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# shortest_augmenting_path

shortest_augmenting_path(G, s, t, capacity='capacity', residual=None, value_only=False, two_phase=False, cutoff=None)

  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

      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.

    value_only bool

      If True compute only the value of the maximum flow. This parameter will be ignored by this algorithm because it is not applicable.

    two_phase bool

      If True, a two-phase variant is used. The two-phase variant improves the running time on unit-capacity networks from \(O(nm)\) to \(O(\min(n^{2/3}, m^{1/2}) m)\). Default value: False.

    cutoff integer, float

      If specified, the algorithm will terminate when the flow value reaches or exceeds the cutoff. In this case, it may be unable to immediately determine a minimum cut. Default value: None.

  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()`
  `preflow_push()`

  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

      >>> 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