def all_pairs_node_connectivity(G, nbunch=None, flow_func=None):
"""Compute node connectivity between all pairs of nodes of G.
Parameters
----------
G : NetworkX graph
Undirected graph
nbunch: container
Container of nodes. If provided node connectivity will be computed
only over pairs of nodes in nbunch.
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
-------
all_pairs : dict
A dictionary with node connectivity between all pairs of nodes
in G, or in nbunch if provided.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`local_edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
"""
if nbunch is None:
nbunch = G
else:
nbunch = set(nbunch)
if G.is_directed():
iter_func = itertools.permutations
else:
iter_func = itertools.combinations
all_pairs = dict.fromkeys(nbunch, dict())
# Reuse auxiliary digraph and residual network
H = build_auxiliary_node_connectivity(G)
mapping = H.graph['mapping']
R = build_residual_network(H, 'capacity')
kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)
for u, v in iter_func(nbunch, 2):
K = local_node_connectivity(G, u, v, **kwargs)
all_pairs[u][v] = K
return all_pairs
def edge_connectivity(G, s=None, t=None, flow_func=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.
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.
>>> nx.edge_connectivity(G, flow_func=nx.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`
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)
# 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(G.degree().values())
nodes = G.nodes()
n = len(nodes)
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(G.degree().values())
# 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 node_connectivity(G, s=None, t=None, flow_func=None):
r"""Returns node connectivity for a graph or digraph G.
Node connectivity is equal to the minimum number of nodes that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local node
connectivity: the minimum number of nodes that must be removed to break
all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected 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
-------
K : integer
Node connectivity of G, or local node connectivity if source
and target are provided.
Examples
--------
>>> # Platonic icosahedral graph is 5-node-connected
>>> G = nx.icosahedral_graph()
>>> nx.node_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.
>>> nx.node_connectivity(G, flow_func=nx.shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local node connectivity.
>>> nx.node_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_node_connectivity` for details.
Notes
-----
This is a flow based implementation of node connectivity. The
algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2)`
maximum flow problems on an auxiliary digraph. Where `\delta`
is the minimum degree of G. For details about the auxiliary
digraph and the computation of local node connectivity see
:meth:`local_node_connectivity`. This implementation is based
on algorithm 11 in [1]_.
See also
--------
:meth:`local_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 node 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_node_connectivity(G, s, t, flow_func=flow_func)
# Global node connectivity
if G.is_directed():
if not nx.is_weakly_connected(G):
return 0
iter_func = itertools.permutations
# It is necessary to consider both predecessors
# and successors for directed graphs
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors_iter(v),
G.successors_iter(v)])
else:
if not nx.is_connected(G):
return 0
iter_func = itertools.combinations
neighbors = G.neighbors_iter
# 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)
# Pick a node with minimum degree
degree = G.degree()
minimum_degree = min(degree.values())
v = next(n for n, d in degree.items() if d == minimum_degree)
# Node connectivity is bounded by degree.
K = minimum_degree
# compute local node connectivity with all its non-neighbors nodes
for w in set(G) - set(neighbors(v)) - set([v]):
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, v, w, **kwargs))
# Also for non adjacent pairs of neighbors of v
for x, y in iter_func(neighbors(v), 2):
if y in G[x]:
continue
kwargs['cutoff'] = K
K = min(K, local_node_connectivity(G, x, y, **kwargs))
return K
def average_node_connectivity(G, flow_func=None):
r"""Returns the average connectivity of a graph G.
The average connectivity `\bar{\kappa}` of a graph G is the average
of local node connectivity over all pairs of nodes of G [1]_ .
.. math::
\bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}
Parameters
----------
G : NetworkX graph
Undirected graph
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:`local_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.
Returns
-------
K : float
Average node connectivity
See also
--------
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average
connectivity of a graph. Discrete mathematics 252(1-3), 31-45.
http://www.sciencedirect.com/science/article/pii/S0012365X01001807
"""
if G.is_directed():
iter_func = itertools.permutations
else:
iter_func = itertools.combinations
# 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)
num, den = 0, 0
for u, v in iter_func(G, 2):
num += local_node_connectivity(G, u, v, **kwargs)
den += 1
if den == 0: # Null Graph
return 0
return num / den
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 contians 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.
>>> len(nx.minimum_edge_cut(G, flow_func=nx.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.
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 algoritm 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 = G.edges(node)
nodes = G.nodes()
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 = 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 dinitz_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
if s == t:
raise nx.NetworkXError('source and sink are the same node')
if residual is None:
R = build_residual_network(G, capacity)
else:
R = residual
# Initialize/reset the residual network.
for u in R:
for e in R[u].values():
e['flow'] = 0
# Use an arbitrary high value as infinite. It is computed
# when building the residual network. Useful when checking
# for infinite capacity paths.
INF = R.graph['inf']
if cutoff is None:
cutoff = INF
R_succ = R.succ
def breath_first_search(G, R, s, t):
rank = {}
rank[s] = 0
queue = deque([s])
while queue:
if t in rank:
break
u = queue.popleft()
for v in R_succ[u]:
attr = R_succ[u][v]
if v not in rank and attr['capacity'] - attr['flow'] > 0:
rank[v] = rank[u] + 1
queue.append(v)
return rank
def depth_first_search(G, R, u, t, flow, rank):
if u == t:
return flow
for v in (n for n in R_succ[u] if n in rank and rank[n] == rank[u] + 1):
attr = R_succ[u][v]
if attr['capacity'] > attr['flow']:
min_flow = min(flow, attr['capacity'] - attr['flow'])
this_flow = depth_first_search(G, R, v, t, min_flow, rank)
if this_flow > 0:
R_succ[u][v]['flow'] += this_flow
R_succ[v][u]['flow'] -= this_flow
return this_flow
return 0
flow_value = 0
while flow_value < cutoff:
rank = breath_first_search(G, R, s, t)
if t not in rank:
break
while flow_value < cutoff:
blocking_flow = depth_first_search(G, R, s, t, INF, rank)
if blocking_flow * 2 > INF:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
elif blocking_flow == 0:
break
flow_value += blocking_flow
R.graph['flow_value'] = flow_value
return R
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 contians 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.
>>> node_cut == nx.minimum_node_cut(G, flow_func=nx.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('node %s not in graph' % s)
if t not in G:
raise nx.NetworkXError('node %s not in graph' % t)
return minimum_st_node_cut(G, s, t, flow_func=flow_func)
# Global minimum node cut.
# Analog to the algoritm 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_iter(v),
G.successors_iter(v)])
else:
if not nx.is_connected(G):
raise nx.NetworkXError('Input graph is not connected')
iter_func = itertools.combinations
neighbors = G.neighbors_iter
# 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)) - set([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
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
if s == t:
raise nx.NetworkXError('source and sink are the same node')
if residual is None:
R = build_residual_network(G, capacity)
else:
R = residual
# Initialize/reset the residual network.
# This is way too slow
#nx.set_edge_attributes(R, 0, 'flow')
for u in R:
for e in R[u].values():
e['flow'] = 0
# Use an arbitrary high value as infinite. It is computed
# when building the residual network.
INF = R.graph['inf']
if cutoff is None:
cutoff = INF
R_succ = R.succ
R_pred = R.pred
def grow():
"""Bidirectional breadth-first search for the growth stage.
Returns a connecting edge, that is and edge that connects
a node from the source search tree with a node from the
target search tree.
The first node in the connecting edge is always from the
source tree and the last node from the target tree.
"""
while active:
u = active[0]
if u in source_tree:
this_tree = source_tree
other_tree = target_tree
neighbors = R_succ
else:
this_tree = target_tree
other_tree = source_tree
neighbors = R_pred
for v, attr in neighbors[u].items():
if attr['capacity'] - attr['flow'] > 0:
if v not in this_tree:
if v in other_tree:
return (u, v) if this_tree is source_tree else (v, u)
this_tree[v] = u
dist[v] = dist[u] + 1
timestamp[v] = timestamp[u]
active.append(v)
elif v in this_tree and _is_closer(u, v):
this_tree[v] = u
dist[v] = dist[u] + 1
timestamp[v] = timestamp[u]
_ = active.popleft()
return None, None
def augment(u, v):
"""Augmentation stage.
Reconstruct path and determine its residual capacity.
We start from a connecting edge, which links a node
from the source tree to a node from the target tree.
The connecting edge is the output of the grow function
and the input of this function.
"""
attr = R_succ[u][v]
flow = min(INF, attr['capacity'] - attr['flow'])
path = [u]
# Trace a path from u to s in source_tree.
w = u
while w != s:
n = w
w = source_tree[n]
attr = R_pred[n][w]
flow = min(flow, attr['capacity'] - attr['flow'])
path.append(w)
path.reverse()
# Trace a path from v to t in target_tree.
path.append(v)
w = v
while w != t:
n = w
w = target_tree[n]
attr = R_succ[n][w]
flow = min(flow, attr['capacity'] - attr['flow'])
path.append(w)
# Augment flow along the path and check for saturated edges.
it = iter(path)
u = next(it)
these_orphans = []
for v in it:
R_succ[u][v]['flow'] += flow
R_succ[v][u]['flow'] -= flow
if R_succ[u][v]['flow'] == R_succ[u][v]['capacity']:
if v in source_tree:
source_tree[v] = None
these_orphans.append(v)
if u in target_tree:
target_tree[u] = None
these_orphans.append(u)
u = v
orphans.extend(sorted(these_orphans, key=dist.get))
return flow
def adopt():
"""Adoption stage.
Reconstruct search trees by adopting or discarding orphans.
During augmentation stage some edges got saturated and thus
the source and target search trees broke down to forests, with
orphans as roots of some of its trees. We have to reconstruct
the search trees rooted to source and target before we can grow
them again.
"""
while orphans:
u = orphans.popleft()
if u in source_tree:
tree = source_tree
neighbors = R_pred
else:
tree = target_tree
neighbors = R_succ
nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items()
if n in tree)
for v, attr, d in sorted(nbrs, key=itemgetter(2)):
if attr['capacity'] - attr['flow'] > 0:
if _has_valid_root(v, tree):
tree[u] = v
dist[u] = dist[v] + 1
timestamp[u] = time
break
else:
nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items()
if n in tree)
for v, attr, d in sorted(nbrs, key=itemgetter(2)):
if attr['capacity'] - attr['flow'] > 0:
if v not in active:
active.append(v)
if tree[v] == u:
tree[v] = None
orphans.appendleft(v)
if u in active:
active.remove(u)
del tree[u]
def _has_valid_root(n, tree):
path = []
v = n
while v is not None:
path.append(v)
if v == s or v == t:
base_dist = 0
break
elif timestamp[v] == time:
base_dist = dist[v]
break
v = tree[v]
else:
return False
length = len(path)
for i, u in enumerate(path, 1):
dist[u] = base_dist + length - i
timestamp[u] = time
return True
def _is_closer(u, v):
return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1
source_tree = {s: None}
target_tree = {t: None}
active = deque([s, t])
orphans = deque()
flow_value = 0
# data structures for the marking heuristic
time = 1
timestamp = {s: time, t: time}
dist = {s: 0, t: 0}
while flow_value < cutoff:
# Growth stage
u, v = grow()
if u is None:
break
time += 1
# Augmentation stage
flow_value += augment(u, v)
# Adoption stage
adopt()
if flow_value * 2 > INF:
raise nx.NetworkXUnbounded('Infinite capacity path, flow unbounded above.')
# Add source and target tree in a graph attribute.
# A partition that defines a minimum cut can be directly
# computed from the search trees as explained in the docstrings.
R.graph['trees'] = (source_tree, target_tree)
# Add the standard flow_value graph attribute.
R.graph['flow_value'] = flow_value
return R
def dinitz_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
if s == t:
raise nx.NetworkXError('source and sink are the same node')
if residual is None:
R = build_residual_network(G, capacity)
else:
R = residual
# Initialize/reset the residual network.
for u in R:
for e in R[u].values():
e['flow'] = 0
# Use an arbitrary high value as infinite. It is computed
# when building the residual network.
INF = R.graph['inf']
if cutoff is None:
cutoff = INF
R_succ = R.succ
R_pred = R.pred
def breath_first_search():
parents = {}
queue = deque([s])
while queue:
if t in parents:
break
u = queue.popleft()
for v in R_succ[u]:
attr = R_succ[u][v]
if v not in parents and attr['capacity'] - attr['flow'] > 0:
parents[v] = u
queue.append(v)
return parents
def depth_first_search(parents):
"""Build a path using DFS starting from the sink"""
path = []
u = t
flow = INF
while u != s:
path.append(u)
v = parents[u]
flow = min(flow, R_pred[u][v]['capacity'] - R_pred[u][v]['flow'])
u = v
path.append(s)
# Augment the flow along the path found
if flow > 0:
for u, v in pairwise(path):
R_pred[u][v]['flow'] += flow
R_pred[v][u]['flow'] -= flow
return flow
flow_value = 0
while flow_value < cutoff:
parents = breath_first_search()
if t not in parents:
break
this_flow = depth_first_search(parents)
if this_flow * 2 > INF:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
flow_value += this_flow
R.graph['flow_value'] = flow_value
return R
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError(f"node {str(s)} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {str(t)} not in graph")
if s == t:
raise nx.NetworkXError("source and sink are the same node")
if residual is None:
R = build_residual_network(G, capacity)
else:
R = residual
# Initialize/reset the residual network.
# This is way too slow
# nx.set_edge_attributes(R, 0, 'flow')
for u in R:
for e in R[u].values():
e["flow"] = 0
# Use an arbitrary high value as infinite. It is computed
# when building the residual network.
INF = R.graph["inf"]
if cutoff is None:
cutoff = INF
R_succ = R.succ
R_pred = R.pred
def grow():
"""Bidirectional breadth-first search for the growth stage.
Returns a connecting edge, that is and edge that connects
a node from the source search tree with a node from the
target search tree.
The first node in the connecting edge is always from the
source tree and the last node from the target tree.
"""
while active:
u = active[0]
if u in source_tree:
this_tree = source_tree
other_tree = target_tree
neighbors = R_succ
else:
this_tree = target_tree
other_tree = source_tree
neighbors = R_pred
for v, attr in neighbors[u].items():
if attr["capacity"] - attr["flow"] > 0:
if v not in this_tree:
if v in other_tree:
return (u, v) if this_tree is source_tree else (v,
u)
this_tree[v] = u
dist[v] = dist[u] + 1
timestamp[v] = timestamp[u]
active.append(v)
elif v in this_tree and _is_closer(u, v):
this_tree[v] = u
dist[v] = dist[u] + 1
timestamp[v] = timestamp[u]
_ = active.popleft()
return None, None
def augment(u, v):
"""Augmentation stage.
Reconstruct path and determine its residual capacity.
We start from a connecting edge, which links a node
from the source tree to a node from the target tree.
The connecting edge is the output of the grow function
and the input of this function.
"""
attr = R_succ[u][v]
flow = min(INF, attr["capacity"] - attr["flow"])
path = [u]
# Trace a path from u to s in source_tree.
w = u
while w != s:
n = w
w = source_tree[n]
attr = R_pred[n][w]
flow = min(flow, attr["capacity"] - attr["flow"])
path.append(w)
path.reverse()
# Trace a path from v to t in target_tree.
path.append(v)
w = v
while w != t:
n = w
w = target_tree[n]
attr = R_succ[n][w]
flow = min(flow, attr["capacity"] - attr["flow"])
path.append(w)
# Augment flow along the path and check for saturated edges.
it = iter(path)
u = next(it)
these_orphans = []
for v in it:
R_succ[u][v]["flow"] += flow
R_succ[v][u]["flow"] -= flow
if R_succ[u][v]["flow"] == R_succ[u][v]["capacity"]:
if v in source_tree:
source_tree[v] = None
these_orphans.append(v)
if u in target_tree:
target_tree[u] = None
these_orphans.append(u)
u = v
orphans.extend(sorted(these_orphans, key=dist.get))
return flow
def adopt():
"""Adoption stage.
Reconstruct search trees by adopting or discarding orphans.
During augmentation stage some edges got saturated and thus
the source and target search trees broke down to forests, with
orphans as roots of some of its trees. We have to reconstruct
the search trees rooted to source and target before we can grow
them again.
"""
while orphans:
u = orphans.popleft()
if u in source_tree:
tree = source_tree
neighbors = R_pred
else:
tree = target_tree
neighbors = R_succ
nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items()
if n in tree)
for v, attr, d in sorted(nbrs, key=itemgetter(2)):
if attr["capacity"] - attr["flow"] > 0:
if _has_valid_root(v, tree):
tree[u] = v
dist[u] = dist[v] + 1
timestamp[u] = time
break
else:
nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items()
if n in tree)
for v, attr, d in sorted(nbrs, key=itemgetter(2)):
if attr["capacity"] - attr["flow"] > 0:
if v not in active:
active.append(v)
if tree[v] == u:
tree[v] = None
orphans.appendleft(v)
if u in active:
active.remove(u)
del tree[u]
def _has_valid_root(n, tree):
path = []
v = n
while v is not None:
path.append(v)
if v == s or v == t:
base_dist = 0
break
elif timestamp[v] == time:
base_dist = dist[v]
break
v = tree[v]
else:
return False
length = len(path)
for i, u in enumerate(path, 1):
dist[u] = base_dist + length - i
timestamp[u] = time
return True
def _is_closer(u, v):
return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1
source_tree = {s: None}
target_tree = {t: None}
active = deque([s, t])
orphans = deque()
flow_value = 0
# data structures for the marking heuristic
time = 1
timestamp = {s: time, t: time}
dist = {s: 0, t: 0}
while flow_value < cutoff:
# Growth stage
u, v = grow()
if u is None:
break
time += 1
# Augmentation stage
flow_value += augment(u, v)
# Adoption stage
adopt()
if flow_value * 2 > INF:
raise nx.NetworkXUnbounded(
"Infinite capacity path, flow unbounded above.")
# Add source and target tree in a graph attribute.
# A partition that defines a minimum cut can be directly
# computed from the search trees as explained in the docstrings.
R.graph["trees"] = (source_tree, target_tree)
# Add the standard flow_value graph attribute.
R.graph["flow_value"] = flow_value
return R
def dinitz_impl(G, s, t, capacity, residual, cutoff):
if s not in G:
raise nx.NetworkXError('node %s not in graph' % str(s))
if t not in G:
raise nx.NetworkXError('node %s not in graph' % str(t))
if s == t:
raise nx.NetworkXError('source and sink are the same node')
if residual is None:
R = build_residual_network(G, capacity)
else:
R = residual
# Initialize/reset the residual network.
for u in R:
for e in R[u].values():
e['flow'] = 0
# Use an arbitrary high value as infinite. It is computed
# when building the residual network.
INF = R.graph['inf']
if cutoff is None:
cutoff = INF
R_succ = R.succ
R_pred = R.pred
def breath_first_search():
parents = {}
queue = deque([s])
while queue:
if t in parents:
break
u = queue.popleft()
for v in R_succ[u]:
attr = R_succ[u][v]
if v not in parents and attr['capacity'] - attr['flow'] > 0:
parents[v] = u
queue.append(v)
return parents
def depth_first_search(parents):
"""Build a path using DFS starting from the sink"""
path = []
u = t
flow = INF
while u != s:
path.append(u)
v = parents[u]
flow = min(flow, R_pred[u][v]['capacity'] - R_pred[u][v]['flow'])
u = v
path.append(s)
# Augment the flow along the path found
if flow > 0:
for u, v in pairwise(path):
R_pred[u][v]['flow'] += flow
R_pred[v][u]['flow'] -= flow
return flow
flow_value = 0
while flow_value < cutoff:
parents = breath_first_search()
if t not in parents:
break
this_flow = depth_first_search(parents)
if this_flow * 2 > INF:
raise nx.NetworkXUnbounded(
'Infinite capacity path, flow unbounded above.')
flow_value += this_flow
R.graph['flow_value'] = flow_value
return R
