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ton
2023-04-06 13:53:47 +07:00
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.CondaPkg/env/Lib/site-packages/networkx/algorithms/cycles.py vendored
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@@ -5,6 +5,7 @@ Cycle finding algorithms
"""
from collections import defaultdict
from itertools import combinations, product
import networkx as nx
from networkx.utils import not_implemented_for, pairwise
@@ -15,6 +16,7 @@ __all__ = [
"recursive_simple_cycles",
"find_cycle",
"minimum_cycle_basis",
"chordless_cycles",
]
@@ -95,22 +97,44 @@ def cycle_basis(G, root=None):
return cycles
@not_implemented_for("undirected")
def simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
def simple_cycles(G, length_bound=None):
"""Find simple cycles (elementary circuits) of a 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.
no node appears twice. In a directed graph, two simple cycles are distinct
if they are not cyclic permutations of each other. In an undirected graph,
two simple cycles are distinct if they are not cyclic permutations of each
other nor of the other's reversal.
This is a nonrecursive, iterator/generator version of Johnson's
algorithm [1]_. There may be better algorithms for some cases [2]_ [3]_.
Optionally, the cycles are bounded in length. In the unbounded case, we use
a nonrecursive, iterator/generator version of Johnson's algorithm [1]_. In
the bounded case, we use a version of the algorithm of Gupta and
Suzumura[2]_. There may be better algorithms for some cases [3]_ [4]_ [5]_.
The algorithms of Johnson, and Gupta and Suzumura, are enhanced by some
well-known preprocessing techniques. When G is directed, we restrict our
attention to strongly connected components of G, generate all simple cycles
containing a certain node, remove that node, and further decompose the
remainder into strongly connected components. When G is undirected, we
restrict our attention to biconnected components, generate all simple cycles
containing a particular edge, remove that edge, and further decompose the
remainder into biconnected components.
Note that multigraphs are supported by this function -- and in undirected
multigraphs, a pair of parallel edges is considered a cycle of length 2.
Likewise, self-loops are considered to be cycles of length 1. We define
cycles as sequences of nodes; so the presence of loops and parallel edges
does not change the number of simple cycles in a graph.
Parameters
----------
G : NetworkX DiGraph
A directed graph
length_bound : int or None, optional (default=None)
If length_bound is an int, generate all simple cycles of G with length at
most length_bound. Otherwise, generate all simple cycles of G.
Yields
------
list of nodes
@@ -134,92 +158,602 @@ def simple_cycles(G):
Notes
-----
The implementation follows pp. 79-80 in [1]_.
When length_bound is None, the time complexity is $O((n+e)(c+1))$ for $n$
nodes, $e$ edges and $c$ simple circuits. Otherwise, when length_bound > 1,
the time complexity is $O((c+n)(k-1)d^k)$ where $d$ is the average degree of
the nodes of G and $k$ = length_bound.
The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
elementary circuits.
Raises
------
ValueError
when length_bound < 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.
https://doi.org/10.1137/0204007
.. [2] Enumerating the cycles of a digraph: a new preprocessing strategy.
.. [2] Finding All Bounded-Length Simple Cycles in a Directed Graph
A. Gupta and T. Suzumura https://arxiv.org/abs/2105.10094
.. [3] 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.
.. [4] 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.
.. [5] Optimal Listing of Cycles and st-Paths in Undirected Graphs
R. Ferreira and R. Grossi and A. Marino and N. Pisanti and R. Rizzi and
G. Sacomoto https://arxiv.org/abs/1205.2766
See Also
--------
cycle_basis
chordless_cycles
"""
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()
if length_bound is not None:
if length_bound == 0:
return
elif length_bound < 0:
raise ValueError("length bound must be non-negative")
# 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]
directed = G.is_directed()
yield from ([v] for v, Gv in G.adj.items() if v in Gv)
# 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)
if length_bound is not None and length_bound == 1:
return
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)
if G.is_multigraph() and not directed:
visited = set()
for u, Gu in G.adj.items():
multiplicity = ((v, len(Guv)) for v, Guv in Gu.items() if v in visited)
yield from ([u, v] for v, m in multiplicity if m > 1)
visited.add(u)
# explicitly filter out loops; implicitly filter out parallel edges
if directed:
G = nx.DiGraph((u, v) for u, Gu in G.adj.items() for v in Gu if v != u)
else:
G = nx.Graph((u, v) for u, Gu in G.adj.items() for v in Gu if v != u)
# this case is not strictly necessary but improves performance
if length_bound is not None and length_bound == 2:
if directed:
visited = set()
for u, Gu in G.adj.items():
yield from (
[v, u] for v in visited.intersection(Gu) if G.has_edge(v, u)
)
visited.add(u)
return
if directed:
yield from _directed_cycle_search(G, length_bound)
else:
yield from _undirected_cycle_search(G, length_bound)
def _directed_cycle_search(G, length_bound):
"""A dispatch function for `simple_cycles` for directed graphs.
We generate all cycles of G through binary partition.
1. Pick a node v in G which belongs to at least one cycle
a. Generate all cycles of G which contain the node v.
b. Recursively generate all cycles of G \\ v.
This is accomplished through the following:
1. Compute the strongly connected components SCC of G.
2. Select and remove a biconnected component C from BCC. Select a
non-tree edge (u, v) of a depth-first search of G[C].
3. For each simple cycle P containing v in G[C], yield P.
4. Add the biconnected components of G[C \\ v] to BCC.
If the parameter length_bound is not None, then step 3 will be limited to
simple cycles of length at most length_bound.
Parameters
----------
G : NetworkX DiGraph
A directed graph
length_bound : int or None
If length_bound is an int, generate all simple cycles of G with length at most length_bound.
Otherwise, generate all simple cycles of G.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
"""
scc = nx.strongly_connected_components
components = [c for c in scc(G) if len(c) >= 2]
while components:
c = components.pop()
Gc = G.subgraph(c)
v = next(iter(c))
if length_bound is None:
yield from _johnson_cycle_search(Gc, [v])
else:
yield from _bounded_cycle_search(Gc, [v], length_bound)
# delete v after searching G, to make sure we can find v
G.remove_node(v)
components.extend(c for c in scc(Gc) if len(c) >= 2)
def _undirected_cycle_search(G, length_bound):
"""A dispatch function for `simple_cycles` for undirected graphs.
We generate all cycles of G through binary partition.
1. Pick an edge (u, v) in G which belongs to at least one cycle
a. Generate all cycles of G which contain the edge (u, v)
b. Recursively generate all cycles of G \\ (u, v)
This is accomplished through the following:
1. Compute the biconnected components BCC of G.
2. Select and remove a biconnected component C from BCC. Select a
non-tree edge (u, v) of a depth-first search of G[C].
3. For each (v -> u) path P remaining in G[C] \\ (u, v), yield P.
4. Add the biconnected components of G[C] \\ (u, v) to BCC.
If the parameter length_bound is not None, then step 3 will be limited to simple paths
of length at most length_bound.
Parameters
----------
G : NetworkX Graph
An undirected graph
length_bound : int or None
If length_bound is an int, generate all simple cycles of G with length at most length_bound.
Otherwise, generate all simple cycles of G.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
"""
bcc = nx.biconnected_components
components = [c for c in bcc(G) if len(c) >= 3]
while components:
c = components.pop()
Gc = G.subgraph(c)
uv = list(next(iter(Gc.edges)))
G.remove_edge(*uv)
# delete (u, v) before searching G, to avoid fake 3-cycles [u, v, u]
if length_bound is None:
yield from _johnson_cycle_search(Gc, uv)
else:
yield from _bounded_cycle_search(Gc, uv, length_bound)
components.extend(c for c in bcc(Gc) if len(c) >= 3)
class _NeighborhoodCache(dict):
"""Very lightweight graph wrapper which caches neighborhoods as list.
This dict subclass uses the __missing__ functionality to query graphs for
their neighborhoods, and store the result as a list. This is used to avoid
the performance penalty incurred by subgraph views.
"""
def __init__(self, G):
self.G = G
def __missing__(self, v):
Gv = self[v] = list(self.G[v])
return Gv
def _johnson_cycle_search(G, path):
"""The main loop of the cycle-enumeration algorithm of Johnson.
Parameters
----------
G : NetworkX Graph or DiGraph
A graph
path : list
A cycle prefix. All cycles generated will begin with this prefix.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
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
"""
G = _NeighborhoodCache(G)
blocked = set(path)
B = defaultdict(set) # graph portions that yield no elementary circuit
start = path[0]
stack = [iter(G[path[-1]])]
closed = [False]
while stack:
nbrs = stack[-1]
for w in nbrs:
if w == start:
yield path[:]
closed[-1] = True
elif w not in blocked:
path.append(w)
closed.append(False)
stack.append(iter(G[w]))
blocked.add(w)
break
else: # no more nbrs
stack.pop()
v = path.pop()
if closed.pop():
if closed:
closed[-1] = True
unblock_stack = {v}
while unblock_stack:
u = unblock_stack.pop()
if u in blocked:
blocked.remove(u)
unblock_stack.update(B[u])
B[u].clear()
else:
for w in G[v]:
B[w].add(v)
def _bounded_cycle_search(G, path, length_bound):
"""The main loop of the cycle-enumeration algorithm of Gupta and Suzumura.
Parameters
----------
G : NetworkX Graph or DiGraph
A graph
path : list
A cycle prefix. All cycles generated will begin with this prefix.
length_bound: int
A length bound. All cycles generated will have length at most length_bound.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
References
----------
.. [1] Finding All Bounded-Length Simple Cycles in a Directed Graph
A. Gupta and T. Suzumura https://arxiv.org/abs/2105.10094
"""
G = _NeighborhoodCache(G)
lock = {v: 0 for v in path}
B = defaultdict(set)
start = path[0]
stack = [iter(G[path[-1]])]
blen = [length_bound]
while stack:
nbrs = stack[-1]
for w in nbrs:
if w == start:
yield path[:]
blen[-1] = 1
elif len(path) < lock.get(w, length_bound):
path.append(w)
blen.append(length_bound)
lock[w] = len(path)
stack.append(iter(G[w]))
break
else:
stack.pop()
v = path.pop()
bl = blen.pop()
if blen:
blen[-1] = min(blen[-1], bl)
if bl < length_bound:
relax_stack = [(bl, v)]
while relax_stack:
bl, u = relax_stack.pop()
if lock.get(u, length_bound) < length_bound - bl + 1:
lock[u] = length_bound - bl + 1
relax_stack.extend((bl + 1, w) for w in B[u].difference(path))
else:
for w in G[v]:
B[w].add(v)
def chordless_cycles(G, length_bound=None):
"""Find simple chordless cycles of a graph.
A `simple cycle` is a closed path where no node appears twice. In a simple
cycle, a `chord` is an additional edge between two nodes in the cycle. A
`chordless cycle` is a simple cycle without chords. Said differently, a
chordless cycle is a cycle C in a graph G where the number of edges in the
induced graph G[C] is equal to the length of `C`.
Note that some care must be taken in the case that G is not a simple graph
nor a simple digraph. Some authors limit the definition of chordless cycles
to have a prescribed minimum length; we do not.
1. We interpret self-loops to be chordless cycles, except in multigraphs
with multiple loops in parallel. Likewise, in a chordless cycle of
length greater than 1, there can be no nodes with self-loops.
2. We interpret directed two-cycles to be chordless cycles, except in
multi-digraphs when any edge in a two-cycle has a parallel copy.
3. We interpret parallel pairs of undirected edges as two-cycles, except
when a third (or more) parallel edge exists between the two nodes.
4. Generalizing the above, edges with parallel clones may not occur in
chordless cycles.
In a directed graph, two chordless cycles are distinct if they are not
cyclic permutations of each other. In an undirected graph, two chordless
cycles are distinct if they are not cyclic permutations of each other nor of
the other's reversal.
Optionally, the cycles are bounded in length.
We use an algorithm strongly inspired by that of Dias et al [1]_. It has
been modified in the following ways:
1. Recursion is avoided, per Python's limitations
2. The labeling function is not necessary, because the starting paths
are chosen (and deleted from the host graph) to prevent multiple
occurrences of the same path
3. The search is optionally bounded at a specified length
4. Support for directed graphs is provided by extending cycles along
forward edges, and blocking nodes along forward and reverse edges
5. Support for multigraphs is provided by omitting digons from the set
of forward edges
Parameters
----------
G : NetworkX DiGraph
A directed graph
length_bound : int or None, optional (default=None)
If length_bound is an int, generate all simple cycles of G with length at
most length_bound. Otherwise, generate all simple cycles of G.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
Examples
--------
>>> sorted(list(nx.chordless_cycles(nx.complete_graph(4))))
[[1, 0, 2], [1, 0, 3], [2, 0, 3], [2, 1, 3]]
Notes
-----
When length_bound is None, and the graph is simple, the time complexity is
$O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$ chordless cycles.
Raises
------
ValueError
when length_bound < 0.
References
----------
.. [1] Efficient enumeration of chordless cycles
E. Dias and D. Castonguay and H. Longo and W.A.R. Jradi
https://arxiv.org/abs/1309.1051
See Also
--------
simple_cycles
"""
if length_bound is not None:
if length_bound == 0:
return
elif length_bound < 0:
raise ValueError("length bound must be non-negative")
directed = G.is_directed()
multigraph = G.is_multigraph()
if multigraph:
yield from ([v] for v, Gv in G.adj.items() if len(Gv.get(v, ())) == 1)
else:
yield from ([v] for v, Gv in G.adj.items() if v in Gv)
if length_bound is not None and length_bound == 1:
return
# Nodes with loops cannot belong to longer cycles. Let's delete them here.
# also, we implicitly reduce the multiplicity of edges down to 1 in the case
# of multiedges.
if directed:
F = nx.DiGraph((u, v) for u, Gu in G.adj.items() if u not in Gu for v in Gu)
B = F.to_undirected(as_view=False)
else:
F = nx.Graph((u, v) for u, Gu in G.adj.items() if u not in Gu for v in Gu)
B = None
# If we're given a multigraph, we have a few cases to consider with parallel
# edges.
#
# 1. If we have 2 or more edges in parallel between the nodes (u, v), we
# must not construct longer cycles along (u, v).
# 2. If G is not directed, then a pair of parallel edges between (u, v) is a
# chordless cycle unless there exists a third (or more) parallel edge.
# 3. If G is directed, then parallel edges do not form cyles, but do
# preclude back-edges from forming cycles (handled in the next section),
# Thus, if an edge (u, v) is duplicated and the reverse (v, u) is also
# present, then we remove both from F.
#
# In directed graphs, we need to consider both directions that edges can
# take, so iterate over all edges (u, v) and possibly (v, u). In undirected
# graphs, we need to be a little careful to only consider every edge once,
# so we use a "visited" set to emulate node-order comparisons.
if multigraph:
if not directed:
B = F.copy()
visited = set()
for u, Gu in G.adj.items():
if directed:
multiplicity = ((v, len(Guv)) for v, Guv in Gu.items())
for v, m in multiplicity:
if m > 1:
F.remove_edges_from(((u, v), (v, u)))
else:
multiplicity = ((v, len(Guv)) for v, Guv in Gu.items() if v in visited)
for v, m in multiplicity:
if m == 2:
yield [u, v]
if m > 1:
F.remove_edge(u, v)
visited.add(u)
# If we're given a directed graphs, we need to think about digons. If we
# have two edges (u, v) and (v, u), then that's a two-cycle. If either edge
# was duplicated above, then we removed both from F. So, any digons we find
# here are chordless. After finding digons, we remove their edges from F
# to avoid traversing them in the search for chordless cycles.
if directed:
for u, Fu in F.adj.items():
digons = [[u, v] for v in Fu if F.has_edge(v, u)]
yield from digons
F.remove_edges_from(digons)
F.remove_edges_from(e[::-1] for e in digons)
if length_bound is not None and length_bound == 2:
return
# Now, we prepare to search for cycles. We have removed all cycles of
# lengths 1 and 2, so F is a simple graph or simple digraph. We repeatedly
# separate digraphs into their strongly connected components, and undirected
# graphs into their biconnected components. For each component, we pick a
# node v, search for chordless cycles based at each "stem" (u, v, w), and
# then remove v from that component before separating the graph again.
if directed:
separate = nx.strongly_connected_components
# Directed stems look like (u -> v -> w), so we use the product of
# predecessors of v with successors of v.
def stems(C, v):
for u, w in product(C.pred[v], C.succ[v]):
if not G.has_edge(u, w): # omit stems with acyclic chords
yield [u, v, w], F.has_edge(w, u)
else:
separate = nx.biconnected_components
# Undirected stems look like (u ~ v ~ w), but we must not also search
# (w ~ v ~ u), so we use combinations of v's neighbors of length 2.
def stems(C, v):
yield from (([u, v, w], F.has_edge(w, u)) for u, w in combinations(C[v], 2))
components = [c for c in separate(F) if len(c) > 2]
while components:
c = components.pop()
v = next(iter(c))
Fc = F.subgraph(c)
Fcc = Bcc = None
for S, is_triangle in stems(Fc, v):
if is_triangle:
yield S
else:
if Fcc is None:
Fcc = _NeighborhoodCache(Fc)
Bcc = Fcc if B is None else _NeighborhoodCache(B.subgraph(c))
yield from _chordless_cycle_search(Fcc, Bcc, S, length_bound)
components.extend(c for c in separate(F.subgraph(c - {v})) if len(c) > 2)
def _chordless_cycle_search(F, B, path, length_bound):
"""The main loop for chordless cycle enumeration.
This algorithm is strongly inspired by that of Dias et al [1]_. It has been
modified in the following ways:
1. Recursion is avoided, per Python's limitations
2. The labeling function is not necessary, because the starting paths
are chosen (and deleted from the host graph) to prevent multiple
occurrences of the same path
3. The search is optionally bounded at a specified length
4. Support for directed graphs is provided by extending cycles along
forward edges, and blocking nodes along forward and reverse edges
5. Support for multigraphs is provided by omitting digons from the set
of forward edges
Parameters
----------
F : _NeighborhoodCache
A graph of forward edges to follow in constructing cycles
B : _NeighborhoodCache
A graph of blocking edges to prevent the production of chordless cycles
path : list
A cycle prefix. All cycles generated will begin with this prefix.
length_bound : int
A length bound. All cycles generated will have length at most length_bound.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
References
----------
.. [1] Efficient enumeration of chordless cycles
E. Dias and D. Castonguay and H. Longo and W.A.R. Jradi
https://arxiv.org/abs/1309.1051
"""
blocked = defaultdict(int)
target = path[0]
blocked[path[1]] = 1
for w in path[1:]:
for v in B[w]:
blocked[v] += 1
stack = [iter(F[path[2]])]
while stack:
nbrs = stack[-1]
for w in nbrs:
if blocked[w] == 1 and (length_bound is None or len(path) < length_bound):
Fw = F[w]
if target in Fw:
yield path + [w]
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)
Bw = B[w]
if target in Bw:
continue
for v in Bw:
blocked[v] += 1
path.append(w)
stack.append(iter(Fw))
break
else:
stack.pop()
for v in B[path.pop()]:
blocked[v] -= 1
@not_implemented_for("undirected")
@@ -269,6 +803,7 @@ def recursive_simple_cycles(G):
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""