



'''simple docstring'''





def 			a_	(				_lowerCAmelCase			) -> int:
  __lowerCamelCase			:			Optional[int]       			=					len(_lowerCAmelCase			)
  __lowerCamelCase			:			Any       			=					len(matrix[0]			)
  __lowerCamelCase			:			List[Any]       			=					min(_lowerCAmelCase			,_lowerCAmelCase			)

  for row in range(_lowerCAmelCase			):
    # Check if diagonal element is not zero
    if matrix[row][row] != 0:
      # Eliminate all the elements below the diagonal
      for col in range(row + 1			,_lowerCAmelCase			):
        __lowerCamelCase			:			Union[str, Any]       			=					matrix[col][row] / matrix[row][row]
        for i in range(_lowerCAmelCase			,_lowerCAmelCase			):
          matrix[col][i] -= multiplier * matrix[row][i]
    else:
      # Find a non-zero diagonal element to swap rows
      __lowerCamelCase			:			Union[str, Any]       			=					True
      for i in range(row + 1			,_lowerCAmelCase			):
        if matrix[i][row] != 0:
          __lowerCamelCase			,__lowerCamelCase			:			List[Any]       			=					matrix[i], matrix[row]
          __lowerCamelCase			:			Tuple       			=					False
          break
      if reduce:
        rank -= 1
        for i in range(_lowerCAmelCase			):
          __lowerCamelCase			:			int       			=					matrix[i][rank]

            # Reduce the row pointer by one to stay on the same row
      row -= 1

  return rank


if __name__ == "__main__":
   import doctest

   doctest.testmod()











'''simple docstring'''





# Algorithm for the pigeonhole sorting



def 			a_	(				_lowerCAmelCase			) -> List[str]:
  __lowerCamelCase			:			int       			=					min(_lowerCAmelCase			)  # min() finds the minimum value
  __lowerCamelCase			:			List[Any]       			=					max(_lowerCAmelCase			)  # max() finds the maximum value

  __lowerCamelCase			:			List[Any]       			=					max_val - min_val + 1  # size is difference of max and min values plus one

  # list of pigeonholes of size equal to the variable size
  __lowerCamelCase			:			List[Any]       			=					[0] * size

  # Populate the pigeonholes.
  for x in a:
    assert isinstance(_lowerCAmelCase			,_lowerCAmelCase			), "integers only please"
    holes[x - min_val] += 1

  # Putting the elements back into the array in an order.
  __lowerCamelCase			:			Tuple       			=					0
  for count in range(_lowerCAmelCase			):
    while holes[count] > 0:
      holes[count] -= 1
      __lowerCamelCase			:			List[Any]       			=					count + min_val
      i += 1

def 			a_	(				) -> str:
  __lowerCamelCase			:			Optional[Any]       			=					[8, 3, 2, 7, 4, 6, 8]
  pigeonhole_sort(_lowerCAmelCase			)
  print('Sorted order is:'			,' '.join(_lowerCAmelCase			)			)


if __name__ == "__main__":
   main()






