"""
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
[docs]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