"""
Algorithms for finding optimum branchings and spanning arborescences.
This implementation is based on:
J. Edmonds, Optimum branchings, J. Res. Natl. Bur. Standards 71B (1967),
233–240. URL: http://archive.org/details/jresv71Bn4p233
"""
# TODO: Implement method from Gabow, Galil, Spence and Tarjan:
#
# @article{
# year={1986},
# issn={0209-9683},
# journal={Combinatorica},
# volume={6},
# number={2},
# doi={10.1007/BF02579168},
# title={Efficient algorithms for finding minimum spanning trees in
# undirected and directed graphs},
# url={https://doi.org/10.1007/BF02579168},
# publisher={Springer-Verlag},
# keywords={68 B 15; 68 C 05},
# author={Gabow, Harold N. and Galil, Zvi and Spencer, Thomas and Tarjan,
# Robert E.},
# pages={109-122},
# language={English}
# }
import string
from dataclasses import dataclass, field
from operator import itemgetter
from queue import PriorityQueue
import networkx as nx
from networkx.utils import py_random_state
from .recognition import is_arborescence, is_branching
__all__ = [
"branching_weight",
"greedy_branching",
"maximum_branching",
"minimum_branching",
"minimal_branching",
"maximum_spanning_arborescence",
"minimum_spanning_arborescence",
"ArborescenceIterator",
"Edmonds",
]
KINDS = {"max", "min"}
STYLES = {
"branching": "branching",
"arborescence": "arborescence",
"spanning arborescence": "arborescence",
}
INF = float("inf")
@py_random_state(1)
def random_string(L=15, seed=None):
return "".join([seed.choice(string.ascii_letters) for n in range(L)])
def _min_weight(weight):
return -weight
def _max_weight(weight):
return weight
[docs]
@nx._dispatchable(edge_attrs={"attr": "default"})
def branching_weight(G, attr="weight", default=1):
"""
Returns the total weight of a branching.
You must access this function through the networkx.algorithms.tree module.
Parameters
----------
G : DiGraph
The directed graph.
attr : str
The attribute to use as weights. If None, then each edge will be
treated equally with a weight of 1.
default : float
When `attr` is not None, then if an edge does not have that attribute,
`default` specifies what value it should take.
Returns
-------
weight: int or float
The total weight of the branching.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from([(0, 1, 2), (1, 2, 4), (2, 3, 3), (3, 4, 2)])
>>> nx.tree.branching_weight(G)
11
"""
return sum(edge[2].get(attr, default) for edge in G.edges(data=True))
[docs]
@py_random_state(4)
@nx._dispatchable(edge_attrs={"attr": "default"}, returns_graph=True)
def greedy_branching(G, attr="weight", default=1, kind="max", seed=None):
"""
Returns a branching obtained through a greedy algorithm.
This algorithm is wrong, and cannot give a proper optimal branching.
However, we include it for pedagogical reasons, as it can be helpful to
see what its outputs are.
The output is a branching, and possibly, a spanning arborescence. However,
it is not guaranteed to be optimal in either case.
Parameters
----------
G : DiGraph
The directed graph to scan.
attr : str
The attribute to use as weights. If None, then each edge will be
treated equally with a weight of 1.
default : float
When `attr` is not None, then if an edge does not have that attribute,
`default` specifies what value it should take.
kind : str
The type of optimum to search for: 'min' or 'max' greedy branching.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
B : directed graph
The greedily obtained branching.
"""
if kind not in KINDS:
raise nx.NetworkXException("Unknown value for `kind`.")
if kind == "min":
reverse = False
else:
reverse = True
if attr is None:
# Generate a random string the graph probably won't have.
attr = random_string(seed=seed)
edges = [(u, v, data.get(attr, default)) for (u, v, data) in G.edges(data=True)]
# We sort by weight, but also by nodes to normalize behavior across runs.
try:
edges.sort(key=itemgetter(2, 0, 1), reverse=reverse)
except TypeError:
# This will fail in Python 3.x if the nodes are of varying types.
# In that case, we use the arbitrary order.
edges.sort(key=itemgetter(2), reverse=reverse)
# The branching begins with a forest of no edges.
B = nx.DiGraph()
B.add_nodes_from(G)
# Now we add edges greedily so long we maintain the branching.
uf = nx.utils.UnionFind()
for i, (u, v, w) in enumerate(edges):
if uf[u] == uf[v]:
# Adding this edge would form a directed cycle.
continue
elif B.in_degree(v) == 1:
# The edge would increase the degree to be greater than one.
continue
else:
# If attr was None, then don't insert weights...
data = {}
if attr is not None:
data[attr] = w
B.add_edge(u, v, **data)
uf.union(u, v)
return B
class MultiDiGraph_EdgeKey(nx.MultiDiGraph):
"""
MultiDiGraph which assigns unique keys to every edge.
Adds a dictionary edge_index which maps edge keys to (u, v, data) tuples.
This is not a complete implementation. For Edmonds algorithm, we only use
add_node and add_edge, so that is all that is implemented here. During
additions, any specified keys are ignored---this means that you also
cannot update edge attributes through add_node and add_edge.
Why do we need this? Edmonds algorithm requires that we track edges, even
as we change the head and tail of an edge, and even changing the weight
of edges. We must reliably track edges across graph mutations.
"""
def __init__(self, incoming_graph_data=None, **attr):
cls = super()
cls.__init__(incoming_graph_data=incoming_graph_data, **attr)
self._cls = cls
self.edge_index = {}
import warnings
msg = "MultiDiGraph_EdgeKey has been deprecated and will be removed in NetworkX 3.4."
warnings.warn(msg, DeprecationWarning)
def remove_node(self, n):
keys = set()
for keydict in self.pred[n].values():
keys.update(keydict)
for keydict in self.succ[n].values():
keys.update(keydict)
for key in keys:
del self.edge_index[key]
self._cls.remove_node(n)
def remove_nodes_from(self, nbunch):
for n in nbunch:
self.remove_node(n)
def add_edge(self, u_for_edge, v_for_edge, key_for_edge, **attr):
"""
Key is now required.
"""
u, v, key = u_for_edge, v_for_edge, key_for_edge
if key in self.edge_index:
uu, vv, _ = self.edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
self._cls.add_edge(u, v, key, **attr)
self.edge_index[key] = (u, v, self.succ[u][v][key])
def add_edges_from(self, ebunch_to_add, **attr):
for u, v, k, d in ebunch_to_add:
self.add_edge(u, v, k, **d)
def remove_edge_with_key(self, key):
try:
u, v, _ = self.edge_index[key]
except KeyError as err:
raise KeyError(f"Invalid edge key {key!r}") from err
else:
del self.edge_index[key]
self._cls.remove_edge(u, v, key)
def remove_edges_from(self, ebunch):
raise NotImplementedError
def get_path(G, u, v):
"""
Returns the edge keys of the unique path between u and v.
This is not a generic function. G must be a branching and an instance of
MultiDiGraph_EdgeKey.
"""
nodes = nx.shortest_path(G, u, v)
# We are guaranteed that there is only one edge connected every node
# in the shortest path.
def first_key(i, vv):
# Needed for 2.x/3.x compatibility
keys = G[nodes[i]][vv].keys()
# Normalize behavior
keys = list(keys)
return keys[0]
edges = [first_key(i, vv) for i, vv in enumerate(nodes[1:])]
return nodes, edges
[docs]
class Edmonds:
"""
Edmonds algorithm [1]_ for finding optimal branchings and spanning
arborescences.
This algorithm can find both minimum and maximum spanning arborescences and
branchings.
Notes
-----
While this algorithm can find a minimum branching, since it isn't required
to be spanning, the minimum branching is always from the set of negative
weight edges which is most likely the empty set for most graphs.
References
----------
.. [1] J. Edmonds, Optimum Branchings, Journal of Research of the National
Bureau of Standards, 1967, Vol. 71B, p.233-240,
https://archive.org/details/jresv71Bn4p233
"""
[docs]
def __init__(self, G, seed=None):
self.G_original = G
# Need to fix this. We need the whole tree.
self.store = True
# The final answer.
self.edges = []
# Since we will be creating graphs with new nodes, we need to make
# sure that our node names do not conflict with the real node names.
self.template = random_string(seed=seed) + "_{0}"
import warnings
msg = "Edmonds has been deprecated and will be removed in NetworkX 3.4. Please use the appropriate minimum or maximum branching or arborescence function directly."
warnings.warn(msg, DeprecationWarning)
def _init(self, attr, default, kind, style, preserve_attrs, seed, partition):
"""
So we need the code in _init and find_optimum to successfully run edmonds algorithm.
Responsibilities of the _init function:
- Check that the kind argument is in {min, max} or raise a NetworkXException.
- Transform the graph if we need a minimum arborescence/branching.
- The current method is to map weight -> -weight. This is NOT a good approach since
the algorithm can and does choose to ignore negative weights when creating a branching
since that is always optimal when maximzing the weights. I think we should set the edge
weights to be (max_weight + 1) - edge_weight.
- Transform the graph into a MultiDiGraph, adding the partition information and potoentially
other edge attributes if we set preserve_attrs = True.
- Setup the buckets and union find data structures required for the algorithm.
"""
if kind not in KINDS:
raise nx.NetworkXException("Unknown value for `kind`.")
# Store inputs.
self.attr = attr
self.default = default
self.kind = kind
self.style = style
# Determine how we are going to transform the weights.
if kind == "min":
self.trans = trans = _min_weight
else:
self.trans = trans = _max_weight
if attr is None:
# Generate a random attr the graph probably won't have.
attr = random_string(seed=seed)
# This is the actual attribute used by the algorithm.
self._attr = attr
# This attribute is used to store whether a particular edge is still
# a candidate. We generate a random attr to remove clashes with
# preserved edges
self.candidate_attr = "candidate_" + random_string(seed=seed)
# The object we manipulate at each step is a multidigraph.
self.G = G = MultiDiGraph_EdgeKey()
self.G.__networkx_cache__ = None # Disable caching
for key, (u, v, data) in enumerate(self.G_original.edges(data=True)):
d = {attr: trans(data.get(attr, default))}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
G.add_edge(u, v, key, **d)
self.level = 0
# These are the "buckets" from the paper.
#
# As in the paper, G^i are modified versions of the original graph.
# D^i and E^i are nodes and edges of the maximal edges that are
# consistent with G^i. These are dashed edges in figures A-F of the
# paper. In this implementation, we store D^i and E^i together as a
# graph B^i. So we will have strictly more B^i than the paper does.
self.B = MultiDiGraph_EdgeKey()
self.B.edge_index = {}
self.graphs = [] # G^i
self.branchings = [] # B^i
self.uf = nx.utils.UnionFind()
# A list of lists of edge indexes. Each list is a circuit for graph G^i.
# Note the edge list will not, in general, be a circuit in graph G^0.
self.circuits = []
# Stores the index of the minimum edge in the circuit found in G^i
# and B^i. The ordering of the edges seems to preserve the weight
# ordering from G^0. So even if the circuit does not form a circuit
# in G^0, it is still true that the minimum edge of the circuit in
# G^i is still the minimum edge in circuit G^0 (despite their weights
# being different).
self.minedge_circuit = []
# TODO: separate each step into an inner function. Then the overall loop would become
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching
# break
[docs]
def find_optimum(
self,
attr="weight",
default=1,
kind="max",
style="branching",
preserve_attrs=False,
partition=None,
seed=None,
):
"""
Returns a branching from G.
Parameters
----------
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
kind : {'min', 'max'}
The type of optimum to search for, either 'min' or 'max'.
style : {'branching', 'arborescence'}
If 'branching', then an optimal branching is found. If `style` is
'arborescence', then a branching is found, such that if the
branching is also an arborescence, then the branching is an
optimal spanning arborescences. A given graph G need not have
an optimal spanning arborescence.
preserve_attrs : bool
If True, preserve the other edge attributes of the original
graph (that are not the one passed to `attr`)
partition : str
The edge attribute holding edge partition data. Used in the
spanning arborescence iterator.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
H : (multi)digraph
The branching.
"""
self._init(attr, default, kind, style, preserve_attrs, seed, partition)
uf = self.uf
# This enormous while loop could use some refactoring...
G, B = self.G, self.B
D = set()
nodes = iter(list(G.nodes()))
attr = self._attr
def desired_edge(v):
"""
Find the edge directed toward v with maximal weight.
If an edge partition exists in this graph, return the included edge
if it exists and no not return any excluded edges. There can only
be one included edge for each vertex otherwise the edge partition is
empty.
"""
edge = None
weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, weight
# Find the best open edge
if new_weight > weight:
weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, weight
while True:
# (I1): Choose a node v in G^i not in D^i.
try:
v = next(nodes)
except StopIteration:
# If there are no more new nodes to consider, then we *should*
# meet the break condition (b) from the paper:
# (b) every node of G^i is in D^i and E^i is a branching
# Construction guarantees that it's a branching.
assert len(G) == len(B)
if len(B):
assert is_branching(B)
if self.store:
self.graphs.append(G.copy())
self.branchings.append(B.copy())
# Add these to keep the lengths equal. Element i is the
# circuit at level i that was merged to form branching i+1.
# There is no circuit for the last level.
self.circuits.append([])
self.minedge_circuit.append(None)
break
else:
if v in D:
# print("v in D", v)
continue
# Put v into bucket D^i.
# print(f"Adding node {v}")
D.add(v)
B.add_node(v)
# End (I1)
# Start cycle detection
edge, weight = desired_edge(v)
# print(f"Max edge is {edge!r}")
if edge is None:
# If there is no edge, continue with a new node at (I1).
continue
else:
# Determine if adding the edge to E^i would mean its no longer
# a branching. Presently, v has indegree 0 in B---it is a root.
u = edge[0]
if uf[u] == uf[v]:
# Then adding the edge will create a circuit. Then B
# contains a unique path P from v to u. So condition (a)
# from the paper does hold. We need to store the circuit
# for future reference.
Q_nodes, Q_edges = get_path(B, v, u)
Q_edges.append(edge[2]) # Edge key
else:
# Then B with the edge is still a branching and condition
# (a) from the paper does not hold.
Q_nodes, Q_edges = None, None
# End cycle detection
# THIS WILL PROBABLY BE REMOVED? MAYBE A NEW ARG FOR THIS FEATURE?
# Conditions for adding the edge.
# If weight < 0, then it cannot help in finding a maximum branching.
# This is the root of the problem with minimum branching.
if self.style == "branching" and weight <= 0:
acceptable = False
else:
acceptable = True
# print(f"Edge is acceptable: {acceptable}")
if acceptable:
dd = {attr: weight}
if edge[4].get(partition) is not None:
dd[partition] = edge[4].get(partition)
B.add_edge(u, v, edge[2], **dd)
G[u][v][edge[2]][self.candidate_attr] = True
uf.union(u, v)
if Q_edges is not None:
# print("Edge introduced a simple cycle:")
# print(Q_nodes, Q_edges)
# Move to method
# Previous meaning of u and v is no longer important.
# Apply (I2).
# Get the edge in the cycle with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B.edge_index[edge_key]
# We cannot remove an included edges, even if it is
# the minimum edge in the circuit
w = data[attr]
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
self.circuits.append(Q_edges)
self.minedge_circuit.append(minedge)
if self.store:
self.graphs.append(G.copy())
# Always need the branching with circuits.
self.branchings.append(B.copy())
# Now we mutate it.
new_node = self.template.format(self.level)
# print(minweight, minedge, Q_incoming_weight)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge, do nothing for now.
# Eventually delete it.
continue
else:
# Outgoing edge. Make it from new node
dd = data.copy()
new_edges.append((new_node, v, key, dd))
else:
if v in Q_incoming_weight:
# Incoming edge. Change its weight
w = data[attr]
w += minweight - Q_incoming_weight[v]
dd = data.copy()
dd[attr] = w
new_edges.append((u, new_node, key, dd))
else:
# Outside edge. No modification necessary.
continue
G.remove_nodes_from(Q_nodes)
B.remove_nodes_from(Q_nodes)
D.difference_update(set(Q_nodes))
for u, v, key, data in new_edges:
G.add_edge(u, v, key, **data)
if self.candidate_attr in data:
del data[self.candidate_attr]
B.add_edge(u, v, key, **data)
uf.union(u, v)
nodes = iter(list(G.nodes()))
self.level += 1
# END STEP (I2)?
# (I3) Branch construction.
# print(self.level)
H = self.G_original.__class__()
def is_root(G, u, edgekeys):
"""
Returns True if `u` is a root node in G.
Node `u` will be a root node if its in-degree, restricted to the
specified edges, is equal to 0.
"""
if u not in G:
# print(G.nodes(), u)
raise Exception(f"{u!r} not in G")
for v in G.pred[u]:
for edgekey in G.pred[u][v]:
if edgekey in edgekeys:
return False, edgekey
else:
return True, None
# Start with the branching edges in the last level.
edges = set(self.branchings[self.level].edge_index)
while self.level > 0:
self.level -= 1
# The current level is i, and we start counting from 0.
# We need the node at level i+1 that results from merging a circuit
# at level i. randomname_0 is the first merged node and this
# happens at level 1. That is, randomname_0 is a node at level 1
# that results from merging a circuit at level 0.
merged_node = self.template.format(self.level)
# The circuit at level i that was merged as a node the graph
# at level i+1.
circuit = self.circuits[self.level]
# print
# print(merged_node, self.level, circuit)
# print("before", edges)
# Note, we ask if it is a root in the full graph, not the branching.
# The branching alone doesn't have all the edges.
isroot, edgekey = is_root(self.graphs[self.level + 1], merged_node, edges)
edges.update(circuit)
if isroot:
minedge = self.minedge_circuit[self.level]
if minedge is None:
raise Exception
# Remove the edge in the cycle with minimum weight.
edges.remove(minedge)
else:
# We have identified an edge at next higher level that
# transitions into the merged node at the level. That edge
# transitions to some corresponding node at the current level.
# We want to remove an edge from the cycle that transitions
# into the corresponding node.
# print("edgekey is: ", edgekey)
# print("circuit is: ", circuit)
# The branching at level i
G = self.graphs[self.level]
# print(G.edge_index)
target = G.edge_index[edgekey][1]
for edgekey in circuit:
u, v, data = G.edge_index[edgekey]
if v == target:
break
else:
raise Exception("Couldn't find edge incoming to merged node.")
edges.remove(edgekey)
self.edges = edges
H.add_nodes_from(self.G_original)
for edgekey in edges:
u, v, d = self.graphs[0].edge_index[edgekey]
dd = {self.attr: self.trans(d[self.attr])}
# Optionally, preserve the other edge attributes of the original
# graph
if preserve_attrs:
for key, value in d.items():
if key not in [self.attr, self.candidate_attr]:
dd[key] = value
# TODO: make this preserve the key.
H.add_edge(u, v, **dd)
return H
[docs]
@nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
def maximum_branching(
G,
attr="weight",
default=1,
preserve_attrs=False,
partition=None,
):
#######################################
### Data Structure Helper Functions ###
#######################################
def edmonds_add_edge(G, edge_index, u, v, key, **d):
"""
Adds an edge to `G` while also updating the edge index.
This algorithm requires the use of an external dictionary to track
the edge keys since it is possible that the source or destination
node of an edge will be changed and the default key-handling
capabilities of the MultiDiGraph class do not account for this.
Parameters
----------
G : MultiDiGraph
The graph to insert an edge into.
edge_index : dict
A mapping from integers to the edges of the graph.
u : node
The source node of the new edge.
v : node
The destination node of the new edge.
key : int
The key to use from `edge_index`.
d : keyword arguments, optional
Other attributes to store on the new edge.
"""
if key in edge_index:
uu, vv, _ = edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
G.add_edge(u, v, key, **d)
edge_index[key] = (u, v, G.succ[u][v][key])
def edmonds_remove_node(G, edge_index, n):
"""
Remove a node from the graph, updating the edge index to match.
Parameters
----------
G : MultiDiGraph
The graph to remove an edge from.
edge_index : dict
A mapping from integers to the edges of the graph.
n : node
The node to remove from `G`.
"""
keys = set()
for keydict in G.pred[n].values():
keys.update(keydict)
for keydict in G.succ[n].values():
keys.update(keydict)
for key in keys:
del edge_index[key]
G.remove_node(n)
#######################
### Algorithm Setup ###
#######################
# Pick an attribute name that the original graph is unlikly to have
candidate_attr = "edmonds' secret candidate attribute"
new_node_base_name = "edmonds new node base name "
G_original = G
G = nx.MultiDiGraph()
G.__networkx_cache__ = None # Disable caching
# A dict to reliably track mutations to the edges using the key of the edge.
G_edge_index = {}
# Each edge is given an arbitrary numerical key
for key, (u, v, data) in enumerate(G_original.edges(data=True)):
d = {attr: data.get(attr, default)}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
edmonds_add_edge(G, G_edge_index, u, v, key, **d)
level = 0 # Stores the number of contracted nodes
# These are the buckets from the paper.
#
# In the paper, G^i are modified versions of the original graph.
# D^i and E^i are the nodes and edges of the maximal edges that are
# consistent with G^i. In this implementation, D^i and E^i are stored
# together as the graph B^i. We will have strictly more B^i then the
# paper will have.
#
# Note that the data in graphs and branchings are tuples with the graph as
# the first element and the edge index as the second.
B = nx.MultiDiGraph()
B_edge_index = {}
graphs = [] # G^i list
branchings = [] # B^i list
selected_nodes = set() # D^i bucket
uf = nx.utils.UnionFind()
# A list of lists of edge indices. Each list is a circuit for graph G^i.
# Note the edge list is not required to be a circuit in G^0.
circuits = []
# Stores the index of the minimum edge in the circuit found in G^i and B^i.
# The ordering of the edges seems to preserver the weight ordering from
# G^0. So even if the circuit does not form a circuit in G^0, it is still
# true that the minimum edges in circuit G^0 (despite their weights being
# different)
minedge_circuit = []
###########################
### Algorithm Structure ###
###########################
# Each step listed in the algorithm is an inner function. Thus, the overall
# loop structure is:
#
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching:
# break
##################################
### Algorithm Helper Functions ###
##################################
def edmonds_find_desired_edge(v):
"""
Find the edge directed towards v with maximal weight.
If an edge partition exists in this graph, return the included
edge if it exists and never return any excluded edge.
Note: There can only be one included edge for each vertex otherwise
the edge partition is empty.
Parameters
----------
v : node
The node to search for the maximal weight incoming edge.
"""
edge = None
max_weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
break
# Find the best open edge
if new_weight > max_weight:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, max_weight
def edmonds_step_I2(v, desired_edge, level):
"""
Perform step I2 from Edmonds' paper
First, check if the last step I1 created a cycle. If it did not, do nothing.
If it did, store the cycle for later reference and contract it.
Parameters
----------
v : node
The current node to consider
desired_edge : edge
The minimum desired edge to remove from the cycle.
level : int
The current level, i.e. the number of cycles that have already been removed.
"""
u = desired_edge[0]
Q_nodes = nx.shortest_path(B, v, u)
Q_edges = [
list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
]
Q_edges.append(desired_edge[2]) # Add the new edge key to complete the circuit
# Get the edge in the circuit with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B_edge_index[edge_key]
w = data[attr]
# We cannot remove an included edge, even if it is the
# minimum edge in the circuit
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
circuits.append(Q_edges)
minedge_circuit.append(minedge)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
# Mutate the graph to contract the circuit
new_node = new_node_base_name + str(level)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge. For the moment do nothing,
# eventually it will be removed.
continue
else:
# Outgoing edge from a node in the circuit.
# Make it come from the new node instead
dd = data.copy()
new_edges.append((new_node, v, key, dd))
else:
if v in Q_incoming_weight:
# Incoming edge to the circuit.
# Update it's weight
w = data[attr]
w += minweight - Q_incoming_weight[v]
dd = data.copy()
dd[attr] = w
new_edges.append((u, new_node, key, dd))
else:
# Outside edge. No modification needed
continue
for node in Q_nodes:
edmonds_remove_node(G, G_edge_index, node)
edmonds_remove_node(B, B_edge_index, node)
selected_nodes.difference_update(set(Q_nodes))
for u, v, key, data in new_edges:
edmonds_add_edge(G, G_edge_index, u, v, key, **data)
if candidate_attr in data:
del data[candidate_attr]
edmonds_add_edge(B, B_edge_index, u, v, key, **data)
uf.union(u, v)
def is_root(G, u, edgekeys):
"""
Returns True if `u` is a root node in G.
Node `u` is a root node if its in-degree over the specified edges is zero.
Parameters
----------
G : Graph
The current graph.
u : node
The node in `G` to check if it is a root.
edgekeys : iterable of edges
The edges for which to check if `u` is a root of.
"""
if u not in G:
raise Exception(f"{u!r} not in G")
for v in G.pred[u]:
for edgekey in G.pred[u][v]:
if edgekey in edgekeys:
return False, edgekey
else:
return True, None
nodes = iter(list(G.nodes))
while True:
try:
v = next(nodes)
except StopIteration:
# If there are no more new nodes to consider, then we should
# meet stopping condition (b) from the paper:
# (b) every node of G^i is in D^i and E^i is a branching
assert len(G) == len(B)
if len(B):
assert is_branching(B)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
circuits.append([])
minedge_circuit.append(None)
break
else:
#####################
### BEGIN STEP I1 ###
#####################
# This is a very simple step, so I don't think it needs a method of it's own
if v in selected_nodes:
continue
selected_nodes.add(v)
B.add_node(v)
desired_edge, desired_edge_weight = edmonds_find_desired_edge(v)
# There might be no desired edge if all edges are excluded or
# v is the last node to be added to B, the ultimate root of the branching
if desired_edge is not None and desired_edge_weight > 0:
u = desired_edge[0]
# Flag adding the edge will create a circuit before merging the two
# connected components of u and v in B
circuit = uf[u] == uf[v]
dd = {attr: desired_edge_weight}
if desired_edge[4].get(partition) is not None:
dd[partition] = desired_edge[4].get(partition)
edmonds_add_edge(B, B_edge_index, u, v, desired_edge[2], **dd)
G[u][v][desired_edge[2]][candidate_attr] = True
uf.union(u, v)
###################
### END STEP I1 ###
###################
#####################
### BEGIN STEP I2 ###
#####################
if circuit:
edmonds_step_I2(v, desired_edge, level)
nodes = iter(list(G.nodes()))
level += 1
###################
### END STEP I2 ###
###################
#####################
### BEGIN STEP I3 ###
#####################
# Create a new graph of the same class as the input graph
H = G_original.__class__()
# Start with the branching edges in the last level.
edges = set(branchings[level][1])
while level > 0:
level -= 1
# The current level is i, and we start counting from 0.
#
# We need the node at level i+1 that results from merging a circuit
# at level i. basename_0 is the first merged node and this happens
# at level 1. That is basename_0 is a node at level 1 that results
# from merging a circuit at level 0.
merged_node = new_node_base_name + str(level)
circuit = circuits[level]
isroot, edgekey = is_root(graphs[level + 1][0], merged_node, edges)
edges.update(circuit)
if isroot:
minedge = minedge_circuit[level]
if minedge is None:
raise Exception
# Remove the edge in the cycle with minimum weight
edges.remove(minedge)
else:
# We have identified an edge at the next higher level that
# transitions into the merged node at this level. That edge
# transitions to some corresponding node at the current level.
#
# We want to remove an edge from the cycle that transitions
# into the corresponding node, otherwise the result would not
# be a branching.
G, G_edge_index = graphs[level]
target = G_edge_index[edgekey][1]
for edgekey in circuit:
u, v, data = G_edge_index[edgekey]
if v == target:
break
else:
raise Exception("Couldn't find edge incoming to merged node.")
edges.remove(edgekey)
H.add_nodes_from(G_original)
for edgekey in edges:
u, v, d = graphs[0][1][edgekey]
dd = {attr: d[attr]}
if preserve_attrs:
for key, value in d.items():
if key not in [attr, candidate_attr]:
dd[key] = value
H.add_edge(u, v, **dd)
###################
### END STEP I3 ###
###################
return H
[docs]
@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
def minimum_branching(
G, attr="weight", default=1, preserve_attrs=False, partition=None
):
for _, _, d in G.edges(data=True):
d[attr] = -d.get(attr, default)
nx._clear_cache(G)
B = maximum_branching(G, attr, default, preserve_attrs, partition)
for _, _, d in G.edges(data=True):
d[attr] = -d.get(attr, default)
nx._clear_cache(G)
for _, _, d in B.edges(data=True):
d[attr] = -d.get(attr, default)
nx._clear_cache(B)
return B
@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
def minimal_branching(
G, /, *, attr="weight", default=1, preserve_attrs=False, partition=None
):
"""
Returns a minimal branching from `G`.
A minimal branching is a branching similar to a minimal arborescence but
without the requirement that the result is actually a spanning arborescence.
This allows minimal branchinges to be computed over graphs which may not
have arborescence (such as multiple components).
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A minimal branching.
"""
max_weight = -INF
min_weight = INF
for _, _, w in G.edges(data=attr, default=default):
if w > max_weight:
max_weight = w
if w < min_weight:
min_weight = w
for _, _, d in G.edges(data=True):
# Transform the weights so that the minimum weight is larger than
# the difference between the max and min weights. This is important
# in order to prevent the edge weights from becoming negative during
# computation
d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
nx._clear_cache(G)
B = maximum_branching(G, attr, default, preserve_attrs, partition)
# Reverse the weight transformations
for _, _, d in G.edges(data=True):
d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
nx._clear_cache(G)
for _, _, d in B.edges(data=True):
d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
nx._clear_cache(B)
return B
[docs]
@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
def maximum_spanning_arborescence(
G, attr="weight", default=1, preserve_attrs=False, partition=None
):
# In order to use the same algorithm is the maximum branching, we need to adjust
# the weights of the graph. The branching algorithm can choose to not include an
# edge if it doesn't help find a branching, mainly triggered by edges with negative
# weights.
#
# To prevent this from happening while trying to find a spanning arborescence, we
# just have to tweak the edge weights so that they are all positive and cannot
# become negative during the branching algorithm, find the maximum branching and
# then return them to their original values.
min_weight = INF
max_weight = -INF
for _, _, w in G.edges(data=attr, default=default):
if w < min_weight:
min_weight = w
if w > max_weight:
max_weight = w
for _, _, d in G.edges(data=True):
d[attr] = d.get(attr, default) - min_weight + 1 - (min_weight - max_weight)
nx._clear_cache(G)
B = maximum_branching(G, attr, default, preserve_attrs, partition)
for _, _, d in G.edges(data=True):
d[attr] = d.get(attr, default) + min_weight - 1 + (min_weight - max_weight)
nx._clear_cache(G)
for _, _, d in B.edges(data=True):
d[attr] = d.get(attr, default) + min_weight - 1 + (min_weight - max_weight)
nx._clear_cache(B)
if not is_arborescence(B):
raise nx.exception.NetworkXException("No maximum spanning arborescence in G.")
return B
[docs]
@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
def minimum_spanning_arborescence(
G, attr="weight", default=1, preserve_attrs=False, partition=None
):
B = minimal_branching(
G,
attr=attr,
default=default,
preserve_attrs=preserve_attrs,
partition=partition,
)
if not is_arborescence(B):
raise nx.exception.NetworkXException("No minimum spanning arborescence in G.")
return B
docstring_branching = """
Returns a {kind} {style} from G.
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A {kind} {style}.
"""
docstring_arborescence = (
docstring_branching
+ """
Raises
------
NetworkXException
If the graph does not contain a {kind} {style}.
"""
)
maximum_branching.__doc__ = docstring_branching.format(
kind="maximum", style="branching"
)
minimum_branching.__doc__ = (
docstring_branching.format(kind="minimum", style="branching")
+ """
See Also
--------
minimal_branching
"""
)
maximum_spanning_arborescence.__doc__ = docstring_arborescence.format(
kind="maximum", style="spanning arborescence"
)
minimum_spanning_arborescence.__doc__ = docstring_arborescence.format(
kind="minimum", style="spanning arborescence"
)
[docs]
class ArborescenceIterator:
"""
Iterate over all spanning arborescences of a graph in either increasing or
decreasing cost.
Notes
-----
This iterator uses the partition scheme from [1]_ (included edges,
excluded edges and open edges). It generates minimum spanning
arborescences using a modified Edmonds' Algorithm which respects the
partition of edges. For arborescences with the same weight, ties are
broken arbitrarily.
References
----------
.. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning
trees in order of increasing cost, Pesquisa Operacional, 2005-08,
Vol. 25 (2), p. 219-229,
https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en
"""
@dataclass(order=True)
class Partition:
"""
This dataclass represents a partition and stores a dict with the edge
data and the weight of the minimum spanning arborescence of the
partition dict.
"""
mst_weight: float
partition_dict: dict = field(compare=False)
def __copy__(self):
return ArborescenceIterator.Partition(
self.mst_weight, self.partition_dict.copy()
)
[docs]
def __init__(self, G, weight="weight", minimum=True, init_partition=None):
"""
Initialize the iterator
Parameters
----------
G : nx.DiGraph
The directed graph which we need to iterate trees over
weight : String, default = "weight"
The edge attribute used to store the weight of the edge
minimum : bool, default = True
Return the trees in increasing order while true and decreasing order
while false.
init_partition : tuple, default = None
In the case that certain edges have to be included or excluded from
the arborescences, `init_partition` should be in the form
`(included_edges, excluded_edges)` where each edges is a
`(u, v)`-tuple inside an iterable such as a list or set.
"""
self.G = G.copy()
self.weight = weight
self.minimum = minimum
self.method = (
minimum_spanning_arborescence if minimum else maximum_spanning_arborescence
)
# Randomly create a key for an edge attribute to hold the partition data
self.partition_key = (
"ArborescenceIterators super secret partition attribute name"
)
if init_partition is not None:
partition_dict = {}
for e in init_partition[0]:
partition_dict[e] = nx.EdgePartition.INCLUDED
for e in init_partition[1]:
partition_dict[e] = nx.EdgePartition.EXCLUDED
self.init_partition = ArborescenceIterator.Partition(0, partition_dict)
else:
self.init_partition = None
def __iter__(self):
"""
Returns
-------
ArborescenceIterator
The iterator object for this graph
"""
self.partition_queue = PriorityQueue()
self._clear_partition(self.G)
# Write the initial partition if it exists.
if self.init_partition is not None:
self._write_partition(self.init_partition)
mst_weight = self.method(
self.G,
self.weight,
partition=self.partition_key,
preserve_attrs=True,
).size(weight=self.weight)
self.partition_queue.put(
self.Partition(
mst_weight if self.minimum else -mst_weight,
{}
if self.init_partition is None
else self.init_partition.partition_dict,
)
)
return self
def __next__(self):
"""
Returns
-------
(multi)Graph
The spanning tree of next greatest weight, which ties broken
arbitrarily.
"""
if self.partition_queue.empty():
del self.G, self.partition_queue
raise StopIteration
partition = self.partition_queue.get()
self._write_partition(partition)
next_arborescence = self.method(
self.G,
self.weight,
partition=self.partition_key,
preserve_attrs=True,
)
self._partition(partition, next_arborescence)
self._clear_partition(next_arborescence)
return next_arborescence
def _partition(self, partition, partition_arborescence):
"""
Create new partitions based of the minimum spanning tree of the
current minimum partition.
Parameters
----------
partition : Partition
The Partition instance used to generate the current minimum spanning
tree.
partition_arborescence : nx.Graph
The minimum spanning arborescence of the input partition.
"""
# create two new partitions with the data from the input partition dict
p1 = self.Partition(0, partition.partition_dict.copy())
p2 = self.Partition(0, partition.partition_dict.copy())
for e in partition_arborescence.edges:
# determine if the edge was open or included
if e not in partition.partition_dict:
# This is an open edge
p1.partition_dict[e] = nx.EdgePartition.EXCLUDED
p2.partition_dict[e] = nx.EdgePartition.INCLUDED
self._write_partition(p1)
try:
p1_mst = self.method(
self.G,
self.weight,
partition=self.partition_key,
preserve_attrs=True,
)
p1_mst_weight = p1_mst.size(weight=self.weight)
p1.mst_weight = p1_mst_weight if self.minimum else -p1_mst_weight
self.partition_queue.put(p1.__copy__())
except nx.NetworkXException:
pass
p1.partition_dict = p2.partition_dict.copy()
def _write_partition(self, partition):
"""
Writes the desired partition into the graph to calculate the minimum
spanning tree. Also, if one incoming edge is included, mark all others
as excluded so that if that vertex is merged during Edmonds' algorithm
we cannot still pick another of that vertex's included edges.
Parameters
----------
partition : Partition
A Partition dataclass describing a partition on the edges of the
graph.
"""
for u, v, d in self.G.edges(data=True):
if (u, v) in partition.partition_dict:
d[self.partition_key] = partition.partition_dict[(u, v)]
else:
d[self.partition_key] = nx.EdgePartition.OPEN
nx._clear_cache(self.G)
for n in self.G:
included_count = 0
excluded_count = 0
for u, v, d in self.G.in_edges(nbunch=n, data=True):
if d.get(self.partition_key) == nx.EdgePartition.INCLUDED:
included_count += 1
elif d.get(self.partition_key) == nx.EdgePartition.EXCLUDED:
excluded_count += 1
# Check that if there is an included edges, all other incoming ones
# are excluded. If not fix it!
if included_count == 1 and excluded_count != self.G.in_degree(n) - 1:
for u, v, d in self.G.in_edges(nbunch=n, data=True):
if d.get(self.partition_key) != nx.EdgePartition.INCLUDED:
d[self.partition_key] = nx.EdgePartition.EXCLUDED
def _clear_partition(self, G):
"""
Removes partition data from the graph
"""
for u, v, d in G.edges(data=True):
if self.partition_key in d:
del d[self.partition_key]
nx._clear_cache(self.G)
