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# Source code for networkx.algorithms.flow.shortestaugmentingpath

    """
    Shortest augmenting path algorithm for maximum flow problems.
    """

    from collections import deque

    import networkx as nx

    from .edmondskarp import edmonds_karp_core
    from .utils import CurrentEdge, build_residual_network

    __all__ = ["shortest_augmenting_path"]

    def shortest_augmenting_path_impl(G, s, t, capacity, residual, two_phase, cutoff):
        """Implementation of the shortest augmenting path 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

        R_nodes = R.nodes
        R_pred = R.pred
        R_succ = R.succ

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

        # Initialize heights of the nodes.
        heights = {t: 0}
        q = deque([(t, 0)])
        while q:
            u, height = q.popleft()
            height += 1
            for v, attr in R_pred[u].items():
                if v not in heights and attr["flow"] < attr["capacity"]:
                    heights[v] = height
                    q.append((v, height))

        if s not in heights:
            # t is not reachable from s in the residual network. The maximum flow
            # must be zero.
            R.graph["flow_value"] = 0
            return R

        n = len(G)
        m = R.size() / 2

        # Initialize heights and 'current edge' data structures of the nodes.
        for u in R:
            R_nodes[u]["height"] = heights[u] if u in heights else n
            R_nodes[u]["curr_edge"] = CurrentEdge(R_succ[u])

        # Initialize counts of nodes in each level.
        counts = [0] * (2 * n - 1)
        for u in R:
            counts[R_nodes[u]["height"]] += 1

        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 relabel(u):
            """Relabel a node to create an admissible edge."""
            height = n - 1
            for v, attr in R_succ[u].items():
                if attr["flow"] < attr["capacity"]:
                    height = min(height, R_nodes[v]["height"])
            return height + 1

        if cutoff is None:
            cutoff = float("inf")

        # Phase 1: Look for shortest augmenting paths using depth-first search.

        flow_value = 0
        path = [s]
        u = s
        d = n if not two_phase else int(min(m**0.5, 2 * n ** (2.0 / 3)))
        done = R_nodes[s]["height"] >= d
        while not done:
            height = R_nodes[u]["height"]
            curr_edge = R_nodes[u]["curr_edge"]
            # Depth-first search for the next node on the path to t.
            while True:
                v, attr = curr_edge.get()
                if height == R_nodes[v]["height"] + 1 and attr["flow"] < attr["capacity"]:
                    # Advance to the next node following an admissible edge.
                    path.append(v)
                    u = v
                    break
                try:
                    curr_edge.move_to_next()
                except StopIteration:
                    counts[height] -= 1
                    if counts[height] == 0:
                        # Gap heuristic: If relabeling causes a level to become
                        # empty, a minimum cut has been identified. The algorithm
                        # can now be terminated.
                        R.graph["flow_value"] = flow_value
                        return R
                    height = relabel(u)
                    if u == s and height >= d:
                        if not two_phase:
                            # t is disconnected from s in the residual network. No
                            # more augmenting paths exist.
                            R.graph["flow_value"] = flow_value
                            return R
                        else:
                            # t is at least d steps away from s. End of phase 1.
                            done = True
                            break
                    counts[height] += 1
                    R_nodes[u]["height"] = height
                    if u != s:
                        # After relabeling, the last edge on the path is no longer
                        # admissible. Retreat one step to look for an alternative.
                        path.pop()
                        u = path[-1]
                        break
            if u == t:
                # t is reached. Augment flow along the path and reset it for a new
                # depth-first search.
                flow_value += augment(path)
                if flow_value >= cutoff:
                    R.graph["flow_value"] = flow_value
                    return R
                path = [s]
                u = s

        # Phase 2: Look for shortest augmenting paths using breadth-first search.
        flow_value += edmonds_karp_core(R, s, t, cutoff - flow_value)

        R.graph["flow_value"] = flow_value
        return R

    @nx._dispatchable(edge_attrs={"capacity": float("inf")}, returns_graph=True)
    def shortest_augmenting_path(
        G,
        s,
        t,
        capacity="capacity",
        residual=None,
        value_only=False,
        two_phase=False,
        cutoff=None,
    ):
        r"""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
        --------
        :meth:`maximum_flow`
        :meth:`minimum_cut`
        :meth:`edmonds_karp`
        :meth:`preflow_push`

        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 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

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