/usr/share/pyshared/networkx/algorithms/flow/maxflow.py is in python-networkx 1.6-2.
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"""
Maximum flow (and minimum cut) algorithms on capacitated graphs.
"""
__author__ = """Loïc Séguin-C. <loicseguin@gmail.com>"""
# Copyright (C) 2010 Loïc Séguin-C. <loicseguin@gmail.com>
# All rights reserved.
# BSD license.
import networkx as nx
__all__ = ['ford_fulkerson',
'ford_fulkerson_flow',
'max_flow',
'min_cut']
def _create_auxiliary_digraph(G, capacity='capacity'):
"""Initialize an auxiliary digraph and dict of infinite capacity
edges for a given graph G.
Ignore edges with capacity <= 0.
"""
auxiliary = nx.DiGraph()
auxiliary.add_nodes_from(G)
inf_capacity_flows = {}
if nx.is_directed(G):
for edge in G.edges(data = True):
if capacity in edge[2]:
if edge[2][capacity] > 0:
auxiliary.add_edge(*edge)
else:
auxiliary.add_edge(*edge)
inf_capacity_flows[(edge[0], edge[1])] = 0
else:
for edge in G.edges(data = True):
if capacity in edge[2]:
if edge[2][capacity] > 0:
auxiliary.add_edge(*edge)
auxiliary.add_edge(edge[1], edge[0], edge[2])
else:
auxiliary.add_edge(*edge)
auxiliary.add_edge(edge[1], edge[0], edge[2])
inf_capacity_flows[(edge[0], edge[1])] = 0
inf_capacity_flows[(edge[1], edge[0])] = 0
return auxiliary, inf_capacity_flows
def _create_flow_dict(G, H, inf_capacity_flows, capacity='capacity'):
"""Creates the flow dict of dicts on G corresponding to the
auxiliary digraph H and infinite capacity edges flows
inf_capacity_flows.
"""
flow = dict([(u, {}) for u in G])
if G.is_directed():
for u, v in G.edges_iter():
if H.has_edge(u, v):
if capacity in G[u][v]:
flow[u][v] = max(0, G[u][v][capacity] - H[u][v][capacity])
elif G.has_edge(v, u) and not capacity in G[v][u]:
flow[u][v] = max(0, inf_capacity_flows[(u, v)] -
inf_capacity_flows[(v, u)])
else:
flow[u][v] = max(0, H[v].get(u, {}).get(capacity, 0) -
G[v].get(u, {}).get(capacity, 0))
else:
flow[u][v] = G[u][v][capacity]
else: # undirected
for u, v in G.edges_iter():
if H.has_edge(u, v):
if capacity in G[u][v]:
flow[u][v] = abs(G[u][v][capacity] - H[u][v][capacity])
else:
flow[u][v] = abs(inf_capacity_flows[(u, v)] -
inf_capacity_flows[(v, u)])
else:
flow[u][v] = abs(G[u][v][capacity] - H[v][u][capacity])
flow[v][u] = flow[u][v]
return flow
def ford_fulkerson(G, s, t, capacity='capacity'):
"""Find a maximum single-commodity flow using the Ford-Fulkerson
algorithm.
This algorithm uses Edmonds-Karp-Dinitz path selection rule which
guarantees a running time of O(nm^2) for n nodes and m edges.
Parameters
----------
G : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity: string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
Returns
-------
flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
flow_dict : dictionary
Dictionary of dictionaries keyed by nodes such that
flow_dict[u][v] is the flow edge (u, v).
Raises
------
NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If
the input graph is an instance of one of these two classes, a
NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a
feasible flow on the graph is unbounded above and the function
raises a NetworkXUnbounded.
Examples
--------
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity=3.0)
>>> G.add_edge('x','b', capacity=1.0)
>>> G.add_edge('a','c', capacity=3.0)
>>> G.add_edge('b','c', capacity=5.0)
>>> G.add_edge('b','d', capacity=4.0)
>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> flow, F = nx.ford_fulkerson(G, 'x', 'y')
>>> flow
3.0
"""
if G.is_multigraph():
raise nx.NetworkXError(
'MultiGraph and MultiDiGraph not supported (yet).')
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)
auxiliary, inf_capacity_flows = _create_auxiliary_digraph(G,
capacity=capacity)
flow_value = 0 # Initial feasible flow.
# As long as there is an (s, t)-path in the auxiliary digraph, find
# the shortest (with respect to the number of arcs) such path and
# augment the flow on this path.
while True:
try:
path_nodes = nx.bidirectional_shortest_path(auxiliary, s, t)
except nx.NetworkXNoPath:
break
# Get the list of edges in the shortest path.
path_edges = list(zip(path_nodes[:-1], path_nodes[1:]))
# Find the minimum capacity of an edge in the path.
try:
path_capacity = min([auxiliary[u][v][capacity]
for u, v in path_edges
if capacity in auxiliary[u][v]])
except ValueError:
# path of infinite capacity implies no max flow
raise nx.NetworkXUnbounded(
"Infinite capacity path, flow unbounded above.")
flow_value += path_capacity
# Augment the flow along the path.
for u, v in path_edges:
edge_attr = auxiliary[u][v]
if capacity in edge_attr:
edge_attr[capacity] -= path_capacity
if edge_attr[capacity] == 0:
auxiliary.remove_edge(u, v)
else:
inf_capacity_flows[(u, v)] += path_capacity
if auxiliary.has_edge(v, u):
if capacity in auxiliary[v][u]:
auxiliary[v][u][capacity] += path_capacity
else:
auxiliary.add_edge(v, u, {capacity: path_capacity})
flow_dict = _create_flow_dict(G, auxiliary, inf_capacity_flows,
capacity=capacity)
return flow_value, flow_dict
def ford_fulkerson_flow(G, s, t, capacity='capacity'):
"""Return a maximum flow for a single-commodity flow problem.
Parameters
----------
G : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity: string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
Returns
-------
flow_dict : dictionary
Dictionary of dictionaries keyed by nodes such that
flow_dict[u][v] is the flow edge (u, v).
Raises
------
NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If
the input graph is an instance of one of these two classes, a
NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a
feasible flow on the graph is unbounded above and the function
raises a NetworkXUnbounded.
Examples
--------
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity=3.0)
>>> G.add_edge('x','b', capacity=1.0)
>>> G.add_edge('a','c', capacity=3.0)
>>> G.add_edge('b','c', capacity=5.0)
>>> G.add_edge('b','d', capacity=4.0)
>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> F = nx.ford_fulkerson_flow(G, 'x', 'y')
>>> for u, v in G.edges_iter():
... print('(%s, %s) %.2f' % (u, v, F[u][v]))
...
(a, c) 2.00
(c, y) 2.00
(b, c) 0.00
(b, d) 1.00
(e, y) 1.00
(d, e) 1.00
(x, a) 2.00
(x, b) 1.00
"""
return ford_fulkerson(G, s, t, capacity=capacity)[1]
def max_flow(G, s, t, capacity='capacity'):
"""Find the value of a maximum single-commodity flow.
Parameters
----------
G : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity: string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
Returns
-------
flow_value : integer, float
Value of the maximum flow, i.e., net outflow from the source.
Raises
------
NetworkXError
The algorithm does not support MultiGraph and MultiDiGraph. If
the input graph is an instance of one of these two classes, a
NetworkXError is raised.
NetworkXUnbounded
If the graph has a path of infinite capacity, the value of a
feasible flow on the graph is unbounded above and the function
raises a NetworkXUnbounded.
Examples
--------
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity=3.0)
>>> G.add_edge('x','b', capacity=1.0)
>>> G.add_edge('a','c', capacity=3.0)
>>> G.add_edge('b','c', capacity=5.0)
>>> G.add_edge('b','d', capacity=4.0)
>>> G.add_edge('d','e', capacity=2.0)
>>> G.add_edge('c','y', capacity=2.0)
>>> G.add_edge('e','y', capacity=3.0)
>>> flow = nx.max_flow(G, 'x', 'y')
>>> flow
3.0
"""
return ford_fulkerson(G, s, t, capacity=capacity)[0]
def min_cut(G, s, t, capacity='capacity'):
"""Compute the value of a minimum (s, t)-cut.
Use the max-flow min-cut theorem, i.e., the capacity of a minimum
capacity cut is equal to the flow value of a maximum flow.
Parameters
----------
G : NetworkX graph
Edges of the graph are expected to have an attribute called
'capacity'. If this attribute is not present, the edge is
considered to have infinite capacity.
s : node
Source node for the flow.
t : node
Sink node for the flow.
capacity: string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
Returns
-------
cutValue : integer, float
Value of the minimum cut.
Raises
------
NetworkXUnbounded
If the graph has a path of infinite capacity, all cuts have
infinite capacity and the function raises a NetworkXError.
Examples
--------
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edge('x','a', capacity = 3.0)
>>> G.add_edge('x','b', capacity = 1.0)
>>> G.add_edge('a','c', capacity = 3.0)
>>> G.add_edge('b','c', capacity = 5.0)
>>> G.add_edge('b','d', capacity = 4.0)
>>> G.add_edge('d','e', capacity = 2.0)
>>> G.add_edge('c','y', capacity = 2.0)
>>> G.add_edge('e','y', capacity = 3.0)
>>> nx.min_cut(G, 'x', 'y')
3.0
"""
try:
return ford_fulkerson(G, s, t, capacity=capacity)[0]
except nx.NetworkXUnbounded:
raise nx.NetworkXUnbounded(
"Infinite capacity path, no minimum cut.")
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