"""
Shortest augmenting path algorithm for maximum flow problems.
"""
from collections import deque
import networkx as nx
from .utils import build_residual_network, CurrentEdge
from .edmondskarp import edmonds_karp_core
__all__ = ["shortest_augmenting_path"]
def shortest_augmenting_path_impl(G, s, t, capacity, residual, two_phase, cutoff):
"""Implementation of the shortest augmenting path 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
R_nodes = R.nodes
R_pred = R.pred
R_succ = R.succ
# Initialize/reset the residual network.
for u in R:
for e in R_succ[u].values():
e["flow"] = 0
# Initialize heights of the nodes.
heights = {t: 0}
q = deque([(t, 0)])
while q:
u, height = q.popleft()
height += 1
for v, attr in R_pred[u].items():
if v not in heights and attr["flow"] < attr["capacity"]:
heights[v] = height
q.append((v, height))
if s not in heights:
# t is not reachable from s in the residual network. The maximum flow
# must be zero.
R.graph["flow_value"] = 0
return R
n = len(G)
m = R.size() / 2
# Initialize heights and 'current edge' data structures of the nodes.
for u in R:
R_nodes[u]["height"] = heights[u] if u in heights else n
R_nodes[u]["curr_edge"] = CurrentEdge(R_succ[u])
# Initialize counts of nodes in each level.
counts = [0] * (2 * n - 1)
for u in R:
counts[R_nodes[u]["height"]] += 1
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 relabel(u):
"""Relabel a node to create an admissible edge.
"""
height = n - 1
for v, attr in R_succ[u].items():
if attr["flow"] < attr["capacity"]:
height = min(height, R_nodes[v]["height"])
return height + 1
if cutoff is None:
cutoff = float("inf")
# Phase 1: Look for shortest augmenting paths using depth-first search.
flow_value = 0
path = [s]
u = s
d = n if not two_phase else int(min(m ** 0.5, 2 * n ** (2.0 / 3)))
done = R_nodes[s]["height"] >= d
while not done:
height = R_nodes[u]["height"]
curr_edge = R_nodes[u]["curr_edge"]
# Depth-first search for the next node on the path to t.
while True:
v, attr = curr_edge.get()
if height == R_nodes[v]["height"] + 1 and attr["flow"] < attr["capacity"]:
# Advance to the next node following an admissible edge.
path.append(v)
u = v
break
try:
curr_edge.move_to_next()
except StopIteration:
counts[height] -= 1
if counts[height] == 0:
# Gap heuristic: If relabeling causes a level to become
# empty, a minimum cut has been identified. The algorithm
# can now be terminated.
R.graph["flow_value"] = flow_value
return R
height = relabel(u)
if u == s and height >= d:
if not two_phase:
# t is disconnected from s in the residual network. No
# more augmenting paths exist.
R.graph["flow_value"] = flow_value
return R
else:
# t is at least d steps away from s. End of phase 1.
done = True
break
counts[height] += 1
R_nodes[u]["height"] = height
if u != s:
# After relabeling, the last edge on the path is no longer
# admissible. Retreat one step to look for an alternative.
path.pop()
u = path[-1]
break
if u == t:
# t is reached. Augment flow along the path and reset it for a new
# depth-first search.
flow_value += augment(path)
if flow_value >= cutoff:
R.graph["flow_value"] = flow_value
return R
path = [s]
u = s
# Phase 2: Look for shortest augmenting paths using breadth-first search.
flow_value += edmonds_karp_core(R, s, t, cutoff - flow_value)
R.graph["flow_value"] = flow_value
return R
[docs]def shortest_augmenting_path(
G,
s,
t,
capacity="capacity",
residual=None,
value_only=False,
two_phase=False,
cutoff=None,
):
r"""Find a maximum single-commodity flow using the shortest augmenting path
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 $O(n^2 m)$ 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.
two_phase : bool
If True, a two-phase variant is used. The two-phase variant improves
the running time on unit-capacity networks from $O(nm)$ to
$O(\min(n^{2/3}, m^{1/2}) m)$. Default value: False.
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:`edmonds_karp`
:meth:`preflow_push`
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 shortest_augmenting_path
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 = shortest_augmenting_path(G, "x", "y")
>>> flow_value = nx.maximum_flow_value(G, "x", "y")
>>> flow_value
3.0
>>> flow_value == R.graph["flow_value"]
True
"""
R = shortest_augmenting_path_impl(G, s, t, capacity, residual, two_phase, cutoff)
R.graph["algorithm"] = "shortest_augmenting_path"
return R