"""
Flow based cut algorithms
"""
import itertools
import networkx as nx
# Define the default maximum flow function to use in all flow based
# cut algorithms.
from networkx.algorithms.flow import edmonds_karp
from networkx.algorithms.flow import build_residual_network
default_flow_func = edmonds_karp
from .utils import build_auxiliary_node_connectivity, build_auxiliary_edge_connectivity
__all__ = [
"minimum_st_node_cut",
"minimum_node_cut",
"minimum_st_edge_cut",
"minimum_edge_cut",
]
[docs]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
[docs]def minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None):
r"""Returns a set of nodes of minimum cardinality that disconnect source
from target in G.
This function returns the set of nodes of minimum cardinality that,
if removed, would destroy all paths among source and target in G.
Parameters
----------
G : NetworkX 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 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.
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 nodes that, if removed, would destroy all paths between
source and target in G.
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_node_cut
We use in this example the platonic icosahedral graph, which has node
connectivity 5.
>>> G = nx.icosahedral_graph()
>>> len(minimum_st_node_cut(G, 0, 6))
5
If you need to compute local st cuts between 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 node connectivity and node cuts, and the
residual network for the underlying maximum flow computation.
Example of how to compute local st node cuts reusing the data
structures:
>>> # You also have to explicitly import the function for
>>> # building the auxiliary digraph from the connectivity package
>>> from networkx.algorithms.connectivity import build_auxiliary_node_connectivity
>>> H = build_auxiliary_node_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")
>>> # Reuse the auxiliary digraph and the residual network by passing them
>>> # as parameters
>>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R))
5
You can also use alternative flow algorithms for computing minimum st
node 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_node_cut(G, 0, 6, flow_func=shortest_augmenting_path))
5
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of maximum flow computations to determine
the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed
and undirected graphs. This implementation is based on algorithm 11
in [1]_.
See also
--------
: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`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if auxiliary is None:
H = build_auxiliary_node_connectivity(G)
else:
H = auxiliary
mapping = H.graph.get("mapping", None)
if mapping is None:
raise nx.NetworkXError("Invalid auxiliary digraph.")
if G.has_edge(s, t) or G.has_edge(t, s):
return {}
kwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H)
# The edge cut in the auxiliary digraph corresponds to the node cut in the
# original graph.
edge_cut = minimum_st_edge_cut(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs)
# Each node in the original graph maps to two nodes of the auxiliary graph
node_cut = {H.nodes[node]["id"] for edge in edge_cut for node in edge}
return node_cut - {s, t}
[docs]def minimum_node_cut(G, s=None, t=None, flow_func=None):
r"""Returns a set of nodes of minimum cardinality that disconnects G.
If source and target nodes are provided, this function returns the
set of nodes of minimum cardinality that, if removed, would destroy
all paths among source and target in G. If not, it returns a set
of nodes 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 nodes that, if removed, would disconnect G. If source
and target nodes are provided, the set contains the nodes that
if removed, would destroy all paths between source and target.
Examples
--------
>>> # Platonic icosahedral graph has node connectivity 5
>>> G = nx.icosahedral_graph()
>>> node_cut = nx.minimum_node_cut(G)
>>> len(node_cut)
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
>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)
True
If you specify a pair of nodes (source and target) as parameters,
this function returns a local st node cut.
>>> len(nx.minimum_node_cut(G, 3, 7))
5
If you need to perform several local st cuts 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:`minimum_st_node_cut` for details.
Notes
-----
This is a flow based implementation of minimum node cut. The algorithm
is based in solving a number of maximum flow computations to determine
the capacity of the minimum cut on an auxiliary directed network that
corresponds to the minimum node cut of G. It handles both directed
and undirected graphs. This implementation is based on algorithm 11
in [1]_.
See also
--------
:meth:`minimum_st_node_cut`
:meth:`minimum_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`
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 minimum node cut.
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError(f"node {s} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {t} not in graph")
return minimum_st_node_cut(G, s, t, flow_func=flow_func)
# Global minimum node cut.
# Analog to the algorithm 11 for global node connectivity in [1].
if G.is_directed():
if not nx.is_weakly_connected(G):
raise nx.NetworkXError("Input graph is not connected")
iter_func = itertools.permutations
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])
else:
if not nx.is_connected(G):
raise nx.NetworkXError("Input graph is not connected")
iter_func = itertools.combinations
neighbors = G.neighbors
# Reuse the auxiliary digraph and the residual network.
H = build_auxiliary_node_connectivity(G)
R = build_residual_network(H, "capacity")
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
# Choose a node with minimum degree.
v = min(G, key=G.degree)
# Initial node cutset is all neighbors of the node with minimum degree.
min_cut = set(G[v])
# Compute st node cuts between v and all its non-neighbors nodes in G.
for w in set(G) - set(neighbors(v)) - {v}:
this_cut = minimum_st_node_cut(G, v, w, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
# Also for non adjacent pairs of neighbors of v.
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
this_cut = minimum_st_node_cut(G, x, y, **kwargs)
if len(min_cut) >= len(this_cut):
min_cut = this_cut
return min_cut
[docs]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(f"node {s} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {t} not in graph")
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