def compute_edge_conn(G):
A = build_auxiliary_edge_connectivity(G)
R = build_residual_network(G, 'weight')
for (u, v) in G.edges:
c = local_edge_connectivity(G, u, v, auxiliary=A, residual=R)
G[u][v].update({"index": c})
return G
def minimum_edge_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of edges of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of edges of minimum cardinality that, if removed, would break
all paths among source and target in G. If not, it returns a set of
edges of minimum cardinality that disconnects G.
Parameters
----------
G : NetworkX graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
cutset : set
Set of edges that, if removed, would disconnect G. If source
and target nodes are provided, the set contains the edges that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has edge connectivity 5
>>> G = nx.icosahedral_graph()
>>> len(nx.minimum_edge_cut(G))
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of minimum edge cut. For
undirected graphs the algorithm works by finding a 'small' dominating
set of nodes of G (see algorithm 7 in [1]_) and computing the maximum
flow between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_. For
directed graphs, the algorithm does n calls to the max flow function.
The function raises an error if the directed graph is not weakly
connected and returns an empty set if it is weakly connected.
It is an implementation of algorithm 8 in [1]_.
See also
--------
:meth:`minimum_st_edge_cut`
:meth:`minimum_node_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if (s is not None and t is None) or (s is None and t is not None):
raise nx.NetworkXError('Both source and target must be specified.')
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)
# Local minimum edge cut if s and t are not None
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
return minimum_st_edge_cut(H, s, t, **kwargs)
# Global minimum edge cut
# Analog to the algorithm for global edge connectivity
if G.is_directed():
# Based on algorithm 8 in [1]
if not nx.is_weakly_connected(G):
raise nx.NetworkXError('Input graph is not connected')
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
nodes = list(G)
n = len(nodes)
for i in range(n):
try:
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1],
**kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
except IndexError: # Last node!
this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
else: # undirected
# Based on algorithm 6 in [1]
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is not connected')
# Initial cutset is all edges of a node with minimum degree
node = min(G, key=G.degree)
min_cut = set(G.edges(node))
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating set will always be of one node
# thus we return min_cut, which now contains the edges of a node
# with minimum degree
return min_cut
for w in D:
this_cut = minimum_st_edge_cut(H, v, w, **kwargs)
if len(this_cut) <= len(min_cut):
min_cut = this_cut
return min_cut
def minimum_st_edge_cut(G,
s,
t,
flow_func=None,
auxiliary=None,
residual=None):
"""Returns the edges of the cut-set of a minimum (s, t)-cut.
This function returns the set of edges of minimum cardinality that,
if removed, would destroy all paths among source and target in G.
Edge weights are not considered. See :meth:`minimum_cut` for
computing minimum cuts considering edge weights.
Parameters
----------
G : NetworkX graph
s : node
Source node for the flow.
t : node
Sink node for the flow.
auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based node connectivity. It has
to have a graph attribute called mapping with a dictionary mapping
node names in G and in the auxiliary digraph. If provided
it will be reused instead of recreated. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See :meth:`node_connectivity` for
details. The choice of the default function may change from version
to version and should not be relied on. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be
reused instead of recreated. Default value: None.
Returns
-------
cutset : set
Set of edges that, if removed from the graph, will disconnect it.
See also
--------
:meth:`minimum_cut`
:meth:`minimum_node_cut`
:meth:`minimum_edge_cut`
:meth:`stoer_wagner`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Examples
--------
This function is not imported in the base NetworkX namespace, so you
have to explicitly import it from the connectivity package:
>>> from networkx.algorithms.connectivity import minimum_st_edge_cut
We use in this example the platonic icosahedral graph, which has edge
connectivity 5.
>>> G = nx.icosahedral_graph()
>>> len(minimum_st_edge_cut(G, 0, 6))
5
If you need to compute local edge cuts on several pairs of
nodes in the same graph, it is recommended that you reuse the
data structures that NetworkX uses in the computation: the
auxiliary digraph for edge connectivity, and the residual
network for the underlying maximum flow computation.
Example of how to compute local edge cuts among all pairs of
nodes of the platonic icosahedral graph reusing the data
structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
... build_auxiliary_edge_connectivity)
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = dict.fromkeys(G, dict())
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> for u, v in itertools.combinations(G, 2):
... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R))
... result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing edge
cuts. For instance, in dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better than
the default :meth:`edmonds_karp` which is faster for sparse
networks with highly skewed degree distributions. Alternative flow
functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path))
5
"""
if flow_func is None:
flow_func = default_flow_func
if auxiliary is None:
H = build_auxiliary_edge_connectivity(G)
else:
H = auxiliary
kwargs = dict(capacity='capacity', flow_func=flow_func, residual=residual)
cut_value, partition = nx.minimum_cut(H, s, t, **kwargs)
reachable, non_reachable = partition
# Any edge in the original graph linking the two sets in the
# partition is part of the edge cutset
cutset = set()
for u, nbrs in ((n, G[n]) for n in reachable):
cutset.update((u, v) for v in nbrs if v in non_reachable)
return cutset
def edge_connectivity(G, s=None, t=None, flow_func=None, cutoff=None):
r"""Returns the edge connectivity of the graph or digraph G.
The edge connectivity is equal to the minimum number of edges that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local edge
connectivity: the minimum number of edges that must be removed to
break all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected or directed graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
cutoff : integer, float
If specified, the maximum flow algorithm will terminate when the
flow value reaches or exceeds the cutoff. This is only for the
algorithms that support the cutoff parameter: :meth:`edmonds_karp`
and :meth:`shortest_augmenting_path`. Other algorithms will ignore
this parameter. Default value: None.
Returns
-------
K : integer
Edge connectivity for G, or local edge connectivity if source
and target were provided
Examples
--------
>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(G)
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of global edge connectivity.
For undirected graphs the algorithm works by finding a 'small'
dominating set of nodes of G (see algorithm 7 in [1]_ ) and
computing local maximum flow (see :meth:`local_edge_connectivity`)
between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_ .
For directed graphs, the algorithm does n calls to the maximum
flow function. This is an implementation of algorithm 8 in [1]_ .
See also
--------
:meth:`local_edge_connectivity`
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
:meth:`k_edge_components`
:meth:`k_edge_subgraphs`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if (s is not None and t is None) or (s is None and t is not None):
raise nx.NetworkXError('Both source and target must be specified.')
# Local edge connectivity
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
return local_edge_connectivity(G,
s,
t,
flow_func=flow_func,
cutoff=cutoff)
# Global edge connectivity
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
if G.is_directed():
# Algorithm 8 in [1]
if not nx.is_weakly_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
nodes = list(G)
n = len(nodes)
if cutoff is not None:
L = min(cutoff, L)
for i in range(n):
kwargs['cutoff'] = L
try:
L = min(
L,
local_edge_connectivity(G, nodes[i], nodes[i + 1],
**kwargs))
except IndexError: # last node!
L = min(
L, local_edge_connectivity(G, nodes[i], nodes[0],
**kwargs))
return L
else: # undirected
# Algorithm 6 in [1]
if not nx.is_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
if cutoff is not None:
L = min(cutoff, L)
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating sets will always be of one node
# thus we return min degree
return L
for w in D:
kwargs['cutoff'] = L
L = min(L, local_edge_connectivity(G, v, w, **kwargs))
return L
def local_edge_connectivity(G,
s,
t,
flow_func=None,
auxiliary=None,
residual=None,
cutoff=None):
r"""Returns local edge connectivity for nodes s and t in G.
Local edge connectivity for two nodes s and t is the minimum number
of edges that must be removed to disconnect them.
This is a flow based implementation of edge connectivity. We compute the
maximum flow on an auxiliary digraph build from the original
network (see below for details). This is equal to the local edge
connectivity because the value of a maximum s-t-flow is equal to the
capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .
Parameters
----------
G : NetworkX graph
Undirected or directed graph
s : node
Source node
t : node
Target node
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
auxiliary : NetworkX DiGraph
Auxiliary digraph for computing flow based edge connectivity. If
provided it will be reused instead of recreated. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be
reused instead of recreated. Default value: None.
cutoff : integer, float
If specified, the maximum flow algorithm will terminate when the
flow value reaches or exceeds the cutoff. This is only for the
algorithms that support the cutoff parameter: :meth:`edmonds_karp`
and :meth:`shortest_augmenting_path`. Other algorithms will ignore
this parameter. Default value: None.
Returns
-------
K : integer
local edge connectivity for nodes s and t.
Examples
--------
This function is not imported in the base NetworkX namespace, so you
have to explicitly import it from the connectivity package:
>>> from networkx.algorithms.connectivity import local_edge_connectivity
We use in this example the platonic icosahedral graph, which has edge
connectivity 5.
>>> G = nx.icosahedral_graph()
>>> local_edge_connectivity(G, 0, 6)
5
If you need to compute local connectivity on several pairs of
nodes in the same graph, it is recommended that you reuse the
data structures that NetworkX uses in the computation: the
auxiliary digraph for edge connectivity, and the residual
network for the underlying maximum flow computation.
Example of how to compute local edge connectivity among
all pairs of nodes of the platonic icosahedral graph reusing
the data structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
... build_auxiliary_edge_connectivity)
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = dict.fromkeys(G, dict())
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> for u, v in itertools.combinations(G, 2):
... k = local_edge_connectivity(G, u, v, auxiliary=H, residual=R)
... result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing edge
connectivity. For instance, in dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better than
the default :meth:`edmonds_karp` which is faster for sparse
networks with highly skewed degree distributions. Alternative flow
functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)
5
Notes
-----
This is a flow based implementation of edge connectivity. We compute the
maximum flow using, by default, the :meth:`edmonds_karp` algorithm on an
auxiliary digraph build from the original input graph:
If the input graph is undirected, we replace each edge (`u`,`v`) with
two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute
'capacity' for each arc to 1. If the input graph is directed we simply
add the 'capacity' attribute. This is an implementation of algorithm 1
in [1]_.
The maximum flow in the auxiliary network is equal to the local edge
connectivity because the value of a maximum s-t-flow is equal to the
capacity of a minimum s-t-cut (Ford and Fulkerson theorem).
See also
--------
:meth:`edge_connectivity`
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if flow_func is None:
flow_func = default_flow_func
if auxiliary is None:
H = build_auxiliary_edge_connectivity(G)
else:
H = auxiliary
kwargs = dict(flow_func=flow_func, residual=residual)
if flow_func is shortest_augmenting_path:
kwargs['cutoff'] = cutoff
kwargs['two_phase'] = True
elif flow_func is edmonds_karp:
kwargs['cutoff'] = cutoff
elif flow_func is dinitz:
kwargs['cutoff'] = cutoff
elif flow_func is boykov_kolmogorov:
kwargs['cutoff'] = cutoff
return nx.maximum_flow_value(H, s, t, **kwargs)
def edge_disjoint_paths(G, s, t, flow_func=None, cutoff=None, auxiliary=None,
residual=None):
"""Returns the edges disjoint paths between source and target.
Edge disjoint paths are paths that do not share any edge. The
number of edge disjoint paths between source and target is equal
to their edge connectivity.
Parameters
----------
G : NetworkX graph
s : node
Source node for the flow.
t : node
Sink node for the flow.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. The choice of the default function
may change from version to version and should not be relied on.
Default value: None.
cutoff : int
Maximum number of paths to yield. Some of the maximum flow
algorithms, such as :meth:`edmonds_karp` (the default) and
:meth:`shortest_augmenting_path` support the cutoff parameter,
and will terminate when the flow value reaches or exceeds the
cutoff. Other algorithms will ignore this parameter.
Default value: None.
auxiliary : NetworkX DiGraph
Auxiliary digraph to compute flow based edge connectivity. It has
to have a graph attribute called mapping with a dictionary mapping
node names in G and in the auxiliary digraph. If provided
it will be reused instead of recreated. Default value: None.
residual : NetworkX DiGraph
Residual network to compute maximum flow. If provided it will be
reused instead of recreated. Default value: None.
Returns
-------
paths : generator
A generator of edge independent paths.
Raises
------
NetworkXNoPath : exception
If there is no path between source and target.
NetworkXError : exception
If source or target are not in the graph G.
See also
--------
:meth:`node_disjoint_paths`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
Examples
--------
We use in this example the platonic icosahedral graph, which has node
edge connectivity 5, thus there are 5 edge disjoint paths between any
pair of nodes.
>>> G = nx.icosahedral_graph()
>>> len(list(nx.edge_disjoint_paths(G, 0, 6)))
5
If you need to compute edge disjoint paths on several pairs of
nodes in the same graph, it is recommended that you reuse the
data structures that NetworkX uses in the computation: the
auxiliary digraph for edge connectivity, and the residual
network for the underlying maximum flow computation.
Example of how to compute edge disjoint paths among all pairs of
nodes of the platonic icosahedral graph reusing the data
structures.
>>> import itertools
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import (
... build_auxiliary_edge_connectivity)
>>> H = build_auxiliary_edge_connectivity(G)
>>> # And the function for building the residual network from the
>>> # flow package
>>> from networkx.algorithms.flow import build_residual_network
>>> # Note that the auxiliary digraph has an edge attribute named capacity
>>> R = build_residual_network(H, 'capacity')
>>> result = {n: {} for n in G}
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as arguments
>>> for u, v in itertools.combinations(G, 2):
... k = len(list(nx.edge_disjoint_paths(G, u, v, auxiliary=H, residual=R)))
... result[u][v] = k
>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
True
You can also use alternative flow algorithms for computing edge disjoint
paths. For instance, in dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better than
the default :meth:`edmonds_karp` which is faster for sparse
networks with highly skewed degree distributions. Alternative flow
functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> len(list(nx.edge_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path)))
5
Notes
-----
This is a flow based implementation of edge disjoint paths. We compute
the maximum flow between source and target on an auxiliary directed
network. The saturated edges in the residual network after running the
maximum flow algorithm correspond to edge disjoint paths between source
and target in the original network. This function handles both directed
and undirected graphs, and can use all flow algorithms from NetworkX flow
package.
"""
if s not in G:
raise nx.NetworkXError('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
if flow_func is None:
flow_func = default_flow_func
if auxiliary is None:
H = build_auxiliary_edge_connectivity(G)
else:
H = auxiliary
# Maximum possible edge disjoint paths
possible = min(H.out_degree(s), H.in_degree(t))
if not possible:
raise NetworkXNoPath
if cutoff is None:
cutoff = possible
else:
cutoff = min(cutoff, possible)
# Compute maximum flow between source and target. Flow functions in
# NetworkX return a residual network.
kwargs = dict(capacity='capacity', residual=residual, cutoff=cutoff,
value_only=True)
if flow_func is preflow_push:
del kwargs['cutoff']
if flow_func is shortest_augmenting_path:
kwargs['two_phase'] = True
R = flow_func(H, s, t, **kwargs)
if R.graph['flow_value'] == 0:
raise NetworkXNoPath
# Saturated edges in the residual network form the edge disjoint paths
# between source and target
cutset = [(u, v) for u, v, d in R.edges(data=True)
if d['capacity'] == d['flow'] and d['flow'] > 0]
# This is equivalent of what flow.utils.build_flow_dict returns, but
# only for the nodes with saturated edges and without reporting 0 flows.
flow_dict = dict((n, {}) for edge in cutset for n in edge)
for u, v in cutset:
flow_dict[u][v] = 1
# Rebuild the edge disjoint paths from the flow dictionary.
paths_found = 0
for v in list(flow_dict[s]):
if paths_found >= cutoff:
# preflow_push does not support cutoff: we have to
# keep track of the paths founds and stop at cutoff.
break
path = [s]
if v == t:
path.append(v)
yield path
continue
u = v
while u != t:
path.append(u)
try:
u, _ = flow_dict[u].popitem()
except KeyError:
break
else:
path.append(t)
yield path
paths_found += 1