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
========================
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
import networkx as nx
from networkx.utils import *
from collections import defaultdict

__all__ = ['cycle_basis','simple_cycles','recursive_simple_cycles']
__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()
>>> 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.

--------
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)
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)])
>>> list(nx.simple_cycles(G))
[[2], [2, 1], [2, 0], [2, 0, 1], [0]]

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.

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)])
>>> list(nx.simple_cycles(copyG))
[[2], [2, 0], [0]]

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.

--------
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
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])) )
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]:
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]]

--------
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

--------
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