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Set operations for arrays based on sorting.

Notes
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For floating point arrays, inaccurate results may appear due to usual round-off
and floating point comparison issues.

Speed could be gained in some operations by an implementation of
`numpy.sort`, that can provide directly the permutation vectors, thus avoiding
calls to `numpy.argsort`.

Original author: Robert Cimrman

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    The differences between consecutive elements of an array.

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    to_end : array_like, optional
        Number(s) to append at the end of the returned differences.
    to_begin : array_like, optional
        Number(s) to prepend at the beginning of the returned differences.

    Returns
    -------
    ediff1d : ndarray
        The differences. Loosely, this is ``ary.flat[1:] - ary.flat[:-1]``.

    See Also
    --------
    diff, gradient

    Notes
    -----
    When applied to masked arrays, this function drops the mask information
    if the `to_begin` and/or `to_end` parameters are used.

    Examples
    --------
    >>> x = np.array([1, 2, 4, 7, 0])
    >>> np.ediff1d(x)
    array([ 1,  2,  3, -7])

    >>> np.ediff1d(x, to_begin=-99, to_end=np.array([88, 99]))
    array([-99,   1,   2, ...,  -7,  88,  99])

    The returned array is always 1D.

    >>> y = [[1, 2, 4], [1, 6, 24]]
    >>> np.ediff1d(y)
    array([ 1,  2, -3,  5, 18])

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    Find the unique elements of an array.

    Returns the sorted unique elements of an array. There are three optional
    outputs in addition to the unique elements:

    * the indices of the input array that give the unique values
    * the indices of the unique array that reconstruct the input array
    * the number of times each unique value comes up in the input array

    Parameters
    ----------
    ar : array_like
        Input array. Unless `axis` is specified, this will be flattened if it
        is not already 1-D.
    return_index : bool, optional
        If True, also return the indices of `ar` (along the specified axis,
        if provided, or in the flattened array) that result in the unique array.
    return_inverse : bool, optional
        If True, also return the indices of the unique array (for the specified
        axis, if provided) that can be used to reconstruct `ar`.
    return_counts : bool, optional
        If True, also return the number of times each unique item appears
        in `ar`.
    axis : int or None, optional
        The axis to operate on. If None, `ar` will be flattened. If an integer,
        the subarrays indexed by the given axis will be flattened and treated
        as the elements of a 1-D array with the dimension of the given axis,
        see the notes for more details.  Object arrays or structured arrays
        that contain objects are not supported if the `axis` kwarg is used. The
        default is None.

        .. versionadded:: 1.13.0

    equal_nan : bool, optional
        If True, collapses multiple NaN values in the return array into one.

        .. versionadded:: 1.24

    Returns
    -------
    unique : ndarray
        The sorted unique values.
    unique_indices : ndarray, optional
        The indices of the first occurrences of the unique values in the
        original array. Only provided if `return_index` is True.
    unique_inverse : ndarray, optional
        The indices to reconstruct the original array from the
        unique array. Only provided if `return_inverse` is True.
    unique_counts : ndarray, optional
        The number of times each of the unique values comes up in the
        original array. Only provided if `return_counts` is True.

        .. versionadded:: 1.9.0

    See Also
    --------
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.
    repeat : Repeat elements of an array.

    Notes
    -----
    When an axis is specified the subarrays indexed by the axis are sorted.
    This is done by making the specified axis the first dimension of the array
    (move the axis to the first dimension to keep the order of the other axes)
    and then flattening the subarrays in C order. The flattened subarrays are
    then viewed as a structured type with each element given a label, with the
    effect that we end up with a 1-D array of structured types that can be
    treated in the same way as any other 1-D array. The result is that the
    flattened subarrays are sorted in lexicographic order starting with the
    first element.

    .. versionchanged: NumPy 1.21
        If nan values are in the input array, a single nan is put
        to the end of the sorted unique values.

        Also for complex arrays all NaN values are considered equivalent
        (no matter whether the NaN is in the real or imaginary part).
        As the representant for the returned array the smallest one in the
        lexicographical order is chosen - see np.sort for how the lexicographical
        order is defined for complex arrays.

    Examples
    --------
    >>> np.unique([1, 1, 2, 2, 3, 3])
    array([1, 2, 3])
    >>> a = np.array([[1, 1], [2, 3]])
    >>> np.unique(a)
    array([1, 2, 3])

    Return the unique rows of a 2D array

    >>> a = np.array([[1, 0, 0], [1, 0, 0], [2, 3, 4]])
    >>> np.unique(a, axis=0)
    array([[1, 0, 0], [2, 3, 4]])

    Return the indices of the original array that give the unique values:

    >>> a = np.array(['a', 'b', 'b', 'c', 'a'])
    >>> u, indices = np.unique(a, return_index=True)
    >>> u
    array(['a', 'b', 'c'], dtype='<U1')
    >>> indices
    array([0, 1, 3])
    >>> a[indices]
    array(['a', 'b', 'c'], dtype='<U1')

    Reconstruct the input array from the unique values and inverse:

    >>> a = np.array([1, 2, 6, 4, 2, 3, 2])
    >>> u, indices = np.unique(a, return_inverse=True)
    >>> u
    array([1, 2, 3, 4, 6])
    >>> indices
    array([0, 1, 4, 3, 1, 2, 1])
    >>> u[indices]
    array([1, 2, 6, 4, 2, 3, 2])

    Reconstruct the input values from the unique values and counts:

    >>> a = np.array([1, 2, 6, 4, 2, 3, 2])
    >>> values, counts = np.unique(a, return_counts=True)
    >>> values
    array([1, 2, 3, 4, 6])
    >>> counts
    array([1, 3, 1, 1, 1])
    >>> np.repeat(values, counts)
    array([1, 2, 2, 2, 3, 4, 6])    # original order not preserved

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    Find the intersection of two arrays.

    Return the sorted, unique values that are in both of the input arrays.

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        Input arrays. Will be flattened if not already 1D.
    assume_unique : bool
        If True, the input arrays are both assumed to be unique, which
        can speed up the calculation.  If True but ``ar1`` or ``ar2`` are not
        unique, incorrect results and out-of-bounds indices could result.
        Default is False.
    return_indices : bool
        If True, the indices which correspond to the intersection of the two
        arrays are returned. The first instance of a value is used if there are
        multiple. Default is False.

        .. versionadded:: 1.15.0

    Returns
    -------
    intersect1d : ndarray
        Sorted 1D array of common and unique elements.
    comm1 : ndarray
        The indices of the first occurrences of the common values in `ar1`.
        Only provided if `return_indices` is True.
    comm2 : ndarray
        The indices of the first occurrences of the common values in `ar2`.
        Only provided if `return_indices` is True.


    See Also
    --------
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.

    Examples
    --------
    >>> np.intersect1d([1, 3, 4, 3], [3, 1, 2, 1])
    array([1, 3])

    To intersect more than two arrays, use functools.reduce:

    >>> from functools import reduce
    >>> reduce(np.intersect1d, ([1, 3, 4, 3], [3, 1, 2, 1], [6, 3, 4, 2]))
    array([3])

    To return the indices of the values common to the input arrays
    along with the intersected values:

    >>> x = np.array([1, 1, 2, 3, 4])
    >>> y = np.array([2, 1, 4, 6])
    >>> xy, x_ind, y_ind = np.intersect1d(x, y, return_indices=True)
    >>> x_ind, y_ind
    (array([0, 2, 4]), array([1, 0, 2]))
    >>> xy, x[x_ind], y[y_ind]
    (array([1, 2, 4]), array([1, 2, 4]), array([1, 2, 4]))

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    Find the set exclusive-or of two arrays.

    Return the sorted, unique values that are in only one (not both) of the
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    Parameters
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    ar1, ar2 : array_like
        Input arrays.
    assume_unique : bool
        If True, the input arrays are both assumed to be unique, which
        can speed up the calculation.  Default is False.

    Returns
    -------
    setxor1d : ndarray
        Sorted 1D array of unique values that are in only one of the input
        arrays.

    Examples
    --------
    >>> a = np.array([1, 2, 3, 2, 4])
    >>> b = np.array([2, 3, 5, 7, 5])
    >>> np.setxor1d(a,b)
    array([1, 4, 5, 7])

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    Test whether each element of a 1-D array is also present in a second array.

    Returns a boolean array the same length as `ar1` that is True
    where an element of `ar1` is in `ar2` and False otherwise.

    We recommend using :func:`isin` instead of `in1d` for new code.

    Parameters
    ----------
    ar1 : (M,) array_like
        Input array.
    ar2 : array_like
        The values against which to test each value of `ar1`.
    assume_unique : bool, optional
        If True, the input arrays are both assumed to be unique, which
        can speed up the calculation.  Default is False.
    invert : bool, optional
        If True, the values in the returned array are inverted (that is,
        False where an element of `ar1` is in `ar2` and True otherwise).
        Default is False. ``np.in1d(a, b, invert=True)`` is equivalent
        to (but is faster than) ``np.invert(in1d(a, b))``.
    kind : {None, 'sort', 'table'}, optional
        The algorithm to use. This will not affect the final result,
        but will affect the speed and memory use. The default, None,
        will select automatically based on memory considerations.

        * If 'sort', will use a mergesort-based approach. This will have
          a memory usage of roughly 6 times the sum of the sizes of
          `ar1` and `ar2`, not accounting for size of dtypes.
        * If 'table', will use a lookup table approach similar
          to a counting sort. This is only available for boolean and
          integer arrays. This will have a memory usage of the
          size of `ar1` plus the max-min value of `ar2`. `assume_unique`
          has no effect when the 'table' option is used.
        * If None, will automatically choose 'table' if
          the required memory allocation is less than or equal to
          6 times the sum of the sizes of `ar1` and `ar2`,
          otherwise will use 'sort'. This is done to not use
          a large amount of memory by default, even though
          'table' may be faster in most cases. If 'table' is chosen,
          `assume_unique` will have no effect.

        .. versionadded:: 1.8.0

    Returns
    -------
    in1d : (M,) ndarray, bool
        The values `ar1[in1d]` are in `ar2`.

    See Also
    --------
    isin                  : Version of this function that preserves the
                            shape of ar1.
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.

    Notes
    -----
    `in1d` can be considered as an element-wise function version of the
    python keyword `in`, for 1-D sequences. ``in1d(a, b)`` is roughly
    equivalent to ``np.array([item in b for item in a])``.
    However, this idea fails if `ar2` is a set, or similar (non-sequence)
    container:  As ``ar2`` is converted to an array, in those cases
    ``asarray(ar2)`` is an object array rather than the expected array of
    contained values.

    Using ``kind='table'`` tends to be faster than `kind='sort'` if the
    following relationship is true:
    ``log10(len(ar2)) > (log10(max(ar2)-min(ar2)) - 2.27) / 0.927``,
    but may use greater memory. The default value for `kind` will
    be automatically selected based only on memory usage, so one may
    manually set ``kind='table'`` if memory constraints can be relaxed.

    .. versionadded:: 1.4.0

    Examples
    --------
    >>> test = np.array([0, 1, 2, 5, 0])
    >>> states = [0, 2]
    >>> mask = np.in1d(test, states)
    >>> mask
    array([ True, False,  True, False,  True])
    >>> test[mask]
    array([0, 2, 0])
    >>> mask = np.in1d(test, states, invert=True)
    >>> mask
    array([False,  True, False,  True, False])
    >>> test[mask]
    array([1, 5])
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    Calculates ``element in test_elements``, broadcasting over `element` only.
    Returns a boolean array of the same shape as `element` that is True
    where an element of `element` is in `test_elements` and False otherwise.

    Parameters
    ----------
    element : array_like
        Input array.
    test_elements : array_like
        The values against which to test each value of `element`.
        This argument is flattened if it is an array or array_like.
        See notes for behavior with non-array-like parameters.
    assume_unique : bool, optional
        If True, the input arrays are both assumed to be unique, which
        can speed up the calculation.  Default is False.
    invert : bool, optional
        If True, the values in the returned array are inverted, as if
        calculating `element not in test_elements`. Default is False.
        ``np.isin(a, b, invert=True)`` is equivalent to (but faster
        than) ``np.invert(np.isin(a, b))``.
    kind : {None, 'sort', 'table'}, optional
        The algorithm to use. This will not affect the final result,
        but will affect the speed and memory use. The default, None,
        will select automatically based on memory considerations.

        * If 'sort', will use a mergesort-based approach. This will have
          a memory usage of roughly 6 times the sum of the sizes of
          `ar1` and `ar2`, not accounting for size of dtypes.
        * If 'table', will use a lookup table approach similar
          to a counting sort. This is only available for boolean and
          integer arrays. This will have a memory usage of the
          size of `ar1` plus the max-min value of `ar2`. `assume_unique`
          has no effect when the 'table' option is used.
        * If None, will automatically choose 'table' if
          the required memory allocation is less than or equal to
          6 times the sum of the sizes of `ar1` and `ar2`,
          otherwise will use 'sort'. This is done to not use
          a large amount of memory by default, even though
          'table' may be faster in most cases. If 'table' is chosen,
          `assume_unique` will have no effect.


    Returns
    -------
    isin : ndarray, bool
        Has the same shape as `element`. The values `element[isin]`
        are in `test_elements`.

    See Also
    --------
    in1d                  : Flattened version of this function.
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.

    Notes
    -----

    `isin` is an element-wise function version of the python keyword `in`.
    ``isin(a, b)`` is roughly equivalent to
    ``np.array([item in b for item in a])`` if `a` and `b` are 1-D sequences.

    `element` and `test_elements` are converted to arrays if they are not
    already. If `test_elements` is a set (or other non-sequence collection)
    it will be converted to an object array with one element, rather than an
    array of the values contained in `test_elements`. This is a consequence
    of the `array` constructor's way of handling non-sequence collections.
    Converting the set to a list usually gives the desired behavior.

    Using ``kind='table'`` tends to be faster than `kind='sort'` if the
    following relationship is true:
    ``log10(len(ar2)) > (log10(max(ar2)-min(ar2)) - 2.27) / 0.927``,
    but may use greater memory. The default value for `kind` will
    be automatically selected based only on memory usage, so one may
    manually set ``kind='table'`` if memory constraints can be relaxed.

    .. versionadded:: 1.13.0

    Examples
    --------
    >>> element = 2*np.arange(4).reshape((2, 2))
    >>> element
    array([[0, 2],
           [4, 6]])
    >>> test_elements = [1, 2, 4, 8]
    >>> mask = np.isin(element, test_elements)
    >>> mask
    array([[False,  True],
           [ True, False]])
    >>> element[mask]
    array([2, 4])

    The indices of the matched values can be obtained with `nonzero`:

    >>> np.nonzero(mask)
    (array([0, 1]), array([1, 0]))

    The test can also be inverted:

    >>> mask = np.isin(element, test_elements, invert=True)
    >>> mask
    array([[ True, False],
           [False,  True]])
    >>> element[mask]
    array([0, 6])

    Because of how `array` handles sets, the following does not
    work as expected:

    >>> test_set = {1, 2, 4, 8}
    >>> np.isin(element, test_set)
    array([[False, False],
           [False, False]])

    Casting the set to a list gives the expected result:

    >>> np.isin(element, list(test_set))
    array([[False,  True],
           [ True, False]])
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    Find the union of two arrays.

    Return the unique, sorted array of values that are in either of the two
    input arrays.

    Parameters
    ----------
    ar1, ar2 : array_like
        Input arrays. They are flattened if they are not already 1D.

    Returns
    -------
    union1d : ndarray
        Unique, sorted union of the input arrays.

    See Also
    --------
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.

    Examples
    --------
    >>> np.union1d([-1, 0, 1], [-2, 0, 2])
    array([-2, -1,  0,  1,  2])

    To find the union of more than two arrays, use functools.reduce:

    >>> from functools import reduce
    >>> reduce(np.union1d, ([1, 3, 4, 3], [3, 1, 2, 1], [6, 3, 4, 2]))
    array([1, 2, 3, 4, 6])
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    Find the set difference of two arrays.

    Return the unique values in `ar1` that are not in `ar2`.

    Parameters
    ----------
    ar1 : array_like
        Input array.
    ar2 : array_like
        Input comparison array.
    assume_unique : bool
        If True, the input arrays are both assumed to be unique, which
        can speed up the calculation.  Default is False.

    Returns
    -------
    setdiff1d : ndarray
        1D array of values in `ar1` that are not in `ar2`. The result
        is sorted when `assume_unique=False`, but otherwise only sorted
        if the input is sorted.

    See Also
    --------
    numpy.lib.arraysetops : Module with a number of other functions for
                            performing set operations on arrays.

    Examples
    --------
    >>> a = np.array([1, 2, 3, 2, 4, 1])
    >>> b = np.array([3, 4, 5, 6])
    >>> np.setdiff1d(a, b)
    array([1, 2])

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