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
s : node
t : node
capacity : string
residual : NetworkX graph
value_only : bool
two_phase : bool
cutoff : integer, float
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Returns: | R : NetworkX DiGraph
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Raises: | NetworkXError :
NetworkXUnbounded :
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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 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']. If cutoff is not specified, 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
>>> import networkx as nx
>>> 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