preflow_push#
- preflow_push(G, s, t, capacity='capacity', residual=None, global_relabel_freq=1, value_only=False)[source]#
Find a maximum single-commodity flow using the highest-label preflow-push 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 \(O(n^2 \sqrt{m})\) for \(n\) nodes and \(m\) edges.
- Parameters:
- GNetworkX 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.
- snode
Source node for the flow.
- tnode
Sink node for the flow.
- capacitystring
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’.
- residualNetworkX graph
Residual network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None.
- global_relabel_freqinteger, float
Relative frequency of applying the global relabeling heuristic to speed up the algorithm. If it is None, the heuristic is disabled. Default value: 1.
- value_onlybool
If False, compute a maximum flow; otherwise, compute a maximum preflow which is enough for computing the maximum flow value. Default value: False.
- Returns:
- RNetworkX 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.
Notes
The residual network
R
from an input graphG
has the same nodes asG
.R
is a DiGraph that contains a pair of edges(u, v)
and(v, u)
iff(u, v)
is not a self-loop, and at least one of(u, v)
and(v, u)
exists inG
. For each nodeu
inR
,R.nodes[u]['excess']
represents the difference between flow intou
and flow out ofu
.For each edge
(u, v)
inR
,R[u][v]['capacity']
is equal to the capacity of(u, v)
inG
if it exists inG
or zero otherwise. If the capacity is infinite,R[u][v]['capacity']
will have a high arbitrary finite value that does not affect the solution of the problem. This value is stored inR.graph['inf']
. For each edge(u, v)
inR
,R[u][v]['flow']
represents the flow function of(u, v)
and satisfiesR[u][v]['flow'] == -R[v][u]['flow']
.The flow value, defined as the total flow into
t
, the sink, is stored inR.graph['flow_value']
. Reachability tot
using only edges(u, v)
such thatR[u][v]['flow'] < R[u][v]['capacity']
induces a minimums
-t
cut.Examples
>>> from networkx.algorithms.flow import preflow_push
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 = preflow_push(G, "x", "y") >>> flow_value = nx.maximum_flow_value(G, "x", "y") >>> flow_value == R.graph["flow_value"] True >>> # preflow_push also stores the maximum flow value >>> # in the excess attribute of the sink node t >>> flow_value == R.nodes["y"]["excess"] True >>> # For some problems, you might only want to compute a >>> # maximum preflow. >>> R = preflow_push(G, "x", "y", value_only=True) >>> flow_value == R.graph["flow_value"] True >>> flow_value == R.nodes["y"]["excess"] True