






def     a_   ()				->						list[list[int]]:



					"""simple docstring"""
					return [list(range(1_000 - i				,       -1_000 - i				,       -1      )      ) for i in range(1_000      )]


UpperCAmelCase__      			=						generate_large_matrix()
UpperCAmelCase__      			=						(
    [[4, 3, 2, -1], [3, 2, 1, -1], [1, 1, -1, -2], [-1, -1, -2, -3]],
    [[3, 2], [1, 0]],
    [[7, 7, 6]],
    [[7, 7, 6], [-1, -2, -3]],
    grid,
)







def     a_   (__A      )				->						None:



					"""simple docstring"""
					assert all(row == sorted(__UpperCAmelCase				,       reverse=__UpperCAmelCase      ) for row in grid      )
					assert all(list(__UpperCAmelCase      ) == sorted(__UpperCAmelCase				,       reverse=__UpperCAmelCase      ) for col in zip(*__UpperCAmelCase      )      )







def     a_   (__A      )				->						int:



					"""simple docstring"""
					__a					:						Optional[int]      =					0
					__a					:						Optional[int]      =					len(__UpperCAmelCase      ) - 1

					# Edge cases such as no values or all numbers are negative.
					if not array or array[0] < 0:
										return 0

					while right + 1 > left:
										__a					:						int      =					(left + right) // 2
										__a					:						List[Any]      =					array[mid]

										# Num must be negative and the index must be greater than or equal to 0.
										if num < 0 and array[mid - 1] >= 0:
															return mid

										if num >= 0:
															__a					:						List[str]      =					mid + 1
										else:
															__a					:						List[str]      =					mid - 1
    # No negative numbers so return the last index of the array + 1 which is the length.
					return len(__UpperCAmelCase      )







def     a_   (__A      )				->						int:



					"""simple docstring"""
					__a					:						str      =					0
					__a					:						List[Any]      =					len(grid[0]      )

					for i in range(len(__UpperCAmelCase      )      ):
										__a					:						Tuple      =					find_negative_index(grid[i][:bound]      )
										total += bound
					return (len(__UpperCAmelCase      ) * len(grid[0]      )) - total







def     a_   (__A      )				->						int:



					"""simple docstring"""
					return len([number for row in grid for number in row if number < 0]      )







def     a_   (__A      )				->						int:



					"""simple docstring"""
					__a					:						int      =					0
					for row in grid:
										for i, number in enumerate(__UpperCAmelCase      ):
															if number < 0:
																				total += len(__UpperCAmelCase      ) - i
																				break
					return total







def     a_   ()				->						None:



					"""simple docstring"""
					from timeit import timeit

					print("Running benchmarks"      )
					__a					:						int      =					(
					    "from __main__ import count_negatives_binary_search, "
					    "count_negatives_brute_force, count_negatives_brute_force_with_break, grid"
					)
					for func in (
					    "count_negatives_binary_search",  # took 0.7727 seconds
					    "count_negatives_brute_force_with_break",  # took 4.6505 seconds
					    "count_negatives_brute_force",  # took 12.8160 seconds
					):
										__a					:						Any      =					timeit(f'{func}(grid=grid)'				,       setup=__UpperCAmelCase				,       number=500      )
										print(f'{func}() took {time:0.4f} seconds'      )


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			benchmark()











from __future__ import annotations

from typing import TypedDict





class 			lowerCamelCase_				( _SCREAMING_SNAKE_CASE   ):


 '''simple docstring'''







 lowercase_							       =       42
 lowercase_							       =       42


def    UpperCAmelCase_				( __UpperCAmelCase		:      str				)   ->					list[str]:
      if not isinstance(__UpperCAmelCase     ,     __UpperCAmelCase				):
            raise TypeError('The parameter s type must be str.'				)

      return [s[i:] + s[:i] for i in range(len(__UpperCAmelCase				)				)]

def    UpperCAmelCase_				( __UpperCAmelCase		:      str				)   ->					BWTTransformDict:
      if not isinstance(__UpperCAmelCase     ,     __UpperCAmelCase				):
            raise TypeError('The parameter s type must be str.'				)
      if not s:
            raise ValueError('The parameter s must not be empty.'				)

      SCREAMING_SNAKE_CASE_      		=     all_rotations(__UpperCAmelCase				)
      rotations.sort()  # sort the list of rotations in alphabetically order
      # make a string composed of the last char of each rotation
      SCREAMING_SNAKE_CASE_      		=     {
          "bwt_string": "".join([word[-1] for word in rotations]				),
          "idx_original_string": rotations.index(__UpperCAmelCase				),
      }
      return response

def    UpperCAmelCase_				( __UpperCAmelCase		:      str     ,     __UpperCAmelCase		:      int				)   ->					str:
      if not isinstance(__UpperCAmelCase     ,     __UpperCAmelCase				):
            raise TypeError('The parameter bwt_string type must be str.'				)
      if not bwt_string:
            raise ValueError('The parameter bwt_string must not be empty.'				)
      try:
            SCREAMING_SNAKE_CASE_      		=     int(__UpperCAmelCase				)
      except ValueError:
            raise TypeError(
                'The parameter idx_original_string type must be int or passive'
                ' of cast to int.'				)
      if idx_original_string < 0:
            raise ValueError('The parameter idx_original_string must not be lower than 0.'				)
      if idx_original_string >= len(__UpperCAmelCase				):
            raise ValueError(
                'The parameter idx_original_string must be lower than' ' len(bwt_string).'				)

      SCREAMING_SNAKE_CASE_      		=     [''] * len(__UpperCAmelCase				)
      for _ in range(len(__UpperCAmelCase				)				):
            for i in range(len(__UpperCAmelCase				)				):
                  SCREAMING_SNAKE_CASE_      		=     bwt_string[i] + ordered_rotations[i]
            ordered_rotations.sort()
      return ordered_rotations[idx_original_string]


if __name__ == "__main__":
   lowerCamelCase__					:  Optional[int]      							=				'Provide a string that I will generate its BWT transform: '
   lowerCamelCase__					:  List[str]      							=				input(entry_msg).strip()
   lowerCamelCase__					:  int      							=				bwt_transform(s)
   print(
       f'''Burrows Wheeler transform for string \'{s}\' results '''
       f'''in \'{result['bwt_string']}\''''
   )
   lowerCamelCase__					:  Dict      							=				reverse_bwt(result['bwt_string'], result['idx_original_string'])
   print(
       f'''Reversing Burrows Wheeler transform for entry \'{result['bwt_string']}\' '''
       f'''we get original string \'{original_string}\''''
   )