preflow_push(G, s, t, capacity='capacity', residual=None, global_relabel_freq=1, value_only=False)
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.
The residual network R
from an input graph G
has the same nodes as G
. 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 in G
. For each node u
in R
, R.nodes[u]['excess']
represents the difference between flow into u
and flow out of u
.
For each edge (u, v)
in R
, R[u][v]['capacity']
is equal to the capacity of (u, v)
in G
if it exists in G
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 in R.graph['inf']
. For each edge (u, v)
in R
, R[u][v]['flow']
represents the flow function of (u, v)
and satisfies R[u][v]['flow'] == -R[v][u]['flow']
.
The flow value, defined as the total flow into t
, the sink, is stored in R.graph['flow_value']
. Reachability to t
using only edges (u, v)
such that R[u][v]['flow'] < R[u][v]['capacity']
induces a minimum s
- t
cut.
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.
Source node for the flow.
Sink node for the flow.
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 network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None.
Relative frequency of applying the global relabeling heuristic to speed up the algorithm. If it is None, the heuristic is disabled. Default value: 1.
If False, compute a maximum flow; otherwise, compute a maximum preflow which is enough for computing the maximum flow value. Default value: False.
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.
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.
Residual network after computing the maximum flow.
Find a maximum single-commodity flow using the highest-label preflow-push algorithm.
edmonds_karp
meth
maximum_flow
meth
minimum_cut
meth
>>> 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"] TrueSee :
The following pages refer to to this document either explicitly or contain code examples using this.
networkx.algorithms.connectivity.cuts.minimum_node_cut
networkx.algorithms.flow.edmondskarp.edmonds_karp
networkx.algorithms.connectivity.connectivity.local_node_connectivity
networkx.algorithms.flow.dinitz_alg.dinitz
networkx.algorithms.connectivity.cuts.minimum_st_edge_cut
networkx.algorithms.flow.maxflow.minimum_cut_value
networkx.algorithms.connectivity.connectivity.local_edge_connectivity
networkx.algorithms.connectivity.connectivity.edge_connectivity
networkx.algorithms.flow.maxflow.maximum_flow
networkx.algorithms.flow.maxflow.maximum_flow_value
networkx.algorithms.connectivity.connectivity.all_pairs_node_connectivity
networkx.algorithms.flow.preflowpush.preflow_push
networkx.algorithms.connectivity.cuts.minimum_edge_cut
networkx.algorithms.connectivity.connectivity.node_connectivity
networkx.algorithms.flow.shortestaugmentingpath.shortest_augmenting_path
networkx.algorithms.connectivity.disjoint_paths.node_disjoint_paths
networkx.algorithms.flow.boykovkolmogorov.boykov_kolmogorov
networkx.algorithms.flow.maxflow.minimum_cut
networkx.algorithms.connectivity.connectivity.average_node_connectivity
networkx.algorithms.connectivity.disjoint_paths.edge_disjoint_paths
networkx.algorithms.connectivity.cuts.minimum_st_node_cut
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them