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# Source code for networkx.algorithms.flow.edmondskarp

    """
    Edmonds-Karp algorithm for maximum flow problems.
    """

    import networkx as nx
    from networkx.algorithms.flow.utils import build_residual_network

    __all__ = ["edmonds_karp"]

    def edmonds_karp_core(R, s, t, cutoff):
        """Implementation of the Edmonds-Karp algorithm."""
        R_nodes = R.nodes
        R_pred = R.pred
        R_succ = R.succ

        inf = R.graph["inf"]

        def augment(path):
            """Augment flow along a path from s to t."""
            # Determine the path residual capacity.
            flow = inf
            it = iter(path)
            u = next(it)
            for v in it:
                attr = R_succ[u][v]
                flow = min(flow, attr["capacity"] - attr["flow"])
                u = v
            if flow * 2 > inf:
                raise nx.NetworkXUnbounded("Infinite capacity path, flow unbounded above.")
            # Augment flow along the path.
            it = iter(path)
            u = next(it)
            for v in it:
                R_succ[u][v]["flow"] += flow
                R_succ[v][u]["flow"] -= flow
                u = v
            return flow

        def bidirectional_bfs():
            """Bidirectional breadth-first search for an augmenting path."""
            pred = {s: None}
            q_s = [s]
            succ = {t: None}
            q_t = [t]
            while True:
                q = []
                if len(q_s) <= len(q_t):
                    for u in q_s:
                        for v, attr in R_succ[u].items():
                            if v not in pred and attr["flow"] < attr["capacity"]:
                                pred[v] = u
                                if v in succ:
                                    return v, pred, succ
                                q.append(v)
                    if not q:
                        return None, None, None
                    q_s = q
                else:
                    for u in q_t:
                        for v, attr in R_pred[u].items():
                            if v not in succ and attr["flow"] < attr["capacity"]:
                                succ[v] = u
                                if v in pred:
                                    return v, pred, succ
                                q.append(v)
                    if not q:
                        return None, None, None
                    q_t = q

        # Look for shortest augmenting paths using breadth-first search.
        flow_value = 0
        while flow_value < cutoff:
            v, pred, succ = bidirectional_bfs()
            if pred is None:
                break
            path = [v]
            # Trace a path from s to v.
            u = v
            while u != s:
                u = pred[u]
                path.append(u)
            path.reverse()
            # Trace a path from v to t.
            u = v
            while u != t:
                u = succ[u]
                path.append(u)
            flow_value += augment(path)

        return flow_value

    def edmonds_karp_impl(G, s, t, capacity, residual, cutoff):
        """Implementation of the Edmonds-Karp algorithm."""
        if s not in G:
            raise nx.NetworkXError(f"node {str(s)} not in graph")
        if t not in G:
            raise nx.NetworkXError(f"node {str(t)} not in graph")
        if s == t:
            raise nx.NetworkXError("source and sink are the same node")

        if residual is None:
            R = build_residual_network(G, capacity)
        else:
            R = residual

        # Initialize/reset the residual network.
        for u in R:
            for e in R[u].values():
                e["flow"] = 0

        if cutoff is None:
            cutoff = float("inf")
        R.graph["flow_value"] = edmonds_karp_core(R, s, t, cutoff)

        return R

    @nx._dispatchable(edge_attrs={"capacity": float("inf")}, returns_graph=True)
    def edmonds_karp(
        G, s, t, capacity="capacity", residual=None, value_only=False, cutoff=None
    ):
        """Find a maximum single-commodity flow using the Edmonds-Karp 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 $O(n m^2)$ 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.

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

        Notes
        -----
        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']`. If :samp:`cutoff` is not
        specified, 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.

        Examples
        --------
        >>> from networkx.algorithms.flow import edmonds_karp

        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 = edmonds_karp(G, "x", "y")
        >>> flow_value = nx.maximum_flow_value(G, "x", "y")
        >>> flow_value
        3.0
        >>> flow_value == R.graph["flow_value"]
        True

        """
        R = edmonds_karp_impl(G, s, t, capacity, residual, cutoff)
        R.graph["algorithm"] = "edmonds_karp"
        nx._clear_cache(R)
        return R