240 lines
7.8 KiB
Python
240 lines
7.8 KiB
Python
|
"""
|
||
|
|
Edmonds-Karp algorithm for maximum flow problems.
|
||
|
|
"""
|
||
|
|
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.algorithms.flow.utils import build_residual_network
|
||
|
|
|
||
|
|
__all__ = ["edmonds_karp"]
|
||
|
|
|
||
|
|
|
||
|
|
def edmonds_karp_core(R, s, t, cutoff):
|
||
|
|
"""Implementation of the Edmonds-Karp algorithm."""
|
||
|
|
R_nodes = R.nodes
|
||
|
|
R_pred = R.pred
|
||
|
|
R_succ = R.succ
|
||
|
|
|
||
|
|
inf = R.graph["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
|
||
|
|
if flow * 2 > inf:
|
||
|
|
raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.")
|
||
|
|
# 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
|
||
|
|
return flow
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
# Look for shortest augmenting paths using breadth-first search.
|
||
|
|
flow_value = 0
|
||
|
|
while flow_value < cutoff:
|
||
|
|
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)
|
||
|
|
flow_value += augment(path)
|
||
|
|
|
||
|
|
return flow_value
|
||
|
|
|
||
|
|
|
||
|
|
def edmonds_karp_impl(G, s, t, capacity, residual, cutoff):
|
||
|
|
"""Implementation of the Edmonds-Karp algorithm."""
|
||
|
|
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.
|
||
|
|
for u in R:
|
||
|
|
for e in R[u].values():
|
||
|
|
e["flow"] = 0
|
||
|
|
|
||
|
|
if cutoff is None:
|
||
|
|
cutoff = float("inf")
|
||
|
|
R.graph["flow_value"] = edmonds_karp_core(R, s, t, cutoff)
|
||
|
|
|
||
|
|
return R
|
||
|
|
|
||
|
|
|
||
|
|
def edmonds_karp(
|
||
|
|
G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None
|
||
|
|
):
|
||
|
|
"""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'.
|
||
|
|
|
||
|
|
residual : NetworkX graph
|
||
|
|
Residual network on which the algorithm is to be executed. If None, a
|
||
|
|
new residual network is created. Default value: None.
|
||
|
|
|
||
|
|
value_only : bool
|
||
|
|
If True compute only the value of the maximum flow. This parameter
|
||
|
|
will be ignored by this algorithm because it is not applicable.
|
||
|
|
|
||
|
|
cutoff : integer, float
|
||
|
|
If specified, the algorithm will terminate when the flow value reaches
|
||
|
|
or exceeds the cutoff. In this case, it may be unable to immediately
|
||
|
|
determine a minimum cut. Default value: None.
|
||
|
|
|
||
|
|
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:`preflow_push`
|
||
|
|
:meth:`shortest_augmenting_path`
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The residual network :samp:`R` from an input graph :samp:`G` has the
|
||
|
|
same nodes as :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 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. This value is stored in
|
||
|
|
:samp:`R.graph['inf']`. 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']`.
|
||
|
|
|
||
|
|
The flow value, defined as the total flow into :samp:`t`, the sink, is
|
||
|
|
stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not
|
||
|
|
specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such
|
||
|
|
that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
|
||
|
|
:samp:`s`-:samp:`t` cut.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms.flow import edmonds_karp
|
||
|
|
|
||
|
|
The functions that implement flow algorithms and output a residual
|
||
|
|
network, such as this one, are not imported to the base NetworkX
|
||
|
|
namespace, so you have to explicitly import them from the flow package.
|
||
|
|
|
||
|
|
>>> 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 = edmonds_karp(G, "x", "y")
|
||
|
|
>>> flow_value = nx.maximum_flow_value(G, "x", "y")
|
||
|
|
>>> flow_value
|
||
|
|
3.0
|
||
|
|
>>> flow_value == R.graph["flow_value"]
|
||
|
|
True
|
||
|
|
|
||
|
|
"""
|
||
|
|
R = edmonds_karp_impl(G, s, t, capacity, residual, cutoff)
|
||
|
|
R.graph["algorithm"] = "edmonds_karp"
|
||
|
|
return R
|