorithms, R. Tarjan
SIAM Journal of Computing 1(2):146-160, (1972).
.. [2] On finding the strongly connected components in a directed graph.
E. Nuutila and E. Soisalon-Soinen
Information Processing Letters 49(1): 9-14, (1994)..
"""
def visit(v,cnt):
root[v]=cnt
visited[v]=cnt
cnt+=1
stack.append(v)
for w in G[v]:
if w not in visited: visit(w,cnt)
if w not in component:
root[v]=min(root[v],root[w])
if root[v]==visited[v]:
component[v]=root[v]
tmpc=[v] # hold nodes in this component
while stack[-1]!=v:
w=stack.pop()
component[w]=root[v]
tmpc.append(w)
stack.remove(v)
scc.append(tmpc) # add to scc list
scc=[]
visited={}
component={}
root={}
cnt=0
stack=[]
for source in G:
if source not in visited:
visit(source,cnt)
scc.sort(key=len,reverse=True)
return scc
def strongly_connected_component_subgraphs(G):
"""Return strongly connected components as subgraphs.
Parameters
----------
G : NetworkX Graph
A graph.
Returns
-------
glist : list
A list of graphs, one for each strongly connected component of G.
See Also
--------
connected_component_subgraphs
Notes
-----
The list is ordered from largest strongly connected component to smallest.
"""
cc=strongly_connected_components(G)
graph_list=[]
for c in cc:
graph_list.append(G.subgraph(c))
return graph_list
def number_strongly_connected_components(G):
"""Return number of strongly connected components in graph.
Parameters
----------
G : NetworkX graph
A directed graph.
Returns
-------
n : integer
Number of strongly connected components
See Also
--------
connected_components
Notes
-----
For directed graphs only.
"""
return len(strongly_connected_components(G))
def is_strongly_connected(G):
"""Test directed graph for strong connectivity.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
connected : bool
True if the graph is strongly connected, False otherwise.
See Also
--------
strongly_connected_components
Notes
-----
For directed graphs only.
"""
if not G.is_directed():
raise nx.NetworkXError("""Not allowed for undirected graph G.
See is_connected() for connectivity test.""")
if len(G)==0:
raise nx.NetworkXPointlessConcept(
"""Connectivity is undefined for the null graph.""")
return len(strongly_connected_components(G)[0])==len(G)
def condensation(G):
"""Returns the condensation of G.
The condensation of G is the graph with each of the strongly connected
components contracted into a single node.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
cG : NetworkX DiGraph
The condensation of G.
Notes
-----
After contracting all strongly connected components to a single node,
the resulting graph is a directed acyclic graph.
"""
scc = strongly_connected_components(G)
mapping = dict([(n,tuple(sorted(c))) for c in scc for n in c])
cG = nx.DiGraph()
for u in mapping:
cG.add_node(mapping[u])
for _,v,d in G.edges_iter(u, data=True):
if v not in mapping[u]:
cG.add_edge(mapping[u], mapping[v])
return cG
|