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# Source code for networkx.algorithms.flow.maxflow

    """
    Maximum flow (and minimum cut) algorithms on capacitated graphs.
    """

    import networkx as nx

    from .boykovkolmogorov import boykov_kolmogorov
    from .dinitz_alg import dinitz
    from .edmondskarp import edmonds_karp
    from .preflowpush import preflow_push
    from .shortestaugmentingpath import shortest_augmenting_path
    from .utils import build_flow_dict

    # Define the default flow function for computing maximum flow.
    default_flow_func = preflow_push

    __all__ = ["maximum_flow", "maximum_flow_value", "minimum_cut", "minimum_cut_value"]

    @nx._dispatchable(graphs="flowG", edge_attrs={"capacity": float("inf")})
    def maximum_flow(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
        """Find a maximum single-commodity flow.

        Parameters
        ----------
        flowG : 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'.

        flow_func : function
            A function for computing the maximum flow among a pair of nodes
            in a capacitated graph. The function has to accept at least three
            parameters: a Graph or Digraph, a source node, and a target node.
            And return a residual network that follows NetworkX conventions
            (see Notes). If flow_func is None, the default maximum
            flow function (:meth:`preflow_push`) is used. See below for
            alternative algorithms. The choice of the default function may change
            from version to version and should not be relied on. Default value:
            None.

        kwargs : Any other keyword parameter is passed to the function that
            computes the maximum flow.

        Returns
        -------
        flow_value : integer, float
            Value of the maximum flow, i.e., net outflow from the source.

        flow_dict : dict
            A dictionary containing the value of the flow that went through
            each edge.

        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
        --------
        :meth:`maximum_flow_value`
        :meth:`minimum_cut`
        :meth:`minimum_cut_value`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        Notes
        -----
        The function used in the flow_func parameter has to return a residual
        network that follows NetworkX conventions:

        The residual network :samp:`R` from an input graph :samp:`G` has the
        same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
        of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
        self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
        in :samp:`G`.

        For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
        is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
        in :samp:`G` or zero otherwise. If the capacity is infinite,
        :samp:`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
        :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
        :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
        satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.

        The flow value, defined as the total flow into :samp:`t`, the sink, is
        stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
        only edges :samp:`(u, v)` such that
        :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
        :samp:`s`-:samp:`t` cut.

        Specific algorithms may store extra data in :samp:`R`.

        The function should supports an optional boolean parameter value_only. When
        True, it can optionally terminate the algorithm as soon as the maximum flow
        value and the minimum cut can be determined.

        Examples
        --------
        >>> 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)

        maximum_flow returns both the value of the maximum flow and a
        dictionary with all flows.

        >>> flow_value, flow_dict = nx.maximum_flow(G, "x", "y")
        >>> flow_value
        3.0
        >>> print(flow_dict["x"]["b"])
        1.0

        You can also use alternative algorithms for computing the
        maximum flow by using the flow_func parameter.

        >>> from networkx.algorithms.flow import shortest_augmenting_path
        >>> flow_value == nx.maximum_flow(G, "x", "y", flow_func=shortest_augmenting_path)[
        ...     0
        ... ]
        True

        """
        if flow_func is None:
            if kwargs:
                raise nx.NetworkXError(
                    "You have to explicitly set a flow_func if"
                    " you need to pass parameters via kwargs."
                )
            flow_func = default_flow_func

        if not callable(flow_func):
            raise nx.NetworkXError("flow_func has to be callable.")

        R = flow_func(flowG, _s, _t, capacity=capacity, value_only=False, **kwargs)
        flow_dict = build_flow_dict(flowG, R)

        return (R.graph["flow_value"], flow_dict)

    @nx._dispatchable(graphs="flowG", edge_attrs={"capacity": float("inf")})
    def maximum_flow_value(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
        """Find the value of maximum single-commodity flow.

        Parameters
        ----------
        flowG : 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'.

        flow_func : function
            A function for computing the maximum flow among a pair of nodes
            in a capacitated graph. The function has to accept at least three
            parameters: a Graph or Digraph, a source node, and a target node.
            And return a residual network that follows NetworkX conventions
            (see Notes). If flow_func is None, the default maximum
            flow function (:meth:`preflow_push`) is used. See below for
            alternative algorithms. The choice of the default function may change
            from version to version and should not be relied on. Default value:
            None.

        kwargs : Any other keyword parameter is passed to the function that
            computes the maximum flow.

        Returns
        -------
        flow_value : integer, float
            Value of the maximum flow, i.e., net outflow from the source.

        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
        --------
        :meth:`maximum_flow`
        :meth:`minimum_cut`
        :meth:`minimum_cut_value`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        Notes
        -----
        The function used in the flow_func parameter has to return a residual
        network that follows NetworkX conventions:

        The residual network :samp:`R` from an input graph :samp:`G` has the
        same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
        of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
        self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
        in :samp:`G`.

        For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
        is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
        in :samp:`G` or zero otherwise. If the capacity is infinite,
        :samp:`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
        :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
        :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
        satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.

        The flow value, defined as the total flow into :samp:`t`, the sink, is
        stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
        only edges :samp:`(u, v)` such that
        :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
        :samp:`s`-:samp:`t` cut.

        Specific algorithms may store extra data in :samp:`R`.

        The function should supports an optional boolean parameter value_only. When
        True, it can optionally terminate the algorithm as soon as the maximum flow
        value and the minimum cut can be determined.

        Examples
        --------
        >>> 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)

        maximum_flow_value computes only the value of the
        maximum flow:

        >>> flow_value = nx.maximum_flow_value(G, "x", "y")
        >>> flow_value
        3.0

        You can also use alternative algorithms for computing the
        maximum flow by using the flow_func parameter.

        >>> from networkx.algorithms.flow import shortest_augmenting_path
        >>> flow_value == nx.maximum_flow_value(
        ...     G, "x", "y", flow_func=shortest_augmenting_path
        ... )
        True

        """
        if flow_func is None:
            if kwargs:
                raise nx.NetworkXError(
                    "You have to explicitly set a flow_func if"
                    " you need to pass parameters via kwargs."
                )
            flow_func = default_flow_func

        if not callable(flow_func):
            raise nx.NetworkXError("flow_func has to be callable.")

        R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)

        return R.graph["flow_value"]

    @nx._dispatchable(graphs="flowG", edge_attrs={"capacity": float("inf")})
    def minimum_cut(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
        """Compute the value and the node partition of a minimum (s, t)-cut.

        Use the max-flow min-cut theorem, i.e., the capacity of a minimum
        capacity cut is equal to the flow value of a maximum flow.

        Parameters
        ----------
        flowG : 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'.

        flow_func : function
            A function for computing the maximum flow among a pair of nodes
            in a capacitated graph. The function has to accept at least three
            parameters: a Graph or Digraph, a source node, and a target node.
            And return a residual network that follows NetworkX conventions
            (see Notes). If flow_func is None, the default maximum
            flow function (:meth:`preflow_push`) is used. See below for
            alternative algorithms. The choice of the default function may change
            from version to version and should not be relied on. Default value:
            None.

        kwargs : Any other keyword parameter is passed to the function that
            computes the maximum flow.

        Returns
        -------
        cut_value : integer, float
            Value of the minimum cut.

        partition : pair of node sets
            A partitioning of the nodes that defines a minimum cut.

        Raises
        ------
        NetworkXUnbounded
            If the graph has a path of infinite capacity, all cuts have
            infinite capacity and the function raises a NetworkXError.

        See also
        --------
        :meth:`maximum_flow`
        :meth:`maximum_flow_value`
        :meth:`minimum_cut_value`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        Notes
        -----
        The function used in the flow_func parameter has to return a residual
        network that follows NetworkX conventions:

        The residual network :samp:`R` from an input graph :samp:`G` has the
        same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
        of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
        self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
        in :samp:`G`.

        For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
        is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
        in :samp:`G` or zero otherwise. If the capacity is infinite,
        :samp:`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
        :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
        :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
        satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.

        The flow value, defined as the total flow into :samp:`t`, the sink, is
        stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
        only edges :samp:`(u, v)` such that
        :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
        :samp:`s`-:samp:`t` cut.

        Specific algorithms may store extra data in :samp:`R`.

        The function should supports an optional boolean parameter value_only. When
        True, it can optionally terminate the algorithm as soon as the maximum flow
        value and the minimum cut can be determined.

        Examples
        --------
        >>> 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)

        minimum_cut computes both the value of the
        minimum cut and the node partition:

        >>> cut_value, partition = nx.minimum_cut(G, "x", "y")
        >>> reachable, non_reachable = partition

        'partition' here is a tuple with the two sets of nodes that define
        the minimum cut. You can compute the cut set of edges that induce
        the minimum cut as follows:

        >>> cutset = set()
        >>> for u, nbrs in ((n, G[n]) for n in reachable):
        ...     cutset.update((u, v) for v in nbrs if v in non_reachable)
        >>> print(sorted(cutset))
        [('c', 'y'), ('x', 'b')]
        >>> cut_value == sum(G.edges[u, v]["capacity"] for (u, v) in cutset)
        True

        You can also use alternative algorithms for computing the
        minimum cut by using the flow_func parameter.

        >>> from networkx.algorithms.flow import shortest_augmenting_path
        >>> cut_value == nx.minimum_cut(G, "x", "y", flow_func=shortest_augmenting_path)[0]
        True

        """
        if flow_func is None:
            if kwargs:
                raise nx.NetworkXError(
                    "You have to explicitly set a flow_func if"
                    " you need to pass parameters via kwargs."
                )
            flow_func = default_flow_func

        if not callable(flow_func):
            raise nx.NetworkXError("flow_func has to be callable.")

        if kwargs.get("cutoff") is not None and flow_func is preflow_push:
            raise nx.NetworkXError("cutoff should not be specified.")

        R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)
        # Remove saturated edges from the residual network
        cutset = [(u, v, d) for u, v, d in R.edges(data=True) if d["flow"] == d["capacity"]]
        R.remove_edges_from(cutset)

        # Then, reachable and non reachable nodes from source in the
        # residual network form the node partition that defines
        # the minimum cut.
        non_reachable = set(dict(nx.shortest_path_length(R, target=_t)))
        partition = (set(flowG) - non_reachable, non_reachable)
        # Finally add again cutset edges to the residual network to make
        # sure that it is reusable.
        R.add_edges_from(cutset)
        return (R.graph["flow_value"], partition)

    @nx._dispatchable(graphs="flowG", edge_attrs={"capacity": float("inf")})
    def minimum_cut_value(flowG, _s, _t, capacity="capacity", flow_func=None, **kwargs):
        """Compute the value of a minimum (s, t)-cut.

        Use the max-flow min-cut theorem, i.e., the capacity of a minimum
        capacity cut is equal to the flow value of a maximum flow.

        Parameters
        ----------
        flowG : 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'.

        flow_func : function
            A function for computing the maximum flow among a pair of nodes
            in a capacitated graph. The function has to accept at least three
            parameters: a Graph or Digraph, a source node, and a target node.
            And return a residual network that follows NetworkX conventions
            (see Notes). If flow_func is None, the default maximum
            flow function (:meth:`preflow_push`) is used. See below for
            alternative algorithms. The choice of the default function may change
            from version to version and should not be relied on. Default value:
            None.

        kwargs : Any other keyword parameter is passed to the function that
            computes the maximum flow.

        Returns
        -------
        cut_value : integer, float
            Value of the minimum cut.

        Raises
        ------
        NetworkXUnbounded
            If the graph has a path of infinite capacity, all cuts have
            infinite capacity and the function raises a NetworkXError.

        See also
        --------
        :meth:`maximum_flow`
        :meth:`maximum_flow_value`
        :meth:`minimum_cut`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        Notes
        -----
        The function used in the flow_func parameter has to return a residual
        network that follows NetworkX conventions:

        The residual network :samp:`R` from an input graph :samp:`G` has the
        same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair
        of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a
        self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists
        in :samp:`G`.

        For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']`
        is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists
        in :samp:`G` or zero otherwise. If the capacity is infinite,
        :samp:`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
        :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`,
        :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and
        satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`.

        The flow value, defined as the total flow into :samp:`t`, the sink, is
        stored in :samp:`R.graph['flow_value']`. Reachability to :samp:`t` using
        only edges :samp:`(u, v)` such that
        :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum
        :samp:`s`-:samp:`t` cut.

        Specific algorithms may store extra data in :samp:`R`.

        The function should supports an optional boolean parameter value_only. When
        True, it can optionally terminate the algorithm as soon as the maximum flow
        value and the minimum cut can be determined.

        Examples
        --------
        >>> 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)

        minimum_cut_value computes only the value of the
        minimum cut:

        >>> cut_value = nx.minimum_cut_value(G, "x", "y")
        >>> cut_value
        3.0

        You can also use alternative algorithms for computing the
        minimum cut by using the flow_func parameter.

        >>> from networkx.algorithms.flow import shortest_augmenting_path
        >>> cut_value == nx.minimum_cut_value(
        ...     G, "x", "y", flow_func=shortest_augmenting_path
        ... )
        True

        """
        if flow_func is None:
            if kwargs:
                raise nx.NetworkXError(
                    "You have to explicitly set a flow_func if"
                    " you need to pass parameters via kwargs."
                )
            flow_func = default_flow_func

        if not callable(flow_func):
            raise nx.NetworkXError("flow_func has to be callable.")

        if kwargs.get("cutoff") is not None and flow_func is preflow_push:
            raise nx.NetworkXError("cutoff should not be specified.")

        R = flow_func(flowG, _s, _t, capacity=capacity, value_only=True, **kwargs)

        return R.graph["flow_value"]