# Copyright 2016 NetworkX developers.
# Copyright (C) 2004-2018 by
#   Aric Hagberg <hagberg@lanl.gov>
#   Dan Schult <dschult@colgate.edu>
#   Pieter Swart <swart@lanl.gov>
#
# Copyright (C) 2008 by
#   Joris van Rantwijk.
#
# Copyright (C) 2011 by
#   Nicholas Mancuso <nick.mancuso@gmail.com>
#
# All rights reserved.
# BSD license.
"""Functions for computing and verifying matchings in a graph."""
from itertools import combinations
from itertools import repeat

__all__ = ['is_matching', 'is_maximal_matching', 'max_weight_matching',
           'maximal_matching']


def maximal_matching(G):
    r"""Find a maximal matching in the graph.

    A matching is a subset of edges in which no node occurs more than once.
    A maximal matching cannot add more edges and still be a matching.

    Parameters
    ----------
    G : NetworkX graph
        Undirected graph

    Returns
    -------
    matching : set
        A maximal matching of the graph.

    Notes
    -----
    The algorithm greedily selects a maximal matching M of the graph G
    (i.e. no superset of M exists). It runs in $O(|E|)$ time.
    """
    matching = set()
    nodes = set()
    for u, v in G.edges():
        # If the edge isn't covered, add it to the matching
        # then remove neighborhood of u and v from consideration.
        if u not in nodes and v not in nodes and u != v:
            matching.add((u, v))
            nodes.add(u)
            nodes.add(v)
    return matching


def matching_dict_to_set(matching):
    """Converts a dictionary representing a matching (as returned by
    :func:`max_weight_matching`) to a set representing a matching (as
    returned by :func:`maximal_matching`).

    In the definition of maximal matching adopted by NetworkX,
    self-loops are not allowed, so the provided dictionary is expected
    to never have any mapping from a key to itself. However, the
    dictionary is expected to have mirrored key/value pairs, for
    example, key ``u`` with value ``v`` and key ``v`` with value ``u``.

    """
    # Need to compensate for the fact that both pairs (u, v) and (v, u)
    # appear in `matching.items()`, so we use a set of sets. This way,
    # only the (frozen)set `{u, v}` appears as an element in the
    # returned set.

    return set((u, v) for (u, v) in set(map(frozenset, matching.items())))


def is_matching(G, matching):
    """Decides whether the given set or dictionary represents a valid
    matching in ``G``.

    A *matching* in a graph is a set of edges in which no two distinct
    edges share a common endpoint.

    Parameters
    ----------
    G : NetworkX graph

    matching : dict or set
        A dictionary or set representing a matching. If a dictionary, it
        must have ``matching[u] == v`` and ``matching[v] == u`` for each
        edge ``(u, v)`` in the matching. If a set, it must have elements
        of the form ``(u, v)``, where ``(u, v)`` is an edge in the
        matching.

    Returns
    -------
    bool
        Whether the given set or dictionary represents a valid matching
        in the graph.

    """
    if isinstance(matching, dict):
        matching = matching_dict_to_set(matching)
    # TODO This is parallelizable.
    return all(len(set(e1) & set(e2)) == 0
               for e1, e2 in combinations(matching, 2))


def is_maximal_matching(G, matching):
    """Decides whether the given set or dictionary represents a valid
    maximal matching in ``G``.

    A *maximal matching* in a graph is a matching in which adding any
    edge would cause the set to no longer be a valid matching.

    Parameters
    ----------
    G : NetworkX graph

    matching : dict or set
        A dictionary or set representing a matching. If a dictionary, it
        must have ``matching[u] == v`` and ``matching[v] == u`` for each
        edge ``(u, v)`` in the matching. If a set, it must have elements
        of the form ``(u, v)``, where ``(u, v)`` is an edge in the
        matching.

    Returns
    -------
    bool
        Whether the given set or dictionary represents a valid maximal
        matching in the graph.

    """
    if isinstance(matching, dict):
        matching = matching_dict_to_set(matching)
    # If the given set is not a matching, then it is not a maximal matching.
    if not is_matching(G, matching):
        return False
    # A matching is maximal if adding any unmatched edge to it causes
    # the resulting set to *not* be a valid matching.
    #
    # HACK Since the `matching_dict_to_set` function returns a set of
    # sets, we need to convert the list of edges to a set of sets in
    # order for the set difference function to work. Ideally, we would
    # just be able to do `set(G.edges()) - matching`.
    all_edges = set(map(frozenset, G.edges()))
    matched_edges = set(map(frozenset, matching))
    unmatched_edges = all_edges - matched_edges
    # TODO This is parallelizable.
    return all(not is_matching(G, matching | {e}) for e in unmatched_edges)


def max_weight_matching(G, maxcardinality=False, weight='weight'):
    """Compute a maximum-weighted matching of G.

    A matching is a subset of edges in which no node occurs more than once.
    The weight of a matching is the sum of the weights of its edges.
    A maximal matching cannot add more edges and still be a matching.
    The cardinality of a matching is the number of matched edges.

    Parameters
    ----------
    G : NetworkX graph
      Undirected graph

    maxcardinality: bool, optional (default=False)
       If maxcardinality is True, compute the maximum-cardinality matching
       with maximum weight among all maximum-cardinality matchings.

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight.
       If key not found, uses 1 as weight.


    Returns
    -------
    matching : set
        A maximal matching of the graph.

    Notes
    -----
    If G has edges with weight attributes the edge data are used as
    weight values else the weights are assumed to be 1.

    This function takes time O(number_of_nodes ** 3).

    If all edge weights are integers, the algorithm uses only integer
    computations.  If floating point weights are used, the algorithm
    could return a slightly suboptimal matching due to numeric
    precision errors.

    This method is based on the "blossom" method for finding augmenting
    paths and the "primal-dual" method for finding a matching of maximum
    weight, both methods invented by Jack Edmonds [1]_.

    Bipartite graphs can also be matched using the functions present in
    :mod:`networkx.algorithms.bipartite.matching`.

    References
    ----------
    .. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
       Zvi Galil, ACM Computing Surveys, 1986.
    """
    #
    # The algorithm is taken from "Efficient Algorithms for Finding Maximum
    # Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
    # It is based on the "blossom" method for finding augmenting paths and
    # the "primal-dual" method for finding a matching of maximum weight, both
    # methods invented by Jack Edmonds.
    #
    # A C program for maximum weight matching by Ed Rothberg was used
    # extensively to validate this new code.
    #
    # Many terms used in the code comments are explained in the paper
    # by Galil. You will probably need the paper to make sense of this code.
    #

    class NoNode:
        """Dummy value which is different from any node."""
        pass

    class Blossom:
        """Representation of a non-trivial blossom or sub-blossom."""

        __slots__ = ['childs', 'edges', 'mybestedges']

        # b.childs is an ordered list of b's sub-blossoms, starting with
        # the base and going round the blossom.

        # b.edges is the list of b's connecting edges, such that
        # b.edges[i] = (v, w) where v is a vertex in b.childs[i]
        # and w is a vertex in b.childs[wrap(i+1)].

        # If b is a top-level S-blossom,
        # b.mybestedges is a list of least-slack edges to neighbouring
        # S-blossoms, or None if no such list has been computed yet.
        # This is used for efficient computation of delta3.

        # Generate the blossom's leaf vertices.
        def leaves(self):
            for t in self.childs:
                if isinstance(t, Blossom):
                    for v in t.leaves():
                        yield v
                else:
                    yield t

    # Get a list of vertices.
    gnodes = list(G)
    if not gnodes:
        return set()  # don't bother with empty graphs

    # Find the maximum edge weight.
    maxweight = 0
    allinteger = True
    for i, j, d in G.edges(data=True):
        wt = d.get(weight, 1)
        if i != j and wt > maxweight:
            maxweight = wt
        allinteger = allinteger and (str(type(wt)).split("'")[1]
                                     in ('int', 'long'))

    # If v is a matched vertex, mate[v] is its partner vertex.
    # If v is a single vertex, v does not occur as a key in mate.
    # Initially all vertices are single; updated during augmentation.
    mate = {}

    # If b is a top-level blossom,
    # label.get(b) is None if b is unlabeled (free),
    #                 1 if b is an S-blossom,
    #                 2 if b is a T-blossom.
    # The label of a vertex is found by looking at the label of its top-level
    # containing blossom.
    # If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
    # from an S-vertex outside the blossom.
    # Labels are assigned during a stage and reset after each augmentation.
    label = {}

    # If b is a labeled top-level blossom,
    # labeledge[b] = (v, w) is the edge through which b obtained its label
    # such that w is a vertex in b, or None if b's base vertex is single.
    # If w is a vertex inside a T-blossom and label[w] == 2,
    # labeledge[w] = (v, w) is an edge through which w is reachable from
    # outside the blossom.
    labeledge = {}

    # If v is a vertex, inblossom[v] is the top-level blossom to which v
    # belongs.
    # If v is a top-level vertex, inblossom[v] == v since v is itself
    # a (trivial) top-level blossom.
    # Initially all vertices are top-level trivial blossoms.
    inblossom = dict(zip(gnodes, gnodes))

    # If b is a sub-blossom,
    # blossomparent[b] is its immediate parent (sub-)blossom.
    # If b is a top-level blossom, blossomparent[b] is None.
    blossomparent = dict(zip(gnodes, repeat(None)))

    # If b is a (sub-)blossom,
    # blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
    blossombase = dict(zip(gnodes, gnodes))

    # If w is a free vertex (or an unreached vertex inside a T-blossom),
    # bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
    # or None if there is no such edge.
    # If b is a (possibly trivial) top-level S-blossom,
    # bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
    # (v inside b), or None if there is no such edge.
    # This is used for efficient computation of delta2 and delta3.
    bestedge = {}

    # If v is a vertex,
    # dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
    # optimization problem (if all edge weights are integers, multiplication
    # by two ensures that all values remain integers throughout the algorithm).
    # Initially, u(v) = maxweight / 2.
    dualvar = dict(zip(gnodes, repeat(maxweight)))

    # If b is a non-trivial blossom,
    # blossomdual[b] = z(b) where z(b) is b's variable in the dual
    # optimization problem.
    blossomdual = {}

    # If (v, w) in allowedge or (w, v) in allowedg, then the edge
    # (v, w) is known to have zero slack in the optimization problem;
    # otherwise the edge may or may not have zero slack.
    allowedge = {}

    # Queue of newly discovered S-vertices.
    queue = []

    # Return 2 * slack of edge (v, w) (does not work inside blossoms).
    def slack(v, w):
        return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)

    # Assign label t to the top-level blossom containing vertex w,
    # coming through an edge from vertex v.
    def assignLabel(w, t, v):
        b = inblossom[w]
        assert label.get(w) is None and label.get(b) is None
        label[w] = label[b] = t
        if v is not None:
            labeledge[w] = labeledge[b] = (v, w)
        else:
            labeledge[w] = labeledge[b] = None
        bestedge[w] = bestedge[b] = None
        if t == 1:
            # b became an S-vertex/blossom; add it(s vertices) to the queue.
            if isinstance(b, Blossom):
                queue.extend(b.leaves())
            else:
                queue.append(b)
        elif t == 2:
            # b became a T-vertex/blossom; assign label S to its mate.
            # (If b is a non-trivial blossom, its base is the only vertex
            # with an external mate.)
            base = blossombase[b]
            assignLabel(mate[base], 1, base)

    # Trace back from vertices v and w to discover either a new blossom
    # or an augmenting path. Return the base vertex of the new blossom,
    # or NoNode if an augmenting path was found.
    def scanBlossom(v, w):
        # Trace back from v and w, placing breadcrumbs as we go.
        path = []
        base = NoNode
        while v is not NoNode:
            # Look for a breadcrumb in v's blossom or put a new breadcrumb.
            b = inblossom[v]
            if label[b] & 4:
                base = blossombase[b]
                break
            assert label[b] == 1
            path.append(b)
            label[b] = 5
            # Trace one step back.
            if labeledge[b] is None:
                # The base of blossom b is single; stop tracing this path.
                assert blossombase[b] not in mate
                v = NoNode
            else:
                assert labeledge[b][0] == mate[blossombase[b]]
                v = labeledge[b][0]
                b = inblossom[v]
                assert label[b] == 2
                # b is a T-blossom; trace one more step back.
                v = labeledge[b][0]
            # Swap v and w so that we alternate between both paths.
            if w is not NoNode:
                v, w = w, v
        # Remove breadcrumbs.
        for b in path:
            label[b] = 1
        # Return base vertex, if we found one.
        return base

    # Construct a new blossom with given base, through S-vertices v and w.
    # Label the new blossom as S; set its dual variable to zero;
    # relabel its T-vertices to S and add them to the queue.
    def addBlossom(base, v, w):
        bb = inblossom[base]
        bv = inblossom[v]
        bw = inblossom[w]
        # Create blossom.
        b = Blossom()
        blossombase[b] = base
        blossomparent[b] = None
        blossomparent[bb] = b
        # Make list of sub-blossoms and their interconnecting edge endpoints.
        b.childs = path = []
        b.edges = edgs = [(v, w)]
        # Trace back from v to base.
        while bv != bb:
            # Add bv to the new blossom.
            blossomparent[bv] = b
            path.append(bv)
            edgs.append(labeledge[bv])
            assert label[bv] == 2 or (label[bv] == 1 and labeledge[
                                      bv][0] == mate[blossombase[bv]])
            # Trace one step back.
            v = labeledge[bv][0]
            bv = inblossom[v]
        # Add base sub-blossom; reverse lists.
        path.append(bb)
        path.reverse()
        edgs.reverse()
        # Trace back from w to base.
        while bw != bb:
            # Add bw to the new blossom.
            blossomparent[bw] = b
            path.append(bw)
            edgs.append((labeledge[bw][1], labeledge[bw][0]))
            assert label[bw] == 2 or (label[bw] == 1 and labeledge[
                                      bw][0] == mate[blossombase[bw]])
            # Trace one step back.
            w = labeledge[bw][0]
            bw = inblossom[w]
        # Set label to S.
        assert label[bb] == 1
        label[b] = 1
        labeledge[b] = labeledge[bb]
        # Set dual variable to zero.
        blossomdual[b] = 0
        # Relabel vertices.
        for v in b.leaves():
            if label[inblossom[v]] == 2:
                # This T-vertex now turns into an S-vertex because it becomes
                # part of an S-blossom; add it to the queue.
                queue.append(v)
            inblossom[v] = b
        # Compute b.mybestedges.
        bestedgeto = {}
        for bv in path:
            if isinstance(bv, Blossom):
                if bv.mybestedges is not None:
                    # Walk this subblossom's least-slack edges.
                    nblist = bv.mybestedges
                    # The sub-blossom won't need this data again.
                    bv.mybestedges = None
                else:
                    # This subblossom does not have a list of least-slack
                    # edges; get the information from the vertices.
                    nblist = [(v, w)
                              for v in bv.leaves()
                              for w in G.neighbors(v)
                              if v != w]
            else:
                nblist = [(bv, w)
                          for w in G.neighbors(bv)
                          if bv != w]
            for k in nblist:
                (i, j) = k
                if inblossom[j] == b:
                    i, j = j, i
                bj = inblossom[j]
                if (bj != b and label.get(bj) == 1 and
                    ((bj not in bestedgeto) or
                     slack(i, j) < slack(*bestedgeto[bj]))):
                    bestedgeto[bj] = k
            # Forget about least-slack edge of the subblossom.
            bestedge[bv] = None
        b.mybestedges = list(bestedgeto.values())
        # Select bestedge[b].
        mybestedge = None
        bestedge[b] = None
        for k in b.mybestedges:
            kslack = slack(*k)
            if mybestedge is None or kslack < mybestslack:
                mybestedge = k
                mybestslack = kslack
        bestedge[b] = mybestedge

    # Expand the given top-level blossom.
    def expandBlossom(b, endstage):
        # Convert sub-blossoms into top-level blossoms.
        for s in b.childs:
            blossomparent[s] = None
            if isinstance(s, Blossom):
                if endstage and blossomdual[s] == 0:
                    # Recursively expand this sub-blossom.
                    expandBlossom(s, endstage)
                else:
                    for v in s.leaves():
                        inblossom[v] = s
            else:
                inblossom[s] = s
        # If we expand a T-blossom during a stage, its sub-blossoms must be
        # relabeled.
        if (not endstage) and label.get(b) == 2:
            # Start at the sub-blossom through which the expanding
            # blossom obtained its label, and relabel sub-blossoms untili
            # we reach the base.
            # Figure out through which sub-blossom the expanding blossom
            # obtained its label initially.
            entrychild = inblossom[labeledge[b][1]]
            # Decide in which direction we will go round the blossom.
            j = b.childs.index(entrychild)
            if j & 1:
                # Start index is odd; go forward and wrap.
                j -= len(b.childs)
                jstep = 1
            else:
                # Start index is even; go backward.
                jstep = -1
            # Move along the blossom until we get to the base.
            v, w = labeledge[b]
            while j != 0:
                # Relabel the T-sub-blossom.
                if jstep == 1:
                    p, q = b.edges[j]
                else:
                    q, p = b.edges[j - 1]
                label[w] = None
                label[q] = None
                assignLabel(w, 2, v)
                # Step to the next S-sub-blossom and note its forward edge.
                allowedge[(p, q)] = allowedge[(q, p)] = True
                j += jstep
                if jstep == 1:
                    v, w = b.edges[j]
                else:
                    w, v = b.edges[j - 1]
                # Step to the next T-sub-blossom.
                allowedge[(v, w)] = allowedge[(w, v)] = True
                j += jstep
            # Relabel the base T-sub-blossom WITHOUT stepping through to
            # its mate (so don't call assignLabel).
            bw = b.childs[j]
            label[w] = label[bw] = 2
            labeledge[w] = labeledge[bw] = (v, w)
            bestedge[bw] = None
            # Continue along the blossom until we get back to entrychild.
            j += jstep
            while b.childs[j] != entrychild:
                # Examine the vertices of the sub-blossom to see whether
                # it is reachable from a neighbouring S-vertex outside the
                # expanding blossom.
                bv = b.childs[j]
                if label.get(bv) == 1:
                    # This sub-blossom just got label S through one of its
                    # neighbours; leave it be.
                    j += jstep
                    continue
                if isinstance(bv, Blossom):
                    for v in bv.leaves():
                        if label.get(v):
                            break
                else:
                    v = bv
                # If the sub-blossom contains a reachable vertex, assign
                # label T to the sub-blossom.
                if label.get(v):
                    assert label[v] == 2
                    assert inblossom[v] == bv
                    label[v] = None
                    label[mate[blossombase[bv]]] = None
                    assignLabel(v, 2, labeledge[v][0])
                j += jstep
        # Remove the expanded blossom entirely.
        label.pop(b, None)
        labeledge.pop(b, None)
        bestedge.pop(b, None)
        del blossomparent[b]
        del blossombase[b]
        del blossomdual[b]

    # Swap matched/unmatched edges over an alternating path through blossom b
    # between vertex v and the base vertex. Keep blossom bookkeeping
    # consistent.
    def augmentBlossom(b, v):
        # Bubble up through the blossom tree from vertex v to an immediate
        # sub-blossom of b.
        t = v
        while blossomparent[t] != b:
            t = blossomparent[t]
        # Recursively deal with the first sub-blossom.
        if isinstance(t, Blossom):
            augmentBlossom(t, v)
        # Decide in which direction we will go round the blossom.
        i = j = b.childs.index(t)
        if i & 1:
            # Start index is odd; go forward and wrap.
            j -= len(b.childs)
            jstep = 1
        else:
            # Start index is even; go backward.
            jstep = -1
        # Move along the blossom until we get to the base.
        while j != 0:
            # Step to the next sub-blossom and augment it recursively.
            j += jstep
            t = b.childs[j]
            if jstep == 1:
                w, x = b.edges[j]
            else:
                x, w = b.edges[j - 1]
            if isinstance(t, Blossom):
                augmentBlossom(t, w)
            # Step to the next sub-blossom and augment it recursively.
            j += jstep
            t = b.childs[j]
            if isinstance(t, Blossom):
                augmentBlossom(t, x)
            # Match the edge connecting those sub-blossoms.
            mate[w] = x
            mate[x] = w
        # Rotate the list of sub-blossoms to put the new base at the front.
        b.childs = b.childs[i:] + b.childs[:i]
        b.edges = b.edges[i:] + b.edges[:i]
        blossombase[b] = blossombase[b.childs[0]]
        assert blossombase[b] == v

    # Swap matched/unmatched edges over an alternating path between two
    # single vertices. The augmenting path runs through S-vertices v and w.
    def augmentMatching(v, w):
        for (s, j) in ((v, w), (w, v)):
            # Match vertex s to vertex j. Then trace back from s
            # until we find a single vertex, swapping matched and unmatched
            # edges as we go.
            while 1:
                bs = inblossom[s]
                assert label[bs] == 1
                assert (
                    labeledge[bs] is None and blossombase[bs] not in mate)\
                    or (labeledge[bs][0] == mate[blossombase[bs]])
                # Augment through the S-blossom from s to base.
                if isinstance(bs, Blossom):
                    augmentBlossom(bs, s)
                # Update mate[s]
                mate[s] = j
                # Trace one step back.
                if labeledge[bs] is None:
                    # Reached single vertex; stop.
                    break
                t = labeledge[bs][0]
                bt = inblossom[t]
                assert label[bt] == 2
                # Trace one more step back.
                s, j = labeledge[bt]
                # Augment through the T-blossom from j to base.
                assert blossombase[bt] == t
                if isinstance(bt, Blossom):
                    augmentBlossom(bt, j)
                # Update mate[j]
                mate[j] = s

    # Verify that the optimum solution has been reached.
    def verifyOptimum():
        if maxcardinality:
            # Vertices may have negative dual;
            # find a constant non-negative number to add to all vertex duals.
            vdualoffset = max(0, -min(dualvar.values()))
        else:
            vdualoffset = 0
        # 0. all dual variables are non-negative
        assert min(dualvar.values()) + vdualoffset >= 0
        assert len(blossomdual) == 0 or min(blossomdual.values()) >= 0
        # 0. all edges have non-negative slack and
        # 1. all matched edges have zero slack;
        for i, j, d in G.edges(data=True):
            wt = d.get(weight, 1)
            if i == j:
                continue  # ignore self-loops
            s = dualvar[i] + dualvar[j] - 2 * wt
            iblossoms = [i]
            jblossoms = [j]
            while blossomparent[iblossoms[-1]] is not None:
                iblossoms.append(blossomparent[iblossoms[-1]])
            while blossomparent[jblossoms[-1]] is not None:
                jblossoms.append(blossomparent[jblossoms[-1]])
            iblossoms.reverse()
            jblossoms.reverse()
            for (bi, bj) in zip(iblossoms, jblossoms):
                if bi != bj:
                    break
                s += 2 * blossomdual[bi]
            assert s >= 0
            if mate.get(i) == j or mate.get(j) == i:
                assert mate[i] == j and mate[j] == i
                assert s == 0
        # 2. all single vertices have zero dual value;
        for v in gnodes:
            assert (v in mate) or dualvar[v] + vdualoffset == 0
        # 3. all blossoms with positive dual value are full.
        for b in blossomdual:
            if blossomdual[b] > 0:
                assert len(b.edges) % 2 == 1
                for (i, j) in b.edges[1::2]:
                    assert mate[i] == j and mate[j] == i
        # Ok.

    # Main loop: continue until no further improvement is possible.
    while 1:

        # Each iteration of this loop is a "stage".
        # A stage finds an augmenting path and uses that to improve
        # the matching.

        # Remove labels from top-level blossoms/vertices.
        label.clear()
        labeledge.clear()

        # Forget all about least-slack edges.
        bestedge.clear()
        for b in blossomdual:
            b.mybestedges = None

        # Loss of labeling means that we can not be sure that currently
        # allowable edges remain allowable througout this stage.
        allowedge.clear()

        # Make queue empty.
        queue[:] = []

        # Label single blossoms/vertices with S and put them in the queue.
        for v in gnodes:
            if (v not in mate) and label.get(inblossom[v]) is None:
                assignLabel(v, 1, None)

        # Loop until we succeed in augmenting the matching.
        augmented = 0
        while 1:

            # Each iteration of this loop is a "substage".
            # A substage tries to find an augmenting path;
            # if found, the path is used to improve the matching and
            # the stage ends. If there is no augmenting path, the
            # primal-dual method is used to pump some slack out of
            # the dual variables.

            # Continue labeling until all vertices which are reachable
            # through an alternating path have got a label.
            while queue and not augmented:

                # Take an S vertex from the queue.
                v = queue.pop()
                assert label[inblossom[v]] == 1

                # Scan its neighbours:
                for w in G.neighbors(v):
                    if w == v:
                        continue  # ignore self-loops
                    # w is a neighbour to v
                    bv = inblossom[v]
                    bw = inblossom[w]
                    if bv == bw:
                        # this edge is internal to a blossom; ignore it
                        continue
                    if (v, w) not in allowedge:
                        kslack = slack(v, w)
                        if kslack <= 0:
                            # edge k has zero slack => it is allowable
                            allowedge[(v, w)] = allowedge[(w, v)] = True
                    if (v, w) in allowedge:
                        if label.get(bw) is None:
                            # (C1) w is a free vertex;
                            # label w with T and label its mate with S (R12).
                            assignLabel(w, 2, v)
                        elif label.get(bw) == 1:
                            # (C2) w is an S-vertex (not in the same blossom);
                            # follow back-links to discover either an
                            # augmenting path or a new blossom.
                            base = scanBlossom(v, w)
                            if base is not NoNode:
                                # Found a new blossom; add it to the blossom
                                # bookkeeping and turn it into an S-blossom.
                                addBlossom(base, v, w)
                            else:
                                # Found an augmenting path; augment the
                                # matching and end this stage.
                                augmentMatching(v, w)
                                augmented = 1
                                break
                        elif label.get(w) is None:
                            # w is inside a T-blossom, but w itself has not
                            # yet been reached from outside the blossom;
                            # mark it as reached (we need this to relabel
                            # during T-blossom expansion).
                            assert label[bw] == 2
                            label[w] = 2
                            labeledge[w] = (v, w)
                    elif label.get(bw) == 1:
                        # keep track of the least-slack non-allowable edge to
                        # a different S-blossom.
                        if bestedge.get(bv) is None or \
                                kslack < slack(*bestedge[bv]):
                            bestedge[bv] = (v, w)
                    elif label.get(w) is None:
                        # w is a free vertex (or an unreached vertex inside
                        # a T-blossom) but we can not reach it yet;
                        # keep track of the least-slack edge that reaches w.
                        if bestedge.get(w) is None or \
                                kslack < slack(*bestedge[w]):
                            bestedge[w] = (v, w)

            if augmented:
                break

            # There is no augmenting path under these constraints;
            # compute delta and reduce slack in the optimization problem.
            # (Note that our vertex dual variables, edge slacks and delta's
            # are pre-multiplied by two.)
            deltatype = -1
            delta = deltaedge = deltablossom = None

            # Compute delta1: the minumum value of any vertex dual.
            if not maxcardinality:
                deltatype = 1
                delta = min(dualvar.values())

            # Compute delta2: the minimum slack on any edge between
            # an S-vertex and a free vertex.
            for v in G.nodes():
                if label.get(inblossom[v]) is None and \
                        bestedge.get(v) is not None:
                    d = slack(*bestedge[v])
                    if deltatype == -1 or d < delta:
                        delta = d
                        deltatype = 2
                        deltaedge = bestedge[v]

            # Compute delta3: half the minimum slack on any edge between
            # a pair of S-blossoms.
            for b in blossomparent:
                if (blossomparent[b] is None and label.get(b) == 1 and
                        bestedge.get(b) is not None):
                    kslack = slack(*bestedge[b])
                    if allinteger:
                        assert (kslack % 2) == 0
                        d = kslack // 2
                    else:
                        d = kslack / 2.0
                    if deltatype == -1 or d < delta:
                        delta = d
                        deltatype = 3
                        deltaedge = bestedge[b]

            # Compute delta4: minimum z variable of any T-blossom.
            for b in blossomdual:
                if (blossomparent[b] is None and label.get(b) == 2 and
                        (deltatype == -1 or blossomdual[b] < delta)):
                    delta = blossomdual[b]
                    deltatype = 4
                    deltablossom = b

            if deltatype == -1:
                # No further improvement possible; max-cardinality optimum
                # reached. Do a final delta update to make the optimum
                # verifyable.
                assert maxcardinality
                deltatype = 1
                delta = max(0, min(dualvar.values()))

            # Update dual variables according to delta.
            for v in gnodes:
                if label.get(inblossom[v]) == 1:
                    # S-vertex: 2*u = 2*u - 2*delta
                    dualvar[v] -= delta
                elif label.get(inblossom[v]) == 2:
                    # T-vertex: 2*u = 2*u + 2*delta
                    dualvar[v] += delta
            for b in blossomdual:
                if blossomparent[b] is None:
                    if label.get(b) == 1:
                        # top-level S-blossom: z = z + 2*delta
                        blossomdual[b] += delta
                    elif label.get(b) == 2:
                        # top-level T-blossom: z = z - 2*delta
                        blossomdual[b] -= delta

            # Take action at the point where minimum delta occurred.
            if deltatype == 1:
                # No further improvement possible; optimum reached.
                break
            elif deltatype == 2:
                # Use the least-slack edge to continue the search.
                (v, w) = deltaedge
                assert label[inblossom[v]] == 1
                allowedge[(v, w)] = allowedge[(w, v)] = True
                queue.append(v)
            elif deltatype == 3:
                # Use the least-slack edge to continue the search.
                (v, w) = deltaedge
                allowedge[(v, w)] = allowedge[(w, v)] = True
                assert label[inblossom[v]] == 1
                queue.append(v)
            elif deltatype == 4:
                # Expand the least-z blossom.
                expandBlossom(deltablossom, False)

            # End of a this substage.

        # Paranoia check that the matching is symmetric.
        for v in mate:
            assert mate[mate[v]] == v

        # Stop when no more augmenting path can be found.
        if not augmented:
            break

        # End of a stage; expand all S-blossoms which have zero dual.
        for b in list(blossomdual.keys()):
            if b not in blossomdual:
                continue  # already expanded
            if (blossomparent[b] is None and label.get(b) == 1 and
                    blossomdual[b] == 0):
                expandBlossom(b, True)

    # Verify that we reached the optimum solution (only for integer weights).
    if allinteger:
        verifyOptimum()

    return matching_dict_to_set(mate)
