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Server: Apache/2.4.46 (Win64) OpenSSL/1.1.1j PHP/8.4.25
System: Windows NT DESKTOP-4TAV2RJ 10.0 build 19045 (Windows 10) AMD64
User: fred (0)
PHP: 8.4.25
Disabled: NONE
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File: C:/Users/fred/anaconda3/Lib/site-packages/scipy/sparse/csgraph/_matching.cp312-win_amd64.pyd
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    maximum_bipartite_matching(graph, perm_type='row')

    Returns a matching of a bipartite graph whose cardinality is as least that
    of any given matching of the graph.

    Parameters
    ----------
    graph : sparse matrix
        Input sparse in CSR format whose rows represent one partition of the
        graph and whose columns represent the other partition. An edge between
        two vertices is indicated by the corresponding entry in the matrix
        existing in its sparse representation.
    perm_type : str, {'row', 'column'}
        Which partition to return the matching in terms of: If ``'row'``, the
        function produces an array whose length is the number of columns in the
        input, and whose :math:`j`'th element is the row matched to the
        :math:`j`'th column. Conversely, if ``perm_type`` is ``'column'``, this
        returns the columns matched to each row.

    Returns
    -------
    perm : ndarray
        A matching of the vertices in one of the two partitions. Unmatched
        vertices are represented by a ``-1`` in the result.

    Notes
    -----
    This function implements the Hopcroft--Karp algorithm [1]_. Its time
    complexity is :math:`O(\lvert E \rvert \sqrt{\lvert V \rvert})`, and its
    space complexity is linear in the number of rows. In practice, this
    asymmetry between rows and columns means that it can be more efficient to
    transpose the input if it contains more columns than rows.

    By Konig's theorem, the cardinality of the matching is also the number of
    vertices appearing in a minimum vertex cover of the graph.

    Note that if the sparse representation contains explicit zeros, these are
    still counted as edges.

    The implementation was changed in SciPy 1.4.0 to allow matching of general
    bipartite graphs, where previous versions would assume that a perfect
    matching existed. As such, code written against 1.4.0 will not necessarily
    work on older versions.

    If multiple valid solutions are possible, output may vary with SciPy and
    Python version.

    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. :doi:`10.1137/0202019`

    Examples
    --------
    >>> from scipy.sparse import csr_matrix
    >>> from scipy.sparse.csgraph import maximum_bipartite_matching

    As a simple example, consider a bipartite graph in which the partitions
    contain 2 and 3 elements respectively. Suppose that one partition contains
    vertices labelled 0 and 1, and that the other partition contains vertices
    labelled A, B, and C. Suppose that there are edges connecting 0 and C,
    1 and A, and 1 and B. This graph would then be represented by the following
    sparse matrix:

    >>> graph = csr_matrix([[0, 0, 1], [1, 1, 0]])

    Here, the 1s could be anything, as long as they end up being stored as
    elements in the sparse matrix. We can now calculate maximum matchings as
    follows:

    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [2 0]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    [ 1 -1  0]

    The first output tells us that 1 and 2 are matched with C and A
    respectively, and the second output tells us that A, B, and C are matched
    with 1, nothing, and 0 respectively.

    Note that explicit zeros are still converted to edges. This means that a
    different way to represent the above graph is by using the CSR structure
    directly as follows:

    >>> data = [0, 0, 0]
    >>> indices = [2, 0, 1]
    >>> indptr = [0, 1, 3]
    >>> graph = csr_matrix((data, indices, indptr))
    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [2 0]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    [ 1 -1  0]

    When one or both of the partitions are empty, the matching is empty as
    well:

    >>> graph = csr_matrix((2, 0))
    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [-1 -1]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    []

    When the input matrix is square, and the graph is known to admit a perfect
    matching, i.e. a matching with the property that every vertex in the graph
    belongs to some edge in the matching, then one can view the output as the
    permutation of rows (or columns) turning the input matrix into one with the
    property that all diagonal elements are non-empty:

    >>> a = [[0, 1, 2, 0], [1, 0, 0, 1], [2, 0, 0, 3], [0, 1, 3, 0]]
    >>> graph = csr_matrix(a)
    >>> perm = maximum_bipartite_matching(graph, perm_type='row')
    >>> print(graph[perm].toarray())
    [[1 0 0 1]
     [0 1 2 0]
     [0 1 3 0]
     [2 0 0 3]]

    maximum_bipartite_matching (line 17)memviewmin
    min_weight_full_bipartite_matching(biadjacency_matrix, maximize=False)

    Returns the minimum weight full matching of a bipartite graph.

    .. versionadded:: 1.6.0

    Parameters
    ----------
    biadjacency_matrix : sparse matrix
        Biadjacency matrix of the bipartite graph: A sparse matrix in CSR, CSC,
        or COO format whose rows represent one partition of the graph and whose
        columns represent the other partition. An edge between two vertices is
        indicated by the corresponding entry in the matrix, and the weight of
        the edge is given by the value of that entry. This should not be
        confused with the full adjacency matrix of the graph, as we only need
        the submatrix defining the bipartite structure.

    maximize : bool (default: False)
        Calculates a maximum weight matching if true.

    Returns
    -------
    row_ind, col_ind : array
        An array of row indices and one of corresponding column indices giving
        the optimal matching. The total weight of the matching can be computed
        as ``graph[row_ind, col_ind].sum()``. The row indices will be
        sorted; in the case of a square matrix they will be equal to
        ``numpy.arange(graph.shape[0])``.

    Notes
    -----

    Let :math:`G = ((U, V), E)` be a weighted bipartite graph with non-zero
    weights :math:`w : E \to \mathbb{R} \setminus \{0\}`. This function then
    produces a matching :math:`M \subseteq E` with cardinality

    .. math::
       \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),

    which minimizes the sum of the weights of the edges included in the
    matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such
    matching exists.

    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*.

    This function implements the LAPJVsp algorithm [2]_, short for "Linear
    assignment problem, Jonker--Volgenant, sparse".

    The problem it solves is equivalent to the rectangular linear assignment
    problem. [3]_ As such, this function can be used to solve the same problems
    as :func:`scipy.optimize.linear_sum_assignment`. That function may perform
    better when the input is dense, or for certain particular types of inputs,
    such as those for which the :math:`(i, j)`'th entry is the distance between
    two points in Euclidean space.

    If no full matching exists, this function raises a ``ValueError``. For
    determining the size of the largest matching in the graph, see
    :func:`maximum_bipartite_matching`.

    We require that weights are non-zero only to avoid issues with the handling
    of explicit zeros when converting between different sparse representations.
    Zero weights can be handled by adding a constant to all weights, so that
    the resulting matrix contains no zeros.

    If multiple valid solutions are possible, output may vary with SciPy and
    Python version.

    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.
    .. [2] Roy Jonker and Anton Volgenant:
       A Shortest Augmenting Path Algorithm for Dense and Sparse Linear
       Assignment Problems.
       Computing 38:325-340, 1987.
    .. [3] https://en.wikipedia.org/wiki/Assignment_problem

    Examples
    --------
    >>> from scipy.sparse import csr_matrix
    >>> from scipy.sparse.csgraph import min_weight_full_bipartite_matching

    Let us first consider an example in which all weights are equal:

    >>> biadjacency_matrix = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]])

    Here, all we get is a perfect matching of the graph:

    >>> print(min_weight_full_bipartite_matching(biadjacency_matrix)[1])
    [2 0 1]

    That is, the first, second, and third rows are matched with the third,
    first, and second column respectively. Note that in this example, the 0
    in the input matrix does *not* correspond to an edge with weight 0, but
    rather a pair of vertices not paired by an edge.

    Note also that in this case, the output matches the result of applying
    :func:`maximum_bipartite_matching`:

    >>> from scipy.sparse.csgraph import maximum_bipartite_matching
    >>> biadjacency = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]])
    >>> print(maximum_bipartite_matching(biadjacency, perm_type='column'))
    [2 0 1]

    When multiple edges are available, the ones with lowest weights are
    preferred:

    >>> biadjacency = csr_matrix([[3, 3, 6], [4, 3, 5], [10, 1, 8]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(col_ind)
    [0 2 1]

    The total weight in this case is :math:`3 + 5 + 1 = 9`:

    >>> print(biadjacency[row_ind, col_ind].sum())
    9

    When the matrix is not square, i.e. when the two partitions have different
    cardinalities, the matching is as large as the smaller of the two
    partitions:

    >>> biadjacency = csr_matrix([[0, 1, 1], [0, 2, 3]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [0 1] [2 1]
    >>> biadjacency = csr_matrix([[0, 1], [3, 1], [1, 4]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [0 2] [1 0]

    When one or both of the partitions are empty, the matching is empty as
    well:

    >>> biadjacency = csr_matrix((2, 0))
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [] []

    In general, we will always reach the same sum of weights as if we had used
    :func:`scipy.optimize.linear_sum_assignment` but note that for that one,
    missing edges are represented by a matrix entry of ``float('inf')``. Let us
    generate a random sparse matrix with integer entries between 1 and 10:

    >>> import numpy as np
    >>> from scipy.sparse import random
    >>> from scipy.optimize import linear_sum_assignment
    >>> sparse = random(10, 10, random_state=42, density=.5, format='coo') * 10
    >>> sparse.data = np.ceil(sparse.data)
    >>> dense = sparse.toarray()
    >>> dense = np.full(sparse.shape, np.inf)
    >>> dense[sparse.row, sparse.col] = sparse.data
    >>> sparse = sparse.tocsr()
    >>> row_ind, col_ind = linear_sum_assignment(dense)
    >>> print(dense[row_ind, col_ind].sum())
    28.0
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(sparse)
    >>> print(sparse[row_ind, col_ind].sum())
    28.0

    min_weight_full_bipartite_matchingmin_weight_full_bipartite_matching (line 288)modename__name__ndim__new__no default __reduce__ due to non-trivial __cinit__no full matching existsnpnumpynumpy.core.multiarray failed to importnumpy.core.umath failed to importobjpackperm_typepickle__pyx_PickleError__pyx_checksum__pyx_result__pyx_state__pyx_type__pyx_unpickle_Enum__pyx_vtable__range__reduce____reduce_cython____reduce_ex__registerrowscipy.sparsescipy.sparse._sputilsscipy.sparse.csgraph._matching__setstate____setstate_cython__shapesize__spec__startstepstop<strided and direct><strided and direct or indirect><strided and indirect><stringsource>structsumsys__test__tocsruint8unable to allocate array data.unable to allocate shape and strides.unpackupdateversion_infowarnwarningsxyzeros00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899
    maximum_bipartite_matching(graph, perm_type='row')

    Returns a matching of a bipartite graph whose cardinality is as least that
    of any given matching of the graph.

    Parameters
    ----------
    graph : sparse matrix
        Input sparse in CSR format whose rows represent one partition of the
        graph and whose columns represent the other partition. An edge between
        two vertices is indicated by the corresponding entry in the matrix
        existing in its sparse representation.
    perm_type : str, {'row', 'column'}
        Which partition to return the matching in terms of: If ``'row'``, the
        function produces an array whose length is the number of columns in the
        input, and whose :math:`j`'th element is the row matched to the
        :math:`j`'th column. Conversely, if ``perm_type`` is ``'column'``, this
        returns the columns matched to each row.

    Returns
    -------
    perm : ndarray
        A matching of the vertices in one of the two partitions. Unmatched
        vertices are represented by a ``-1`` in the result.

    Notes
    -----
    This function implements the Hopcroft--Karp algorithm [1]_. Its time
    complexity is :math:`O(\lvert E \rvert \sqrt{\lvert V \rvert})`, and its
    space complexity is linear in the number of rows. In practice, this
    asymmetry between rows and columns means that it can be more efficient to
    transpose the input if it contains more columns than rows.

    By Konig's theorem, the cardinality of the matching is also the number of
    vertices appearing in a minimum vertex cover of the graph.

    Note that if the sparse representation contains explicit zeros, these are
    still counted as edges.

    The implementation was changed in SciPy 1.4.0 to allow matching of general
    bipartite graphs, where previous versions would assume that a perfect
    matching existed. As such, code written against 1.4.0 will not necessarily
    work on older versions.

    If multiple valid solutions are possible, output may vary with SciPy and
    Python version.

    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. :doi:`10.1137/0202019`

    Examples
    --------
    >>> from scipy.sparse import csr_matrix
    >>> from scipy.sparse.csgraph import maximum_bipartite_matching

    As a simple example, consider a bipartite graph in which the partitions
    contain 2 and 3 elements respectively. Suppose that one partition contains
    vertices labelled 0 and 1, and that the other partition contains vertices
    labelled A, B, and C. Suppose that there are edges connecting 0 and C,
    1 and A, and 1 and B. This graph would then be represented by the following
    sparse matrix:

    >>> graph = csr_matrix([[0, 0, 1], [1, 1, 0]])

    Here, the 1s could be anything, as long as they end up being stored as
    elements in the sparse matrix. We can now calculate maximum matchings as
    follows:

    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [2 0]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    [ 1 -1  0]

    The first output tells us that 1 and 2 are matched with C and A
    respectively, and the second output tells us that A, B, and C are matched
    with 1, nothing, and 0 respectively.

    Note that explicit zeros are still converted to edges. This means that a
    different way to represent the above graph is by using the CSR structure
    directly as follows:

    >>> data = [0, 0, 0]
    >>> indices = [2, 0, 1]
    >>> indptr = [0, 1, 3]
    >>> graph = csr_matrix((data, indices, indptr))
    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [2 0]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    [ 1 -1  0]

    When one or both of the partitions are empty, the matching is empty as
    well:

    >>> graph = csr_matrix((2, 0))
    >>> print(maximum_bipartite_matching(graph, perm_type='column'))
    [-1 -1]
    >>> print(maximum_bipartite_matching(graph, perm_type='row'))
    []

    When the input matrix is square, and the graph is known to admit a perfect
    matching, i.e. a matching with the property that every vertex in the graph
    belongs to some edge in the matching, then one can view the output as the
    permutation of rows (or columns) turning the input matrix into one with the
    property that all diagonal elements are non-empty:

    >>> a = [[0, 1, 2, 0], [1, 0, 0, 1], [2, 0, 0, 3], [0, 1, 3, 0]]
    >>> graph = csr_matrix(a)
    >>> perm = maximum_bipartite_matching(graph, perm_type='row')
    >>> print(graph[perm].toarray())
    [[1 0 0 1]
     [0 1 2 0]
     [0 1 3 0]
     [2 0 0 3]]

    
    min_weight_full_bipartite_matching(biadjacency_matrix, maximize=False)

    Returns the minimum weight full matching of a bipartite graph.

    .. versionadded:: 1.6.0

    Parameters
    ----------
    biadjacency_matrix : sparse matrix
        Biadjacency matrix of the bipartite graph: A sparse matrix in CSR, CSC,
        or COO format whose rows represent one partition of the graph and whose
        columns represent the other partition. An edge between two vertices is
        indicated by the corresponding entry in the matrix, and the weight of
        the edge is given by the value of that entry. This should not be
        confused with the full adjacency matrix of the graph, as we only need
        the submatrix defining the bipartite structure.

    maximize : bool (default: False)
        Calculates a maximum weight matching if true.

    Returns
    -------
    row_ind, col_ind : array
        An array of row indices and one of corresponding column indices giving
        the optimal matching. The total weight of the matching can be computed
        as ``graph[row_ind, col_ind].sum()``. The row indices will be
        sorted; in the case of a square matrix they will be equal to
        ``numpy.arange(graph.shape[0])``.

    Notes
    -----

    Let :math:`G = ((U, V), E)` be a weighted bipartite graph with non-zero
    weights :math:`w : E \to \mathbb{R} \setminus \{0\}`. This function then
    produces a matching :math:`M \subseteq E` with cardinality

    .. math::
       \lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),

    which minimizes the sum of the weights of the edges included in the
    matching, :math:`\sum_{e \in M} w(e)`, or raises an error if no such
    matching exists.

    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*.

    This function implements the LAPJVsp algorithm [2]_, short for "Linear
    assignment problem, Jonker--Volgenant, sparse".

    The problem it solves is equivalent to the rectangular linear assignment
    problem. [3]_ As such, this function can be used to solve the same problems
    as :func:`scipy.optimize.linear_sum_assignment`. That function may perform
    better when the input is dense, or for certain particular types of inputs,
    such as those for which the :math:`(i, j)`'th entry is the distance between
    two points in Euclidean space.

    If no full matching exists, this function raises a ``ValueError``. For
    determining the size of the largest matching in the graph, see
    :func:`maximum_bipartite_matching`.

    We require that weights are non-zero only to avoid issues with the handling
    of explicit zeros when converting between different sparse representations.
    Zero weights can be handled by adding a constant to all weights, so that
    the resulting matrix contains no zeros.

    If multiple valid solutions are possible, output may vary with SciPy and
    Python version.

    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.
    .. [2] Roy Jonker and Anton Volgenant:
       A Shortest Augmenting Path Algorithm for Dense and Sparse Linear
       Assignment Problems.
       Computing 38:325-340, 1987.
    .. [3] https://en.wikipedia.org/wiki/Assignment_problem

    Examples
    --------
    >>> from scipy.sparse import csr_matrix
    >>> from scipy.sparse.csgraph import min_weight_full_bipartite_matching

    Let us first consider an example in which all weights are equal:

    >>> biadjacency_matrix = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]])

    Here, all we get is a perfect matching of the graph:

    >>> print(min_weight_full_bipartite_matching(biadjacency_matrix)[1])
    [2 0 1]

    That is, the first, second, and third rows are matched with the third,
    first, and second column respectively. Note that in this example, the 0
    in the input matrix does *not* correspond to an edge with weight 0, but
    rather a pair of vertices not paired by an edge.

    Note also that in this case, the output matches the result of applying
    :func:`maximum_bipartite_matching`:

    >>> from scipy.sparse.csgraph import maximum_bipartite_matching
    >>> biadjacency = csr_matrix([[1, 1, 1], [1, 0, 0], [0, 1, 0]])
    >>> print(maximum_bipartite_matching(biadjacency, perm_type='column'))
    [2 0 1]

    When multiple edges are available, the ones with lowest weights are
    preferred:

    >>> biadjacency = csr_matrix([[3, 3, 6], [4, 3, 5], [10, 1, 8]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(col_ind)
    [0 2 1]

    The total weight in this case is :math:`3 + 5 + 1 = 9`:

    >>> print(biadjacency[row_ind, col_ind].sum())
    9

    When the matrix is not square, i.e. when the two partitions have different
    cardinalities, the matching is as large as the smaller of the two
    partitions:

    >>> biadjacency = csr_matrix([[0, 1, 1], [0, 2, 3]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [0 1] [2 1]
    >>> biadjacency = csr_matrix([[0, 1], [3, 1], [1, 4]])
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [0 2] [1 0]

    When one or both of the partitions are empty, the matching is empty as
    well:

    >>> biadjacency = csr_matrix((2, 0))
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(biadjacency)
    >>> print(row_ind, col_ind)
    [] []

    In general, we will always reach the same sum of weights as if we had used
    :func:`scipy.optimize.linear_sum_assignment` but note that for that one,
    missing edges are represented by a matrix entry of ``float('inf')``. Let us
    generate a random sparse matrix with integer entries between 1 and 10:

    >>> import numpy as np
    >>> from scipy.sparse import random
    >>> from scipy.optimize import linear_sum_assignment
    >>> sparse = random(10, 10, random_state=42, density=.5, format='coo') * 10
    >>> sparse.data = np.ceil(sparse.data)
    >>> dense = sparse.toarray()
    >>> dense = np.full(sparse.shape, np.inf)
    >>> dense[sparse.row, sparse.col] = sparse.data
    >>> sparse = sparse.tocsr()
    >>> row_ind, col_ind = linear_sum_assignment(dense)
    >>> print(dense[row_ind, col_ind].sum())
    28.0
    >>> row_ind, col_ind = min_weight_full_bipartite_matching(sparse)
    >>> print(sparse[row_ind, col_ind].sum())
    28.0

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