"""
========================
Cycle finding algorithms
========================
"""
# Copyright (C) 2010 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
import networkx as nx
from networkx.utils import *
from collections import defaultdict
__all__ = ['cycle_basis','simple_cycles']
__author__ = "\n".join(['Jon Olav Vik <jonovik@gmail.com>',
'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
"""
# if G.is_directed():
# e='cycle_basis() not implemented for directed graphs'
# raise Exception(e)
# if G.is_multigraph():
# e='cycle_basis() not implemented for multigraphs'
# raise Exception(e)
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 is 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.
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.simple_cycles(G)
[[0, 0], [0, 1, 2, 0], [0, 2, 0], [1, 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
--------
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 + [startnode])
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
# if not G.is_directed():
# raise nx.NetworkXError(\
# "simple_cycles() not implemented for undirected graphs.")
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