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# Source code for networkx.algorithms.connectivity.stoerwagner

    """
    Stoer-Wagner minimum cut algorithm.
    """

    from itertools import islice

    import networkx as nx

    from ...utils import BinaryHeap, arbitrary_element, not_implemented_for

    __all__ = ["stoer_wagner"]

    @not_implemented_for("directed")
    @not_implemented_for("multigraph")
    @nx._dispatchable(edge_attrs="weight")
    def stoer_wagner(G, weight="weight", heap=BinaryHeap):
        r"""Returns the weighted minimum edge cut using the Stoer-Wagner algorithm.

        Determine the minimum edge cut of a connected graph using the
        Stoer-Wagner algorithm. In weighted cases, all weights must be
        nonnegative.

        The running time of the algorithm depends on the type of heaps used:

        ============== =============================================
        Type of heap   Running time
        ============== =============================================
        Binary heap    $O(n (m + n) \log n)$
        Fibonacci heap $O(nm + n^2 \log n)$
        Pairing heap   $O(2^{2 \sqrt{\log \log n}} nm + n^2 \log n)$
        ============== =============================================

        Parameters
        ----------
        G : NetworkX graph
            Edges of the graph are expected to have an attribute named by the
            weight parameter below. If this attribute is not present, the edge is
            considered to have unit weight.

        weight : string
            Name of the weight attribute of the edges. If the attribute is not
            present, unit weight is assumed. Default value: 'weight'.

        heap : class
            Type of heap to be used in the algorithm. It should be a subclass of
            :class:`MinHeap` or implement a compatible interface.

            If a stock heap implementation is to be used, :class:`BinaryHeap` is
            recommended over :class:`PairingHeap` for Python implementations without
            optimized attribute accesses (e.g., CPython) despite a slower
            asymptotic running time. For Python implementations with optimized
            attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better
            performance. Default value: :class:`BinaryHeap`.

        Returns
        -------
        cut_value : integer or float
            The sum of weights of edges in a minimum cut.

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

        Raises
        ------
        NetworkXNotImplemented
            If the graph is directed or a multigraph.

        NetworkXError
            If the graph has less than two nodes, is not connected or has a
            negative-weighted edge.

        Examples
        --------
        >>> G = nx.Graph()
        >>> G.add_edge("x", "a", weight=3)
        >>> G.add_edge("x", "b", weight=1)
        >>> G.add_edge("a", "c", weight=3)
        >>> G.add_edge("b", "c", weight=5)
        >>> G.add_edge("b", "d", weight=4)
        >>> G.add_edge("d", "e", weight=2)
        >>> G.add_edge("c", "y", weight=2)
        >>> G.add_edge("e", "y", weight=3)
        >>> cut_value, partition = nx.stoer_wagner(G)
        >>> cut_value
        4
        """
        n = len(G)
        if n < 2:
            raise nx.NetworkXError("graph has less than two nodes.")
        if not nx.is_connected(G):
            raise nx.NetworkXError("graph is not connected.")

        # Make a copy of the graph for internal use.
        G = nx.Graph(
            (u, v, {"weight": e.get(weight, 1)}) for u, v, e in G.edges(data=True) if u != v
        )
        G.__networkx_cache__ = None  # Disable caching

        for u, v, e in G.edges(data=True):
            if e["weight"] < 0:
                raise nx.NetworkXError("graph has a negative-weighted edge.")

        cut_value = float("inf")
        nodes = set(G)
        contractions = []  # contracted node pairs

        # Repeatedly pick a pair of nodes to contract until only one node is left.
        for i in range(n - 1):
            # Pick an arbitrary node u and create a set A = {u}.
            u = arbitrary_element(G)
            A = {u}
            # Repeatedly pick the node "most tightly connected" to A and add it to
            # A. The tightness of connectivity of a node not in A is defined by the
            # of edges connecting it to nodes in A.
            h = heap()  # min-heap emulating a max-heap
            for v, e in G[u].items():
                h.insert(v, -e["weight"])
            # Repeat until all but one node has been added to A.
            for j in range(n - i - 2):
                u = h.pop()[0]
                A.add(u)
                for v, e in G[u].items():
                    if v not in A:
                        h.insert(v, h.get(v, 0) - e["weight"])
            # A and the remaining node v define a "cut of the phase". There is a
            # minimum cut of the original graph that is also a cut of the phase.
            # Due to contractions in earlier phases, v may in fact represent
            # multiple nodes in the original graph.
            v, w = h.min()
            w = -w
            if w < cut_value:
                cut_value = w
                best_phase = i
            # Contract v and the last node added to A.
            contractions.append((u, v))
            for w, e in G[v].items():
                if w != u:
                    if w not in G[u]:
                        G.add_edge(u, w, weight=e["weight"])
                    else:
                        G[u][w]["weight"] += e["weight"]
            G.remove_node(v)

        # Recover the optimal partitioning from the contractions.
        G = nx.Graph(islice(contractions, best_phase))
        v = contractions[best_phase][1]
        G.add_node(v)
        reachable = set(nx.single_source_shortest_path_length(G, v))
        partition = (list(reachable), list(nodes - reachable))

        return cut_value, partition