preflow_push

preflow_push(G, s, t, capacity='capacity', residual=None, global_relabel_freq=1, value_only=False)

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 or function (default= ‘capacity’)

If this is a string, then edge capacity will be accessed via the edge attribute with this key (that is, the capacity of the edge joining u to v will be G.edges[u, v][capacity]). If no such edge attribute exists, the capacity of the edge is assumed to be infinite.

If this is a function, the capacity of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number or None to indicate a hidden edge.

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.

See also

maximum_flow()

minimum_cut()

edmonds_karp()

shortest_augmenting_path()

Notes

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.

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