# matching.py - bipartite graph maximum matching algorithms
#
# Copyright 2015 Jeffrey Finkelstein <jeffrey.finkelstein@gmail.com>,
# Copyright 2019 Søren Fuglede Jørgensen
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
#
# This module uses material from the Wikipedia article Hopcroft--Karp algorithm
# <https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>, accessed on
# January 3, 2015, which is released under the Creative Commons
# Attribution-Share-Alike License 3.0
# <http://creativecommons.org/licenses/by-sa/3.0/>. That article includes
# pseudocode, which has been translated into the corresponding Python code.
#
# Portions of this module use code from David Eppstein's Python Algorithms and
# Data Structures (PADS) library, which is dedicated to the public domain (for
# proof, see <http://www.ics.uci.edu/~eppstein/PADS/ABOUT-PADS.txt>).
"""Provides functions for computing maximum cardinality matchings and minimum
weight full matchings in a bipartite graph.

If you don't care about the particular implementation of the maximum matching
algorithm, simply use the :func:`maximum_matching`. If you do care, you can
import one of the named maximum matching algorithms directly.

For example, to find a maximum matching in the complete bipartite graph with
two vertices on the left and three vertices on the right:

>>> import networkx as nx
>>> G = nx.complete_bipartite_graph(2, 3)
>>> left, right = nx.bipartite.sets(G)
>>> list(left)
[0, 1]
>>> list(right)
[2, 3, 4]
>>> nx.bipartite.maximum_matching(G)
{0: 2, 1: 3, 2: 0, 3: 1}

The dictionary returned by :func:`maximum_matching` includes a mapping for
vertices in both the left and right vertex sets.

Similarly, :func:`minimum_weight_full_matching` produces, for a complete
weighted bipartite graph, a matching whose cardinality is the cardinality of
the smaller of the two partitions, and for which the sum of the weights of the
edges included in the matching is minimal.

"""
import collections
import itertools

from networkx.algorithms.bipartite.matrix import biadjacency_matrix
from networkx.algorithms.bipartite import sets as bipartite_sets
import networkx as nx

__all__ = ['maximum_matching', 'hopcroft_karp_matching', 'eppstein_matching',
           'to_vertex_cover', 'minimum_weight_full_matching']

INFINITY = float('inf')


def hopcroft_karp_matching(G, top_nodes=None):
    """Returns the maximum cardinality matching of the bipartite graph `G`.

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

      Undirected bipartite graph

    top_nodes : container

      Container with all nodes in one bipartite node set. If not supplied
      it will be computed. But if more than one solution exists an exception
      will be raised.

    Returns
    -------
    matches : dictionary

      The matching is returned as a dictionary, `matches`, such that
      ``matches[v] == w`` if node `v` is matched to node `w`. Unmatched
      nodes do not occur as a key in mate.

    Raises
    ------
    AmbiguousSolution : Exception

      Raised if the input bipartite graph is disconnected and no container
      with all nodes in one bipartite set is provided. When determining
      the nodes in each bipartite set more than one valid solution is
      possible if the input graph is disconnected.

    Notes
    -----
    This function is implemented with the `Hopcroft--Karp matching algorithm
    <https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm>`_ for
    bipartite graphs.

    See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
    for further details on how bipartite graphs are handled in NetworkX.

    See Also
    --------

    eppstein_matching

    References
    ----------
    .. [1] John E. Hopcroft and Richard M. Karp. "An n^{5 / 2} Algorithm for
       Maximum Matchings in Bipartite Graphs" In: **SIAM Journal of Computing**
       2.4 (1973), pp. 225--231. <https://doi.org/10.1137/0202019>.

    """
    # First we define some auxiliary search functions.
    #
    # If you are a human reading these auxiliary search functions, the "global"
    # variables `leftmatches`, `rightmatches`, `distances`, etc. are defined
    # below the functions, so that they are initialized close to the initial
    # invocation of the search functions.
    def breadth_first_search():
        for v in left:
            if leftmatches[v] is None:
                distances[v] = 0
                queue.append(v)
            else:
                distances[v] = INFINITY
        distances[None] = INFINITY
        while queue:
            v = queue.popleft()
            if distances[v] < distances[None]:
                for u in G[v]:
                    if distances[rightmatches[u]] is INFINITY:
                        distances[rightmatches[u]] = distances[v] + 1
                        queue.append(rightmatches[u])
        return distances[None] is not INFINITY

    def depth_first_search(v):
        if v is not None:
            for u in G[v]:
                if distances[rightmatches[u]] == distances[v] + 1:
                    if depth_first_search(rightmatches[u]):
                        rightmatches[u] = v
                        leftmatches[v] = u
                        return True
            distances[v] = INFINITY
            return False
        return True

    # Initialize the "global" variables that maintain state during the search.
    left, right = bipartite_sets(G, top_nodes)
    leftmatches = {v: None for v in left}
    rightmatches = {v: None for v in right}
    distances = {}
    queue = collections.deque()

    # Implementation note: this counter is incremented as pairs are matched but
    # it is currently not used elsewhere in the computation.
    num_matched_pairs = 0
    while breadth_first_search():
        for v in left:
            if leftmatches[v] is None:
                if depth_first_search(v):
                    num_matched_pairs += 1

    # Strip the entries matched to `None`.
    leftmatches = {k: v for k, v in leftmatches.items() if v is not None}
    rightmatches = {k: v for k, v in rightmatches.items() if v is not None}

    # At this point, the left matches and the right matches are inverses of one
    # another. In other words,
    #
    #     leftmatches == {v, k for k, v in rightmatches.items()}
    #
    # Finally, we combine both the left matches and right matches.
    return dict(itertools.chain(leftmatches.items(), rightmatches.items()))


def eppstein_matching(G, top_nodes=None):
    """Returns the maximum cardinality matching of the bipartite graph `G`.

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

      Undirected bipartite graph

    top_nodes : container

      Container with all nodes in one bipartite node set. If not supplied
      it will be computed. But if more than one solution exists an exception
      will be raised.

    Returns
    -------
    matches : dictionary

      The matching is returned as a dictionary, `matching`, such that
      ``matching[v] == w`` if node `v` is matched to node `w`. Unmatched
      nodes do not occur as a key in mate.

    Raises
    ------
    AmbiguousSolution : Exception

      Raised if the input bipartite graph is disconnected and no container
      with all nodes in one bipartite set is provided. When determining
      the nodes in each bipartite set more than one valid solution is
      possible if the input graph is disconnected.

    Notes
    -----
    This function is implemented with David Eppstein's version of the algorithm
    Hopcroft--Karp algorithm (see :func:`hopcroft_karp_matching`), which
    originally appeared in the `Python Algorithms and Data Structures library
    (PADS) <http://www.ics.uci.edu/~eppstein/PADS/ABOUT-PADS.txt>`_.

    See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
    for further details on how bipartite graphs are handled in NetworkX.

    See Also
    --------

    hopcroft_karp_matching

    """
    # Due to its original implementation, a directed graph is needed
    # so that the two sets of bipartite nodes can be distinguished
    left, right = bipartite_sets(G, top_nodes)
    G = nx.DiGraph(G.edges(left))
    # initialize greedy matching (redundant, but faster than full search)
    matching = {}
    for u in G:
        for v in G[u]:
            if v not in matching:
                matching[v] = u
                break
    while True:
        # structure residual graph into layers
        # pred[u] gives the neighbor in the previous layer for u in U
        # preds[v] gives a list of neighbors in the previous layer for v in V
        # unmatched gives a list of unmatched vertices in final layer of V,
        # and is also used as a flag value for pred[u] when u is in the first
        # layer
        preds = {}
        unmatched = []
        pred = {u: unmatched for u in G}
        for v in matching:
            del pred[matching[v]]
        layer = list(pred)

        # repeatedly extend layering structure by another pair of layers
        while layer and not unmatched:
            newLayer = {}
            for u in layer:
                for v in G[u]:
                    if v not in preds:
                        newLayer.setdefault(v, []).append(u)
            layer = []
            for v in newLayer:
                preds[v] = newLayer[v]
                if v in matching:
                    layer.append(matching[v])
                    pred[matching[v]] = v
                else:
                    unmatched.append(v)

        # did we finish layering without finding any alternating paths?
        if not unmatched:
            unlayered = {}
            for u in G:
                # TODO Why is extra inner loop necessary?
                for v in G[u]:
                    if v not in preds:
                        unlayered[v] = None
            # TODO Originally, this function returned a three-tuple:
            #
            #     return (matching, list(pred), list(unlayered))
            #
            # For some reason, the documentation for this function
            # indicated that the second and third elements of the returned
            # three-tuple would be the vertices in the left and right vertex
            # sets, respectively, that are also in the maximum independent set.
            # However, what I think the author meant was that the second
            # element is the list of vertices that were unmatched and the third
            # element was the list of vertices that were matched. Since that
            # seems to be the case, they don't really need to be returned,
            # since that information can be inferred from the matching
            # dictionary.

            # All the matched nodes must be a key in the dictionary
            for key in matching.copy():
                matching[matching[key]] = key
            return matching

        # recursively search backward through layers to find alternating paths
        # recursion returns true if found path, false otherwise
        def recurse(v):
            if v in preds:
                L = preds.pop(v)
                for u in L:
                    if u in pred:
                        pu = pred.pop(u)
                        if pu is unmatched or recurse(pu):
                            matching[v] = u
                            return True
            return False

        for v in unmatched:
            recurse(v)


def _is_connected_by_alternating_path(G, v, matched_edges, unmatched_edges,
                                      targets):
    """Returns True if and only if the vertex `v` is connected to one of
    the target vertices by an alternating path in `G`.

    An *alternating path* is a path in which every other edge is in the
    specified maximum matching (and the remaining edges in the path are not in
    the matching). An alternating path may have matched edges in the even
    positions or in the odd positions, as long as the edges alternate between
    'matched' and 'unmatched'.

    `G` is an undirected bipartite NetworkX graph.

    `v` is a vertex in `G`.

    `matched_edges` is a set of edges present in a maximum matching in `G`.

    `unmatched_edges` is a set of edges not present in a maximum
    matching in `G`.

    `targets` is a set of vertices.

    """
    def _alternating_dfs(u, along_matched=True):
        """Returns True if and only if `u` is connected to one of the
        targets by an alternating path.

        `u` is a vertex in the graph `G`.

        If `along_matched` is True, this step of the depth-first search
        will continue only through edges in the given matching. Otherwise, it
        will continue only through edges *not* in the given matching.

        """
        if along_matched:
            edges = itertools.cycle([matched_edges, unmatched_edges])
        else:
            edges = itertools.cycle([unmatched_edges, matched_edges])
        visited = set()
        stack = [(u, iter(G[u]), next(edges))]
        while stack:
            parent, children, valid_edges = stack[-1]
            try:
                child = next(children)
                if child not in visited:
                    if ((parent, child) in valid_edges
                            or (child, parent) in valid_edges):
                        if child in targets:
                            return True
                        visited.add(child)
                        stack.append((child, iter(G[child]), next(edges)))
            except StopIteration:
                stack.pop()
        return False

    # Check for alternating paths starting with edges in the matching, then
    # check for alternating paths starting with edges not in the
    # matching.
    return (_alternating_dfs(v, along_matched=True) or
            _alternating_dfs(v, along_matched=False))


def _connected_by_alternating_paths(G, matching, targets):
    """Returns the set of vertices that are connected to one of the target
    vertices by an alternating path in `G` or are themselves a target.

    An *alternating path* is a path in which every other edge is in the
    specified maximum matching (and the remaining edges in the path are not in
    the matching). An alternating path may have matched edges in the even
    positions or in the odd positions, as long as the edges alternate between
    'matched' and 'unmatched'.

    `G` is an undirected bipartite NetworkX graph.

    `matching` is a dictionary representing a maximum matching in `G`, as
    returned by, for example, :func:`maximum_matching`.

    `targets` is a set of vertices.

    """
    # Get the set of matched edges and the set of unmatched edges. Only include
    # one version of each undirected edge (for example, include edge (1, 2) but
    # not edge (2, 1)). Using frozensets as an intermediary step we do not
    # require nodes to be orderable.
    edge_sets = {frozenset((u, v)) for u, v in matching.items()}
    matched_edges = {tuple(edge) for edge in edge_sets}
    unmatched_edges = {(u, v) for (u, v) in G.edges()
                       if frozenset((u, v)) not in edge_sets}

    return {v for v in G if v in targets or
            _is_connected_by_alternating_path(G, v, matched_edges,
                                              unmatched_edges, targets)}


def to_vertex_cover(G, matching, top_nodes=None):
    """Returns the minimum vertex cover corresponding to the given maximum
    matching of the bipartite graph `G`.

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

      Undirected bipartite graph

    matching : dictionary

      A dictionary whose keys are vertices in `G` and whose values are the
      distinct neighbors comprising the maximum matching for `G`, as returned
      by, for example, :func:`maximum_matching`. The dictionary *must*
      represent the maximum matching.

    top_nodes : container

      Container with all nodes in one bipartite node set. If not supplied
      it will be computed. But if more than one solution exists an exception
      will be raised.

    Returns
    -------
    vertex_cover : :class:`set`

      The minimum vertex cover in `G`.

    Raises
    ------
    AmbiguousSolution : Exception

      Raised if the input bipartite graph is disconnected and no container
      with all nodes in one bipartite set is provided. When determining
      the nodes in each bipartite set more than one valid solution is
      possible if the input graph is disconnected.

    Notes
    -----
    This function is implemented using the procedure guaranteed by `Konig's
    theorem
    <https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29>`_,
    which proves an equivalence between a maximum matching and a minimum vertex
    cover in bipartite graphs.

    Since a minimum vertex cover is the complement of a maximum independent set
    for any graph, one can compute the maximum independent set of a bipartite
    graph this way:

    >>> import networkx as nx
    >>> G = nx.complete_bipartite_graph(2, 3)
    >>> matching = nx.bipartite.maximum_matching(G)
    >>> vertex_cover = nx.bipartite.to_vertex_cover(G, matching)
    >>> independent_set = set(G) - vertex_cover
    >>> print(list(independent_set))
    [2, 3, 4]

    See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
    for further details on how bipartite graphs are handled in NetworkX.

    """
    # This is a Python implementation of the algorithm described at
    # <https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_%28graph_theory%29#Proof>.
    L, R = bipartite_sets(G, top_nodes)
    # Let U be the set of unmatched vertices in the left vertex set.
    unmatched_vertices = set(G) - set(matching)
    U = unmatched_vertices & L
    # Let Z be the set of vertices that are either in U or are connected to U
    # by alternating paths.
    Z = _connected_by_alternating_paths(G, matching, U)
    # At this point, every edge either has a right endpoint in Z or a left
    # endpoint not in Z. This gives us the vertex cover.
    return (L - Z) | (R & Z)


#: Returns the maximum cardinality matching in the given bipartite graph.
#:
#: This function is simply an alias for :func:`hopcroft_karp_matching`.
maximum_matching = hopcroft_karp_matching


def minimum_weight_full_matching(G, top_nodes=None, weight='weight'):
    r"""Returns the minimum weight full matching of the bipartite graph `G`.

    Let :math:`G = ((U, V), E)` be a complete weighted bipartite graph with
    real weights :math:`w : E \to \mathbb{R}`. This function then produces
    a maximum matching :math:`M \subseteq E` which, since the graph is
    assumed to be complete, has cardinality
   
    .. math::
       \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),

    and which minimizes the sum of the weights of the edges included in the
    matching, :math:`\sum_{e \in M} w(e)`.
    
    When :math:`\lvert U \rvert = \lvert V \rvert`, this is commonly
    referred to as a perfect matching; here, since we allow
    :math:`\lvert U \rvert` and :math:`\lvert V \rvert` to differ, we
    follow Karp [1]_ and refer to the matching as *full*.

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

      Undirected bipartite graph

    top_nodes : container

      Container with all nodes in one bipartite node set. If not supplied
      it will be computed.

    weight : string, optional (default='weight')

       The edge data key used to provide each value in the matrix.

    Returns
    -------
    matches : dictionary

      The matching is returned as a dictionary, `matches`, such that
      ``matches[v] == w`` if node `v` is matched to node `w`. Unmatched
      nodes do not occur as a key in matches.

    Raises
    ------
    ValueError : Exception

      Raised if the input bipartite graph is not complete.

    ImportError : Exception

      Raised if SciPy is not available.

    Notes
    -----
    The problem of determining a minimum weight full matching is also known as
    the rectangular linear assignment problem. This implementation defers the
    calculation of the assignment to SciPy.

    References
    ----------
    .. [1] Richard Manning Karp:
       An algorithm to Solve the m x n Assignment Problem in Expected Time
       O(mn log n).
       Networks, 10(2):143–152, 1980.

    """
    try:
        import scipy.optimize
    except ImportError:
        raise ImportError('minimum_weight_full_matching requires SciPy: ' +
                          'https://scipy.org/')
    left, right = nx.bipartite.sets(G, top_nodes)
    # Ensure that the graph is complete. This is currently a requirement in
    # the underlying  optimization algorithm from SciPy, but the constraint
    # will be removed in SciPy 1.4.0, at which point it can also be removed
    # here.
    for (u, v) in itertools.product(left, right):
        # As the graph is undirected, make sure to check for edges in
        # both directions
        if (u, v) not in G.edges() and (v, u) not in G.edges():
            raise ValueError('The bipartite graph must be complete.')
    U = list(left)
    V = list(right)
    weights = biadjacency_matrix(G, row_order=U,
                                 column_order=V, weight=weight).toarray()
    left_matches = scipy.optimize.linear_sum_assignment(weights)
    d = {U[u]: V[v] for u, v in zip(*left_matches)}
    # d will contain the matching from edges in left to right; we need to
    # add the ones from right to left as well.
    d.update({v: u for u, v in d.items()})
    return d
