Warning
This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.
Source code for networkx.algorithms.cycles
"""
========================
Cycle finding algorithms
========================
"""
# Copyright (C) 2010-2012 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
from collections import defaultdict
import networkx as nx
from networkx.utils import *
from networkx.algorithms.traversal.edgedfs import helper_funcs, edge_dfs
__all__ = [
'cycle_basis','simple_cycles','recursive_simple_cycles', 'find_cycle'
]
__author__ = "\n".join(['Jon Olav Vik <jonovik@gmail.com>',
'Dan Schult <dschult@colgate.edu>',
'Aric Hagberg <hagberg@lanl.gov>'])
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def cycle_basis(G,root=None):
""" Returns a list of cycles which form a basis for cycles of G.
A basis for cycles of a network is a minimal collection of
cycles such that any cycle in the network can be written
as a sum of cycles in the basis. Here summation of cycles
is defined as "exclusive or" of the edges. Cycle bases are
useful, e.g. when deriving equations for electric circuits
using Kirchhoff's Laws.
Parameters
----------
G : NetworkX Graph
root : node, optional
Specify starting node for basis.
Returns
-------
A list of cycle lists. Each cycle list is a list of nodes
which forms a cycle (loop) in G.
Examples
--------
>>> G=nx.Graph()
>>> G.add_cycle([0,1,2,3])
>>> G.add_cycle([0,3,4,5])
>>> print(nx.cycle_basis(G,0))
[[3, 4, 5, 0], [1, 2, 3, 0]]
Notes
-----
This is adapted from algorithm CACM 491 [1]_.
References
----------
.. [1] Paton, K. An algorithm for finding a fundamental set of
cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.
See Also
--------
simple_cycles
"""
gnodes=set(G.nodes())
cycles=[]
while gnodes: # loop over connected components
if root is None:
root=gnodes.pop()
stack=[root]
pred={root:root}
used={root:set()}
while stack: # walk the spanning tree finding cycles
z=stack.pop() # use last-in so cycles easier to find
zused=used[z]
for nbr in G[z]:
if nbr not in used: # new node
pred[nbr]=z
stack.append(nbr)
used[nbr]=set([z])
elif nbr == z: # self loops
cycles.append([z])
elif nbr not in zused:# found a cycle
pn=used[nbr]
cycle=[nbr,z]
p=pred[z]
while p not in pn:
cycle.append(p)
p=pred[p]
cycle.append(p)
cycles.append(cycle)
used[nbr].add(z)
gnodes-=set(pred)
root=None
return cycles
@not_implemented_for('undirected')
[docs]def simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
An simple cycle, or elementary circuit, is a closed path where no
node appears twice, except that the first and last node are the same.
Two elementary circuits are distinct if they are not cyclic permutations
of each other.
This is a nonrecursive, iterator/generator version of Johnson's
algorithm [1]_. There may be better algorithms for some cases [2]_ [3]_.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
cycle_generator: generator
A generator that produces elementary cycles of the graph. Each cycle is
a list of nodes with the first and last nodes being the same.
Examples
--------
>>> G = nx.DiGraph([(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)])
>>> len(list(nx.simple_cycles(G)))
5
To filter the cycles so that they don't include certain nodes or edges,
copy your graph and eliminate those nodes or edges before calling
>>> copyG = G.copy()
>>> copyG.remove_nodes_from([1])
>>> copyG.remove_edges_from([(0, 1)])
>>> len(list(nx.simple_cycles(copyG)))
3
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is `O((n+e)(c+1))` for `n` nodes, `e` edges and `c`
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
http://dx.doi.org/10.1137/0204007
.. [2] Enumerating the cycles of a digraph: a new preprocessing strategy.
G. Loizou and P. Thanish, Information Sciences, v. 27, 163-182, 1982.
.. [3] A search strategy for the elementary cycles of a directed graph.
J.L. Szwarcfiter and P.E. Lauer, BIT NUMERICAL MATHEMATICS,
v. 16, no. 2, 192-204, 1976.
See Also
--------
cycle_basis
"""
def _unblock(thisnode,blocked,B):
stack=set([thisnode])
while stack:
node=stack.pop()
if node in blocked:
blocked.remove(node)
stack.update(B[node])
B[node].clear()
# Johnson's algorithm requires some ordering of the nodes.
# We assign the arbitrary ordering given by the strongly connected comps
# There is no need to track the ordering as each node removed as processed.
subG = type(G)(G.edges_iter()) # save the actual graph so we can mutate it here
# We only take the edges because we do not want to
# copy edge and node attributes here.
sccs = list(nx.strongly_connected_components(subG))
while sccs:
scc=sccs.pop()
# order of scc determines ordering of nodes
startnode = scc.pop()
# Processing node runs "circuit" routine from recursive version
path=[startnode]
blocked = set() # vertex: blocked from search?
closed = set() # nodes involved in a cycle
blocked.add(startnode)
B=defaultdict(set) # graph portions that yield no elementary circuit
stack=[ (startnode,list(subG[startnode])) ] # subG gives component nbrs
while stack:
thisnode,nbrs = stack[-1]
if nbrs:
nextnode = nbrs.pop()
# print thisnode,nbrs,":",nextnode,blocked,B,path,stack,startnode
# f=raw_input("pause")
if nextnode == startnode:
yield path[:]
closed.update(path)
# print "Found a cycle",path,closed
elif nextnode not in blocked:
path.append(nextnode)
stack.append( (nextnode,list(subG[nextnode])) )
closed.discard(nextnode)
blocked.add(nextnode)
continue
# done with nextnode... look for more neighbors
if not nbrs: # no more nbrs
if thisnode in closed:
_unblock(thisnode,blocked,B)
else:
for nbr in subG[thisnode]:
if thisnode not in B[nbr]:
B[nbr].add(thisnode)
stack.pop()
# assert path[-1]==thisnode
path.pop()
# done processing this node
subG.remove_node(startnode)
H=subG.subgraph(scc) # make smaller to avoid work in SCC routine
sccs.extend(list(nx.strongly_connected_components(H)))
@not_implemented_for('undirected')
def recursive_simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
A simple cycle, or elementary circuit, is a closed path where no
node appears twice, except that the first and last node are the same.
Two elementary circuits are distinct if they are not cyclic permutations
of each other.
This version uses a recursive algorithm to build a list of cycles.
You should probably use the iterator version caled simple_cycles().
Warning: This recursive version uses lots of RAM!
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of circuits, where each circuit is a list of nodes, with the first
and last node being the same.
Example:
>>> G = nx.DiGraph([(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)])
>>> nx.recursive_simple_cycles(G)
[[0], [0, 1, 2], [0, 2], [1, 2], [2]]
See Also
--------
cycle_basis (for undirected graphs)
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is `O((n+e)(c+1))` for `n` nodes, `e` edges and `c`
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
http://dx.doi.org/10.1137/0204007
See Also
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""
if blocked[thisnode]:
blocked[thisnode] = False
while B[thisnode]:
_unblock(B[thisnode].pop())
def circuit(thisnode, startnode, component):
closed = False # set to True if elementary path is closed
path.append(thisnode)
blocked[thisnode] = True
for nextnode in component[thisnode]: # direct successors of thisnode
if nextnode == startnode:
result.append(path[:])
closed = True
elif not blocked[nextnode]:
if circuit(nextnode, startnode, component):
closed = True
if closed:
_unblock(thisnode)
else:
for nextnode in component[thisnode]:
if thisnode not in B[nextnode]: # TODO: use set for speedup?
B[nextnode].append(thisnode)
path.pop() # remove thisnode from path
return closed
path = [] # stack of nodes in current path
blocked = defaultdict(bool) # vertex: blocked from search?
B = defaultdict(list) # graph portions that yield no elementary circuit
result = [] # list to accumulate the circuits found
# Johnson's algorithm requires some ordering of the nodes.
# They might not be sortable so we assign an arbitrary ordering.
ordering=dict(zip(G,range(len(G))))
for s in ordering:
# Build the subgraph induced by s and following nodes in the ordering
subgraph = G.subgraph(node for node in G
if ordering[node] >= ordering[s])
# Find the strongly connected component in the subgraph
# that contains the least node according to the ordering
strongcomp = nx.strongly_connected_components(subgraph)
mincomp=min(strongcomp,
key=lambda nodes: min(ordering[n] for n in nodes))
component = G.subgraph(mincomp)
if component:
# smallest node in the component according to the ordering
startnode = min(component,key=ordering.__getitem__)
for node in component:
blocked[node] = False
B[node][:] = []
dummy=circuit(startnode, startnode, component)
return result
[docs]def find_cycle(G, source=None, orientation='original'):
"""
Returns the edges of a cycle found via a directed, depth-first traversal.
Parameters
----------
G : graph
A directed/undirected graph/multigraph.
source : node, list of nodes
The node from which the traversal begins. If ``None``, then a source
is chosen arbitrarily and repeatedly until all edges from each node in
the graph are searched.
orientation : 'original' | 'reverse' | 'ignore'
For directed graphs and directed multigraphs, edge traversals need not
respect the original orientation of the edges. When set to 'reverse',
then every edge will be traversed in the reverse direction. When set to
'ignore', then each directed edge is treated as a single undirected
edge that can be traversed in either direction. For undirected graphs
and undirected multigraphs, this parameter is meaningless and is not
consulted by the algorithm.
Returns
-------
edges : directed edges
A list of directed edges indicating the path taken for the loop. If
no cycle is found, then ``edges`` will be an empty list. For graphs, an
edge is of the form (u, v) where ``u`` and ``v`` are the tail and head
of the edge as determined by the traversal. For multigraphs, an edge is
of the form (u, v, key), where ``key`` is the key of the edge. When the
graph is directed, then ``u`` and ``v`` are always in the order of the
actual directed edge. If orientation is 'ignore', then an edge takes
the form (u, v, key, direction) where direction indicates if the edge
was followed in the forward (tail to head) or reverse (head to tail)
direction. When the direction is forward, the value of ``direction``
is 'forward'. When the direction is reverse, the value of ``direction``
is 'reverse'.
Examples
--------
In this example, we construct a DAG and find, in the first call, that there
are no directed cycles, and so an exception is raised. In the second call,
we ignore edge orientations and find that there is an undirected cycle.
Note that the second call finds a directed cycle while effectively
traversing an undirected graph, and so, we found an "undirected cycle".
This means that this DAG structure does not form a directed tree (which
is also known as a polytree).
>>> import networkx as nx
>>> G = nx.DiGraph([(0,1), (0,2), (1,2)])
>>> try:
... find_cycle(G, orientation='original')
... except:
... pass
...
>>> list(find_cycle(G, orientation='ignore'))
[(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]
"""
out_edge, key, tailhead = helper_funcs(G, orientation)
explored = set()
cycle = []
final_node = None
for start_node in G.nbunch_iter(source):
if start_node in explored:
# No loop is possible.
continue
edges = []
# All nodes seen in this iteration of edge_dfs
seen = {start_node}
# Nodes in active path.
active_nodes = {start_node}
previous_node = None
for edge in edge_dfs(G, start_node, orientation):
# Determine if this edge is a continuation of the active path.
tail, head = tailhead(edge)
if previous_node is not None and tail != previous_node:
# This edge results from backtracking.
# Pop until we get a node whose head equals the current tail.
# So for example, we might have:
# (0,1), (1,2), (2,3), (1,4)
# which must become:
# (0,1), (1,4)
while True:
try:
popped_edge = edges.pop()
except IndexError:
edges = []
active_nodes = {tail}
break
else:
popped_head = tailhead(popped_edge)[1]
active_nodes.remove(popped_head)
if edges:
last_head = tailhead(edges[-1])[1]
if tail == last_head:
break
edges.append(edge)
if head in active_nodes:
# We have a loop!
cycle.extend(edges)
final_node = head
break
elif head in explored:
# Then we've already explored it. No loop is possible.
break
else:
seen.add(head)
active_nodes.add(head)
previous_node = head
if cycle:
break
else:
explored.update(seen)
else:
assert(len(cycle) == 0)
raise nx.exception.NetworkXNoCycle('No cycle found.')
# We now have a list of edges which ends on a cycle.
# So we need to remove from the beginning edges that are not relevant.
for i, edge in enumerate(cycle):
tail, head = tailhead(edge)
if tail == final_node:
break
return cycle[i:]