





'''simple docstring'''






from __future__ import annotations

from math import gcd
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase = 2							,	lowerCAmelCase = 1							,	lowerCAmelCase = 3							,	):





      """simple docstring"""


      if num < 2:
            raise ValueError("""The input value cannot be less than 2"""  )

      # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
      # algorithm struggles to find factors that are divisible by two.
      # As a workaround, we specifically check for two and even inputs.
      #   See: https://math.stackexchange.com/a/2856214/165820
      if num > 2 and num % 2 == 0:
            return 2

      # Pollard's Rho algorithm requires a function that returns pseudorandom
      # values between 0 <= X < ``num``.  It doesn't need to be random in the
      # sense that the output value is cryptographically secure or difficult
      # to calculate, it only needs to be random in the sense that all output
      # values should be equally likely to appear.
      # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
      # However, the success of Pollard's algorithm isn't guaranteed and is
      # determined in part by the initial seed and the chosen random function.
      # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
      # where ``C`` is a value that we can modify between each attempt.
      def rand_fn(lowerCAmelCase							,	lowerCAmelCase							,	lowerCAmelCase  ) -> int:
            return (pow(UpperCamelCase_							,	2  ) + step) % modulus

      for _ in range(UpperCamelCase_  ):
            # These track the position within the cycle detection logic.
            _lowerCAmelCase							     =							seed
            _lowerCAmelCase							     =							seed

            while True:
                  # At each iteration, the tortoise moves one step and the hare moves two.
                  _lowerCAmelCase							     =							rand_fn(UpperCamelCase_							,	UpperCamelCase_							,	UpperCamelCase_  )
                  _lowerCAmelCase							     =							rand_fn(UpperCamelCase_							,	UpperCamelCase_							,	UpperCamelCase_  )
                  _lowerCAmelCase							     =							rand_fn(UpperCamelCase_							,	UpperCamelCase_							,	UpperCamelCase_  )

                  # At some point both the tortoise and the hare will enter a cycle whose
                  # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
                  # the tortoise and hare will end up on the same value modulo ``p``.
                  # We can detect when this happens because the position difference between
                  # the tortoise and the hare will share a common divisor with ``num``.
                  _lowerCAmelCase							     =							gcd(hare - tortoise							,	UpperCamelCase_  )

                  if divisor == 1:
                        # No common divisor yet, just keep searching.
                        continue
                  else:
                        # We found a common divisor!
                        if divisor == num:
                              # Unfortunately, the divisor is ``num`` itself and is useless.
                              break
                        else:
                              # The divisor is a nontrivial factor of ``num``!
                              return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
            _lowerCAmelCase							     =							hare

            # The new step value for the random function can just be incremented.
            # At first the results will be similar to what the old function would
            # have produced, but the value will quickly diverge after a bit.
            step += 1

      # We haven't found a divisor within the requested number of attempts.
      # We were unlucky or ``num`` itself is actually prime.
      return None


if __name__ == "__main__":
      import argparse

      A__				:   Dict      =argparse.ArgumentParser()
      parser.add_argument(
          '''num''',
          type=int,
          help='''The value to find a divisor of''',
      )
      parser.add_argument(
          '''--attempts''',
          type=int,
          default=3,
          help='''The number of attempts before giving up''',
      )
      A__				:   int      =parser.parse_args()

      A__				:   int      =pollard_rho(args.num, attempts=args.attempts)
      if divisor is None:
            print(F"""{args.num} is probably prime""")
      else:
            A__				:   List[str]      =args.num // divisor
            print(F"""{args.num} = {divisor} * {quotient}""")








def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)		->		int:
					'''simple docstring'''


					def update_area_of_max_square(UpperCamelCase_	,	UpperCamelCase_		) -> int:
										# BASE CASE
										if row >= rows or col >= cols:
															return 0

										SCREAMING_SNAKE_CASE__													= update_area_of_max_square(UpperCamelCase_	,	col + 1		)
										SCREAMING_SNAKE_CASE__													= update_area_of_max_square(row + 1	,	col + 1		)
										SCREAMING_SNAKE_CASE__													= update_area_of_max_square(row + 1	,	UpperCamelCase_		)

										if mat[row][col]:
															SCREAMING_SNAKE_CASE__													= 1 + min([right, diagonal, down]		)
															SCREAMING_SNAKE_CASE__													= max(largest_square_area[0]	,	UpperCamelCase_		)
															return sub_problem_sol
										else:
															return 0

					SCREAMING_SNAKE_CASE__													= [0]
					update_area_of_max_square(0	,	0		)
					return largest_square_area[0]






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)		->		int:
					'''simple docstring'''


					def update_area_of_max_square_using_dp_array(
					    UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		) -> int:
										if row >= rows or col >= cols:
															return 0
										if dp_array[row][col] != -1:
															return dp_array[row][col]

										SCREAMING_SNAKE_CASE__													= update_area_of_max_square_using_dp_array(UpperCamelCase_	,	col + 1	,	UpperCamelCase_		)
										SCREAMING_SNAKE_CASE__													= update_area_of_max_square_using_dp_array(row + 1	,	col + 1	,	UpperCamelCase_		)
										SCREAMING_SNAKE_CASE__													= update_area_of_max_square_using_dp_array(row + 1	,	UpperCamelCase_	,	UpperCamelCase_		)

										if mat[row][col]:
															SCREAMING_SNAKE_CASE__													= 1 + min([right, diagonal, down]		)
															SCREAMING_SNAKE_CASE__													= max(largest_square_area[0]	,	UpperCamelCase_		)
															SCREAMING_SNAKE_CASE__													= sub_problem_sol
															return sub_problem_sol
										else:
															return 0

					SCREAMING_SNAKE_CASE__													= [0]
					SCREAMING_SNAKE_CASE__													= [[-1] * cols for _ in range(UpperCamelCase_		)]
					update_area_of_max_square_using_dp_array(0	,	0	,	UpperCamelCase_		)

					return largest_square_area[0]






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)		->		int:
					'''simple docstring'''


					SCREAMING_SNAKE_CASE__													= [[0] * (cols + 1) for _ in range(rows + 1		)]
					SCREAMING_SNAKE_CASE__													= 0
					for row in range(rows - 1	,	-1	,	-1		):
										for col in range(cols - 1	,	-1	,	-1		):
															SCREAMING_SNAKE_CASE__													= dp_array[row][col + 1]
															SCREAMING_SNAKE_CASE__													= dp_array[row + 1][col + 1]
															SCREAMING_SNAKE_CASE__													= dp_array[row + 1][col]

															if mat[row][col] == 1:
																				SCREAMING_SNAKE_CASE__													= 1 + min(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)
																				SCREAMING_SNAKE_CASE__													= max(dp_array[row][col]	,	UpperCamelCase_		)
															else:
																				SCREAMING_SNAKE_CASE__													= 0

					return largest_square_area






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)		->		int:
					'''simple docstring'''


					SCREAMING_SNAKE_CASE__													= [0] * (cols + 1)
					SCREAMING_SNAKE_CASE__													= [0] * (cols + 1)
					SCREAMING_SNAKE_CASE__													= 0
					for row in range(rows - 1	,	-1	,	-1		):
										for col in range(cols - 1	,	-1	,	-1		):
															SCREAMING_SNAKE_CASE__													= current_row[col + 1]
															SCREAMING_SNAKE_CASE__													= next_row[col + 1]
															SCREAMING_SNAKE_CASE__													= next_row[col]

															if mat[row][col] == 1:
																				SCREAMING_SNAKE_CASE__													= 1 + min(UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)
																				SCREAMING_SNAKE_CASE__													= max(current_row[col]	,	UpperCamelCase_		)
															else:
																				SCREAMING_SNAKE_CASE__													= 0
										SCREAMING_SNAKE_CASE__													= current_row

					return largest_square_area


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				print(largest_square_area_in_matrix_bottom_up(2, 2, [[1, 1], [1, 1]]))





