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# 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:
    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.

    global_relabel_freq integer, 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_only bool

      If False, compute a maximum flow; otherwise, compute a maximum preflow which is enough for computing the maximum flow value. Default value: False.

  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

  `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