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from . import maxflow, mincost, preflow_push, shortest_augmenting_path | ||
from . import (maxflow, mincost, edmonds_karp, ford_fulkerson, preflow_push, | ||
shortest_augmenting_path) | ||
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__all__ = sum([maxflow.__all__, mincost.__all__, preflow_push.__all__, | ||
shortest_augmenting_path.__all__], []) | ||
__all__ = sum([maxflow.__all__, | ||
mincost.__all__, | ||
edmonds_karp.__all__, | ||
ford_fulkerson.__all__, | ||
preflow_push.__all__, | ||
shortest_augmenting_path.__all__, | ||
], []) | ||
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from .maxflow import * | ||
from .mincost import * | ||
from .edmonds_karp import * | ||
from .ford_fulkerson import * | ||
from .preflow_push import * | ||
from .shortest_augmenting_path import * |
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# -*- coding: utf-8 -*- | ||
""" | ||
Edmonds-Karp algorithm for maximum flow problems. | ||
""" | ||
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__author__ = """ysitu <ysitu@users.noreply.github.com>""" | ||
# Copyright (C) 2014 ysitu <ysitu@users.noreply.github.com> | ||
# All rights reserved. | ||
# BSD license. | ||
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import networkx as nx | ||
from networkx.algorithms.flow.utils import * | ||
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__all__ = ['edmonds_karp'] | ||
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def edmonds_karp_core(R, s, t): | ||
"""Implementation of the Edmonds-Karp algorithm. | ||
""" | ||
R_node = R.node | ||
R_pred = R.pred | ||
R_succ = R.succ | ||
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inf = float('inf') | ||
def augment(path): | ||
"""Augment flow along a path from s to t. | ||
""" | ||
# Determine the path residual capacity. | ||
flow = inf | ||
it = iter(path) | ||
u = next(it) | ||
for v in it: | ||
attr = R_succ[u][v] | ||
flow = min(flow, attr['capacity'] - attr['flow']) | ||
u = v | ||
# Augment flow along the path. | ||
it = iter(path) | ||
u = next(it) | ||
for v in it: | ||
R_succ[u][v]['flow'] += flow | ||
R_succ[v][u]['flow'] -= flow | ||
u = v | ||
# Accumulate the flow values. | ||
R_node[s]['excess'] -= flow | ||
R_node[t]['excess'] += flow | ||
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def bidirectional_bfs(): | ||
"""Bidirectional breadth-first search for an augmenting path. | ||
""" | ||
pred = {s: None} | ||
q_s = [s] | ||
succ = {t: None} | ||
q_t = [t] | ||
while True: | ||
q = [] | ||
if len(q_s) <= len(q_t): | ||
for u in q_s: | ||
for v, attr in R_succ[u].items(): | ||
if v not in pred and attr['flow'] < attr['capacity']: | ||
pred[v] = u | ||
if v in succ: | ||
return v, pred, succ | ||
q.append(v) | ||
if not q: | ||
return None, None, None | ||
q_s = q | ||
else: | ||
for u in q_t: | ||
for v, attr in R_pred[u].items(): | ||
if v not in succ and attr['flow'] < attr['capacity']: | ||
succ[v] = u | ||
if v in pred: | ||
return v, pred, succ | ||
q.append(v) | ||
if not q: | ||
return None, None, None | ||
q_t = q | ||
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# Look for shortest augmenting paths using breadth-first search. | ||
while True: | ||
v, pred, succ = bidirectional_bfs() | ||
if pred is None: | ||
break | ||
path = [v] | ||
# Trace a path from s to v. | ||
u = v | ||
while u != s: | ||
u = pred[u] | ||
path.append(u) | ||
path.reverse() | ||
# Trace a path from v to t. | ||
u = v | ||
while u != t: | ||
u = succ[u] | ||
path.append(u) | ||
augment(path) | ||
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def edmonds_karp_impl(G, s, t, capacity): | ||
"""Implementation of the Edmonds-Karp algorithm. | ||
""" | ||
R = build_residual_network(G, s, t, capacity) | ||
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edmonds_karp_core(R, s, t) | ||
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return R | ||
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def edmonds_karp(G, s, t, capacity='capacity', value_only=False): | ||
"""Find a maximum single-commodity flow using the Edmonds-Karp algorithm. | ||
This function returns the residual network resulting after computing | ||
the maximum flow. See below for details about the conventions | ||
NetworkX uses for defining residual networks. | ||
This algorithm has a running time of `O(n m^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'. | ||
value_only : bool | ||
If True compute only the value of the maximum flow. This parameter | ||
will be ignored by this algorithm because is not applicable. | ||
Returns | ||
------- | ||
R : NetworkX DiGraph | ||
Residual network after computing the maximum flow. | ||
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. | ||
See also | ||
-------- | ||
:meth:`maximum_flow` | ||
:meth:`minimum_cut` | ||
:meth:`ford_fulkerson` | ||
:meth:`preflow_push` | ||
:meth:`shortest_augmenting_path` | ||
Notes | ||
----- | ||
The residual network :samp:`R` from an input graph :samp:`G` has the | ||
same nodes than :samp:`G`. :samp:`R` is a DiGraph that contains a pair | ||
of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a | ||
self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists | ||
in :samp:`G`. For each node :samp:`u` in :samp:`R`, | ||
:samp:`R.node[u]['excess']` represents the difference between flow into | ||
:samp:`u` and flow out of :samp:`u`. Thus the maximum flow value is | ||
stored in :samp:`R.node[t]['excess']`, where :samp:`t` is the sink node. | ||
For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` | ||
is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists | ||
in :samp:`G` or zero otherwise. If the capacity is infinite, | ||
:samp:`R[u][v]['capacity']` will have a high arbitrary finite value | ||
that does not affect the solution of the problem. For each edge | ||
:samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['flow']` represents | ||
the flow function of :samp:`(u, v)` and satisfies | ||
:samp:`R[u][v]['flow'] == -R[v][u]['flow']`. | ||
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) | ||
>>> R = nx.edmonds_karp(G, 'x', 'y') | ||
>>> flow_value = nx.maximum_flow(G, 'x', 'y') | ||
>>> flow_value | ||
3.0 | ||
>>> assert(flow_value == R.node['y']['excess']) | ||
""" | ||
R = edmonds_karp_impl(G, s, t, capacity) | ||
R.graph['algorithm'] = 'edmonds_karp' | ||
return R |
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