prompt: You will be provided with a partial code base and an issue statement explaining a problem to resolve.
  <issue>
  Make cupy.sort support arrays with rank two or more.
  # Background
  Arrays sorted with `cupy.sort` operation have some properties such as dtype, rank, sorting axis and C/F-contiguousness. Currently, `cupy.sort` supports sorting arrays only with the rank of one because of its implementation reason, see #55.
  
  # Problem
  This issue addresses a problem that makes `cupy.sort` support sorting arrays with the rank of two or more, with the last axis and C-contiguousness.
  
  # Approach
  
  **Rank two**
  
  For an array with the rank of two, 
  
  ```
  [[4, 3]
   [2, 1]]
  ```
  
  treating the array as flattened one, `[4, 3, 2 ,1]`, and providing the following comparator in pseudo code to underlying Thrust library:
  
  ```
  if floor(i / 2) < floor(j / 2) then return true;
  else if floor(i / 2) > floor(j / 2) then return false;
  else return data[i] < data[j];
  ```
  
  where `i` and `j` are array indices, and `data[i]` represents `i` th element of array `data`,
  
  we get the C-contiguous array sorted with the last axis.
  
  ```
  [[3, 4]
   [1, 2]]
  ```
  
  **Rank N**
  
  Generalized to the rank of N with shape `(d_0, d_1, ..., d_n-1)`, the following comparator works:
  
  ```
  if floor(i / d_n-1) < floor(j / d_n-1) then return true;
  else if floor(i / d_n-1) > floor(j / d_n-1) then return false;
  else return data[i] < data[j];
  ```
  
  
  </issue>
  <code>
  [start of cupy/sorting/sort.py]
  1 import cupy
  2 import numpy
  3 
  4 if cupy.cuda.thrust_enabled:
  5     from cupy.cuda import thrust
  6 
  7 
  8 def sort(a):
  9     """Returns a sorted copy of an array with a stable sorting algorithm.
  10 
  11     Args:
  12         a (cupy.ndarray): Array to be sorted.
  13 
  14     Returns:
  15         cupy.ndarray: Array of the same type and shape as ``a``.
  16 
  17     .. note::
  18        For its implementation reason, ``cupy.sort`` currently supports only
  19        arrays with their rank of one and does not support ``axis``, ``kind``
  20        and ``order`` parameters that ``numpy.sort`` does support.
  21 
  22     .. seealso:: :func:`numpy.sort`
  23 
  24     """
  25     ret = a.copy()
  26     ret.sort()
  27     return ret
  28 
  29 
  30 def lexsort(keys):
  31     """Perform an indirect sort using an array of keys.
  32 
  33     Args:
  34         keys (cupy.ndarray): ``(k, N)`` array containing ``k`` ``(N,)``-shaped
  35             arrays. The ``k`` different "rows" to be sorted. The last row is
  36             the primary sort key.
  37 
  38     Returns:
  39         cupy.ndarray: Array of indices that sort the keys.
  40 
  41     .. note::
  42         For its implementation reason, ``cupy.lexsort`` currently supports only
  43         keys with their rank of one or two and does not support ``axis``
  44         parameter that ``numpy.lexsort`` supports.
  45 
  46     .. seealso:: :func:`numpy.lexsort`
  47 
  48     """
  49 
  50     # TODO(takagi): Support axis argument.
  51 
  52     if not cupy.cuda.thrust_enabled:
  53         raise RuntimeError('Thrust is needed to use cupy.lexsort. Please '
  54                            'install CUDA Toolkit with Thrust then reinstall '
  55                            'CuPy after uninstalling it.')
  56 
  57     if keys.ndim == ():
  58         # as numpy.lexsort() raises
  59         raise TypeError('need sequence of keys with len > 0 in lexsort')
  60 
  61     if keys.ndim == 1:
  62         return 0
  63 
  64     # TODO(takagi): Support ranks of three or more.
  65     if keys.ndim > 2:
  66         raise NotImplementedError('Keys with the rank of three or more is not '
  67                                   'supported in lexsort')
  68 
  69     idx_array = cupy.ndarray(keys._shape[1:], dtype=numpy.intp)
  70     k = keys._shape[0]
  71     n = keys._shape[1]
  72     thrust.lexsort(keys.dtype, idx_array.data.ptr, keys.data.ptr, k, n)
  73 
  74     return idx_array
  75 
  76 
  77 def argsort(a):
  78     """Return the indices that would sort an array with a stable sorting.
  79 
  80     Args:
  81         a (cupy.ndarray): Array to sort.
  82 
  83     Returns:
  84         cupy.ndarray: Array of indices that sort ``a``.
  85 
  86     .. note::
  87        For its implementation reason, ``cupy.argsort`` currently supports only
  88        arrays with their rank of one and does not support ``axis``, ``kind``
  89        and ``order`` parameters that ``numpy.argsort`` supports.
  90 
  91     .. seealso:: :func:`numpy.argsort`
  92 
  93     """
  94     return a.argsort()
  95 
  96 
  97 # TODO(okuta): Implement msort
  98 
  99 
  100 # TODO(okuta): Implement sort_complex
  101 
  102 
  103 # TODO(okuta): Implement partition
  104 
  105 
  106 # TODO(okuta): Implement argpartition
  107
  [end of cupy/sorting/sort.py]
  </code>
  
  I need you to solve this issue by generating a single patch file that I can apply directly to this repository using git apply. Please respond with a single patch file in the following format.
  <patch>
  diff --git a/file.py b/file.py
  --- a/file.py
  +++ b/file.py
  @@ -1,27 +1,35 @@
   def euclidean(a, b):
  -    while b:
  -        a, b = b, a % b
  -    return a
  +    if b == 0:
  +        return a
  +    return euclidean(b, a % b)
   
   
   def bresenham(x0, y0, x1, y1):
       points = []
       dx = abs(x1 - x0)
       dy = abs(y1 - y0)
  -    sx = 1 if x0 < x1 else -1
  -    sy = 1 if y0 < y1 else -1
  -    err = dx - dy
  +    x, y = x0, y0
  +    sx = -1 if x0 > x1 else 1
  +    sy = -1 if y0 > y1 else 1
   
  -    while True:
  -        points.append((x0, y0))
  -        if x0 == x1 and y0 == y1:
  -            break
  -        e2 = 2 * err
  -        if e2 > -dy:
  -            err -= dy
  -            x0 += sx
  -        if e2 < dx:
  -            err += dx
  -            y0 += sy
  +    if dx > dy:
  +        err = dx / 2.0
  +        while x != x1:
  +            points.append((x, y))
  +            err -= dy
  +            if err < 0:
  +                y += sy
  +                err += dx
  +            x += sx
  +    else:
  +        err = dy / 2.0
  +        while y != y1:
  +            points.append((x, y))
  +            err -= dx
  +            if err < 0:
  +                x += sx
  +                err += dy
  +            y += sy
  +
  +    points.append((x, y))
       return points
  
  </patch>
golden_diff: diff --git a/cupy/sorting/sort.py b/cupy/sorting/sort.py
  --- a/cupy/sorting/sort.py
  +++ b/cupy/sorting/sort.py
  @@ -15,9 +15,9 @@
           cupy.ndarray: Array of the same type and shape as ``a``.
   
       .. note::
  -       For its implementation reason, ``cupy.sort`` currently supports only
  -       arrays with their rank of one and does not support ``axis``, ``kind``
  -       and ``order`` parameters that ``numpy.sort`` does support.
  +       For its implementation reason, ``cupy.sort`` currently does not support
  +       ``axis``, ``kind`` and ``order`` parameters that ``numpy.sort`` does
  +       support.
   
       .. seealso:: :func:`numpy.sort`
  
verification_info: {"golden_diff": "diff --git a/cupy/sorting/sort.py b/cupy/sorting/sort.py\n--- a/cupy/sorting/sort.py\n+++ b/cupy/sorting/sort.py\n@@ -15,9 +15,9 @@\n         cupy.ndarray: Array of the same type and shape as ``a``.\n \n     .. note::\n-       For its implementation reason, ``cupy.sort`` currently supports only\n-       arrays with their rank of one and does not support ``axis``, ``kind``\n-       and ``order`` parameters that ``numpy.sort`` does support.\n+       For its implementation reason, ``cupy.sort`` currently does not support\n+       ``axis``, ``kind`` and ``order`` parameters that ``numpy.sort`` does\n+       support.\n \n     .. seealso:: :func:`numpy.sort`\n", "issue": "Make cupy.sort support arrays with rank two or more.\n# Background\r\nArrays sorted with `cupy.sort` operation have some properties such as dtype, rank, sorting axis and C/F-contiguousness. Currently, `cupy.sort` supports sorting arrays only with the rank of one because of its implementation reason, see #55.\r\n\r\n# Problem\r\nThis issue addresses a problem that makes `cupy.sort` support sorting arrays with the rank of two or more, with the last axis and C-contiguousness.\r\n\r\n# Approach\r\n\r\n**Rank two**\r\n\r\nFor an array with the rank of two, \r\n\r\n```\r\n[[4, 3]\r\n [2, 1]]\r\n```\r\n\r\ntreating the array as flattened one, `[4, 3, 2 ,1]`, and providing the following comparator in pseudo code to underlying Thrust library:\r\n\r\n```\r\nif floor(i / 2) < floor(j / 2) then return true;\r\nelse if floor(i / 2) > floor(j / 2) then return false;\r\nelse return data[i] < data[j];\r\n```\r\n\r\nwhere `i` and `j` are array indices, and `data[i]` represents `i` th element of array `data`,\r\n\r\nwe get the C-contiguous array sorted with the last axis.\r\n\r\n```\r\n[[3, 4]\r\n [1, 2]]\r\n```\r\n\r\n**Rank N**\r\n\r\nGeneralized to the rank of N with shape `(d_0, d_1, ..., d_n-1)`, the following comparator works:\r\n\r\n```\r\nif floor(i / d_n-1) < floor(j / d_n-1) then return true;\r\nelse if floor(i / d_n-1) > floor(j / d_n-1) then return false;\r\nelse return data[i] < data[j];\r\n```\r\n\n", "before_files": [{"content": "import cupy\nimport numpy\n\nif cupy.cuda.thrust_enabled:\n    from cupy.cuda import thrust\n\n\ndef sort(a):\n    \"\"\"Returns a sorted copy of an array with a stable sorting algorithm.\n\n    Args:\n        a (cupy.ndarray): Array to be sorted.\n\n    Returns:\n        cupy.ndarray: Array of the same type and shape as ``a``.\n\n    .. note::\n       For its implementation reason, ``cupy.sort`` currently supports only\n       arrays with their rank of one and does not support ``axis``, ``kind``\n       and ``order`` parameters that ``numpy.sort`` does support.\n\n    .. seealso:: :func:`numpy.sort`\n\n    \"\"\"\n    ret = a.copy()\n    ret.sort()\n    return ret\n\n\ndef lexsort(keys):\n    \"\"\"Perform an indirect sort using an array of keys.\n\n    Args:\n        keys (cupy.ndarray): ``(k, N)`` array containing ``k`` ``(N,)``-shaped\n            arrays. The ``k`` different \"rows\" to be sorted. The last row is\n            the primary sort key.\n\n    Returns:\n        cupy.ndarray: Array of indices that sort the keys.\n\n    .. note::\n        For its implementation reason, ``cupy.lexsort`` currently supports only\n        keys with their rank of one or two and does not support ``axis``\n        parameter that ``numpy.lexsort`` supports.\n\n    .. seealso:: :func:`numpy.lexsort`\n\n    \"\"\"\n\n    # TODO(takagi): Support axis argument.\n\n    if not cupy.cuda.thrust_enabled:\n        raise RuntimeError('Thrust is needed to use cupy.lexsort. Please '\n                           'install CUDA Toolkit with Thrust then reinstall '\n                           'CuPy after uninstalling it.')\n\n    if keys.ndim == ():\n        # as numpy.lexsort() raises\n        raise TypeError('need sequence of keys with len > 0 in lexsort')\n\n    if keys.ndim == 1:\n        return 0\n\n    # TODO(takagi): Support ranks of three or more.\n    if keys.ndim > 2:\n        raise NotImplementedError('Keys with the rank of three or more is not '\n                                  'supported in lexsort')\n\n    idx_array = cupy.ndarray(keys._shape[1:], dtype=numpy.intp)\n    k = keys._shape[0]\n    n = keys._shape[1]\n    thrust.lexsort(keys.dtype, idx_array.data.ptr, keys.data.ptr, k, n)\n\n    return idx_array\n\n\ndef argsort(a):\n    \"\"\"Return the indices that would sort an array with a stable sorting.\n\n    Args:\n        a (cupy.ndarray): Array to sort.\n\n    Returns:\n        cupy.ndarray: Array of indices that sort ``a``.\n\n    .. note::\n       For its implementation reason, ``cupy.argsort`` currently supports only\n       arrays with their rank of one and does not support ``axis``, ``kind``\n       and ``order`` parameters that ``numpy.argsort`` supports.\n\n    .. seealso:: :func:`numpy.argsort`\n\n    \"\"\"\n    return a.argsort()\n\n\n# TODO(okuta): Implement msort\n\n\n# TODO(okuta): Implement sort_complex\n\n\n# TODO(okuta): Implement partition\n\n\n# TODO(okuta): Implement argpartition\n", "path": "cupy/sorting/sort.py"}]}