"""
========================
Cycle finding algorithms
========================
"""
from collections import defaultdict
import networkx as nx
from networkx.utils import not_implemented_for, pairwise
__all__ = [
"cycle_basis",
"simple_cycles",
"recursive_simple_cycles",
"find_cycle",
"minimum_cycle_basis",
]
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
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()
>>> nx.add_cycle(G, [0, 1, 2, 3])
>>> nx.add_cycle(G, [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] = {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
[docs]@not_implemented_for("undirected")
def 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. 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 represented by a list of nodes along the cycle.
Examples
--------
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> 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.
https://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 = {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.
# Also we save the actual graph so we can mutate it. We only take the
# edges because we do not want to copy edge and node attributes here.
subG = type(G)(G.edges())
sccs = [scc for scc in nx.strongly_connected_components(subG) if len(scc) > 1]
# Johnson's algorithm exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in subG:
if subG.has_edge(v, v):
yield [v]
subG.remove_edge(v, v)
while sccs:
scc = sccs.pop()
sccG = subG.subgraph(scc)
# 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(sccG[startnode]))] # sccG gives comp nbrs
while stack:
thisnode, nbrs = stack[-1]
if nbrs:
nextnode = nbrs.pop()
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(sccG[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 sccG[thisnode]:
if thisnode not in B[nbr]:
B[nbr].add(thisnode)
stack.pop()
# assert path[-1] == thisnode
path.pop()
# done processing this node
H = subG.subgraph(scc) # make smaller to avoid work in SCC routine
sccs.extend(scc for scc in nx.strongly_connected_components(H) if len(scc) > 1)
@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. 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 called simple_cycles().
Warning: This recursive version uses lots of RAM!
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of cycles, where each cycle is represented by a list of nodes
along the cycle.
Example:
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> nx.recursive_simple_cycles(G)
[[0], [2], [0, 1, 2], [0, 2], [1, 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.
https://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 exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in G:
if G.has_edge(v, v):
result.append([v])
G.remove_edge(v, v)
# 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 ns: min(ordering[n] for n in ns))
component = G.subgraph(mincomp)
if len(component) > 1:
# 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=None):
"""Returns a cycle found via depth-first traversal.
The cycle is a list of edges indicating the cyclic path.
Orientation of directed edges is controlled by `orientation`.
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 : None | 'original' | 'reverse' | 'ignore' (default: None)
For directed graphs and directed multigraphs, edge traversals need not
respect the original orientation of the edges.
When set to 'reverse' every edge is traversed in the reverse direction.
When set to 'ignore', every edge is treated as undirected.
When set to 'original', every edge is treated as directed.
In all three cases, the yielded edge tuples add a last entry to
indicate the direction in which that edge was traversed.
If orientation is None, the yielded edge has no direction indicated.
The direction is respected, but not reported.
Returns
-------
edges : directed edges
A list of directed edges indicating the path taken for the loop.
If no cycle is found, then an exception is raised.
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 not None then the edge tuple is extended to include
the direction of traversal ('forward' or 'reverse') on that edge.
Raises
------
NetworkXNoCycle
If no cycle was found.
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).
>>> G = nx.DiGraph([(0, 1), (0, 2), (1, 2)])
>>> try:
... nx.find_cycle(G, orientation="original")
... except:
... pass
...
>>> list(nx.find_cycle(G, orientation="ignore"))
[(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]
See Also
--------
simple_cycles
"""
if not G.is_directed() or orientation in (None, "original"):
def tailhead(edge):
return edge[:2]
elif orientation == "reverse":
def tailhead(edge):
return edge[1], edge[0]
elif orientation == "ignore":
def tailhead(edge):
if edge[-1] == "reverse":
return edge[1], edge[0]
return edge[:2]
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_head = None
for edge in nx.edge_dfs(G, start_node, orientation):
# Determine if this edge is a continuation of the active path.
tail, head = tailhead(edge)
if head in explored:
# Then we've already explored it. No loop is possible.
continue
if previous_head is not None and tail != previous_head:
# 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
else:
seen.add(head)
active_nodes.add(head)
previous_head = 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:]
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def minimum_cycle_basis(G, weight=None):
""" Returns a minimum weight cycle basis for G
Minimum weight means a cycle basis for which the total weight
(length for unweighted graphs) of all the cycles is minimum.
Parameters
----------
G : NetworkX Graph
weight: string
name of the edge attribute to use for edge weights
Returns
-------
A list of cycle lists. Each cycle list is a list of nodes
which forms a cycle (loop) in G. Note that the nodes are not
necessarily returned in a order by which they appear in the cycle
Examples
--------
>>> G = nx.Graph()
>>> nx.add_cycle(G, [0, 1, 2, 3])
>>> nx.add_cycle(G, [0, 3, 4, 5])
>>> print([sorted(c) for c in nx.minimum_cycle_basis(G)])
[[0, 1, 2, 3], [0, 3, 4, 5]]
References:
[1] Kavitha, Telikepalli, et al. "An O(m^2n) Algorithm for
Minimum Cycle Basis of Graphs."
http://link.springer.com/article/10.1007/s00453-007-9064-z
[2] de Pina, J. 1995. Applications of shortest path methods.
Ph.D. thesis, University of Amsterdam, Netherlands
See Also
--------
simple_cycles, cycle_basis
"""
# We first split the graph in commected subgraphs
return sum(
(_min_cycle_basis(G.subgraph(c), weight) for c in nx.connected_components(G)),
[],
)
def _min_cycle_basis(comp, weight):
cb = []
# We extract the edges not in a spanning tree. We do not really need a
# *minimum* spanning tree. That is why we call the next function with
# weight=None. Depending on implementation, it may be faster as well
spanning_tree_edges = list(nx.minimum_spanning_edges(comp, weight=None, data=False))
edges_excl = [frozenset(e) for e in comp.edges() if e not in spanning_tree_edges]
N = len(edges_excl)
# We maintain a set of vectors orthogonal to sofar found cycles
set_orth = [{edge} for edge in edges_excl]
for k in range(N):
# kth cycle is "parallel" to kth vector in set_orth
new_cycle = _min_cycle(comp, set_orth[k], weight=weight)
cb.append(list(set().union(*new_cycle)))
# now update set_orth so that k+1,k+2... th elements are
# orthogonal to the newly found cycle, as per [p. 336, 1]
base = set_orth[k]
set_orth[k + 1 :] = [
orth ^ base if len(orth & new_cycle) % 2 else orth
for orth in set_orth[k + 1 :]
]
return cb
def _min_cycle(G, orth, weight=None):
"""
Computes the minimum weight cycle in G,
orthogonal to the vector orth as per [p. 338, 1]
"""
T = nx.Graph()
nodes_idx = {node: idx for idx, node in enumerate(G.nodes())}
idx_nodes = {idx: node for node, idx in nodes_idx.items()}
nnodes = len(nodes_idx)
# Add 2 copies of each edge in G to T. If edge is in orth, add cross edge;
# otherwise in-plane edge
for u, v, data in G.edges(data=True):
uidx, vidx = nodes_idx[u], nodes_idx[v]
edge_w = data.get(weight, 1)
if frozenset((u, v)) in orth:
T.add_edges_from(
[(uidx, nnodes + vidx), (nnodes + uidx, vidx)], weight=edge_w
)
else:
T.add_edges_from(
[(uidx, vidx), (nnodes + uidx, nnodes + vidx)], weight=edge_w
)
all_shortest_pathlens = dict(nx.shortest_path_length(T, weight=weight))
cross_paths_w_lens = {
n: all_shortest_pathlens[n][nnodes + n] for n in range(nnodes)
}
# Now compute shortest paths in T, which translates to cyles in G
start = min(cross_paths_w_lens, key=cross_paths_w_lens.get)
end = nnodes + start
min_path = nx.shortest_path(T, source=start, target=end, weight="weight")
# Now we obtain the actual path, re-map nodes in T to those in G
min_path_nodes = [node if node < nnodes else node - nnodes for node in min_path]
# Now remove the edges that occur two times
mcycle_pruned = _path_to_cycle(min_path_nodes)
return {frozenset((idx_nodes[u], idx_nodes[v])) for u, v in mcycle_pruned}
def _path_to_cycle(path):
"""
Removes the edges from path that occur even number of times.
Returns a set of edges
"""
edges = set()
for edge in pairwise(path):
# Toggle whether to keep the current edge.
edges ^= {edge}
return edges