xhs-note-crawling/python/py/Lib/site-packages/networkx/algorithms/connectivity/stoerwagner.py

"""
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