





def 							_lowerCAmelCase (  _a   :					int					,   _a   :					int					,   _a   :					list[list[int]]			)	->			int:

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

          lowerCAmelCase_					:       Optional[int]    =		update_area_of_max_square(_a					,   col + 1			)
          lowerCAmelCase_					:       List[Any]    =		update_area_of_max_square(row + 1					,   col + 1			)
          lowerCAmelCase_					:       Any    =		update_area_of_max_square(row + 1					,   _a			)

          if mat[row][col]:
               lowerCAmelCase_					:       Optional[Any]    =		1 + min([right, diagonal, down]			)
               lowerCAmelCase_					:       Optional[Any]    =		max(largest_square_area[0]					,   _a			)
               return sub_problem_sol
          else:
               return 0

     lowerCAmelCase_					:       Optional[int]    =		[0]
     update_area_of_max_square(0					,   0			)
     return largest_square_area[0]






def 							_lowerCAmelCase (  _a   :					int					,   _a   :					int					,   _a   :					list[list[int]]			)	->			int:

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

          lowerCAmelCase_					:       Optional[Any]    =		update_area_of_max_square_using_dp_array(_a					,   col + 1					,   _a			)
          lowerCAmelCase_					:       Optional[Any]    =		update_area_of_max_square_using_dp_array(row + 1					,   col + 1					,   _a			)
          lowerCAmelCase_					:       int    =		update_area_of_max_square_using_dp_array(row + 1					,   _a					,   _a			)

          if mat[row][col]:
               lowerCAmelCase_					:       Tuple    =		1 + min([right, diagonal, down]			)
               lowerCAmelCase_					:       List[Any]    =		max(largest_square_area[0]					,   _a			)
               lowerCAmelCase_					:       Tuple    =		sub_problem_sol
               return sub_problem_sol
          else:
               return 0

     lowerCAmelCase_					:       Tuple    =		[0]
     lowerCAmelCase_					:       str    =		[[-1] * cols for _ in range(_a			)]
     update_area_of_max_square_using_dp_array(0					,   0					,   _a			)

     return largest_square_area[0]






def 							_lowerCAmelCase (  _a   :					int					,   _a   :					int					,   _a   :					list[list[int]]			)	->			int:
     lowerCAmelCase_					:       Any    =		[[0] * (cols + 1) for _ in range(rows + 1			)]
     lowerCAmelCase_					:       int    =		0
     for row in range(rows - 1					,   -1					,   -1			):
          for col in range(cols - 1					,   -1					,   -1			):
               lowerCAmelCase_					:       Optional[int]    =		dp_array[row][col + 1]
               lowerCAmelCase_					:       int    =		dp_array[row + 1][col + 1]
               lowerCAmelCase_					:       List[str]    =		dp_array[row + 1][col]

               if mat[row][col] == 1:
                    lowerCAmelCase_					:       int    =		1 + min(_a					,   _a					,   _a			)
                    lowerCAmelCase_					:       Tuple    =		max(dp_array[row][col]					,   _a			)
               else:
                    lowerCAmelCase_					:       Union[str, Any]    =		0

     return largest_square_area






def 							_lowerCAmelCase (  _a   :					int					,   _a   :					int					,   _a   :					list[list[int]]			)	->			int:
     lowerCAmelCase_					:       Dict    =		[0] * (cols + 1)
     lowerCAmelCase_					:       int    =		[0] * (cols + 1)
     lowerCAmelCase_					:       Optional[int]    =		0
     for row in range(rows - 1					,   -1					,   -1			):
          for col in range(cols - 1					,   -1					,   -1			):
               lowerCAmelCase_					:       List[str]    =		current_row[col + 1]
               lowerCAmelCase_					:       Any    =		next_row[col + 1]
               lowerCAmelCase_					:       Dict    =		next_row[col]

               if mat[row][col] == 1:
                    lowerCAmelCase_					:       str    =		1 + min(_a					,   _a					,   _a			)
                    lowerCAmelCase_					:       Optional[Any]    =		max(current_row[col]					,   _a			)
               else:
                    lowerCAmelCase_					:       Optional[int]    =		0
          lowerCAmelCase_					:       Tuple    =		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]]))







def 							_lowerCAmelCase (  _a   :					int					,   _a   :					bool = False			)	->			bool:
     if n == 2:
          return True
     if not n % 2 or n < 2:
          return False
     if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
          return False
     if n > 3_31_70_44_06_46_79_88_73_85_96_19_81 and not allow_probable:
          raise ValueError(
              """Warning: upper bound of deterministic test is exceeded. """
              """Pass allow_probable=True to allow probabilistic test. """
              """A return value of True indicates a probable prime."""			)
     # array bounds provided by analysis
     lowerCAmelCase_					:       Optional[int]    =		[
         20_47,
         1_37_36_53,
         25_32_60_01,
         32_15_03_17_51,
         2_15_23_02_89_87_47,
         3_47_47_49_66_03_83,
         3_41_55_00_71_72_83_21,
         1,
         3_82_51_23_05_65_46_41_30_51,
         1,
         1,
         31_86_65_85_78_34_03_11_51_16_74_61,
         3_31_70_44_06_46_79_88_73_85_96_19_81,
     ]

     lowerCAmelCase_					:       Dict    =		[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
     for idx, _p in enumerate(_a					,   1			):
          if n < _p:
               # then we have our last prime to check
               lowerCAmelCase_					:       str    =		primes[:idx]
               break
     lowerCAmelCase_   ,  lowerCAmelCase_					:       Optional[Any]    =		n - 1, 0
     # break up n -1 into a power of 2 (s) and
     # remaining odd component
     # essentially, solve for d * 2 ** s == n - 1
     while d % 2 == 0:
          d //= 2
          s += 1
     for prime in plist:
          lowerCAmelCase_					:       Tuple    =		False
          for r in range(_a			):
               lowerCAmelCase_					:       Tuple    =		pow(_a					,   d * 2**r					,   _a			)
               # see article for analysis explanation for m
               if (r == 0 and m == 1) or ((m + 1) % n == 0):
                    lowerCAmelCase_					:       List[Any]    =		True
                    # this loop will not determine compositeness
                    break
          if pr:
               continue
          # if pr is False, then the above loop never evaluated to true,
          # and the n MUST be composite
          return False
     return True






def 							_lowerCAmelCase (  )	->			None:
     assert not miller_rabin(5_61			)
     assert miller_rabin(5_63			)
     # 2047

     assert not miller_rabin(83_82_01			)
     assert miller_rabin(83_82_07			)
     # 1_373_653

     assert not miller_rabin(17_31_60_01			)
     assert miller_rabin(17_31_60_17			)
     # 25_326_001

     assert not miller_rabin(30_78_38_66_41			)
     assert miller_rabin(30_78_38_66_53			)
     # 3_215_031_751

     assert not miller_rabin(1_71_30_45_57_48_01			)
     assert miller_rabin(1_71_30_45_57_48_19			)
     # 2_152_302_898_747

     assert not miller_rabin(2_77_97_99_72_83_07			)
     assert miller_rabin(2_77_97_99_72_83_27			)
     # 3_474_749_660_383

     assert not miller_rabin(1_13_85_00_23_90_94_41			)
     assert miller_rabin(1_13_85_00_23_90_95_27			)
     # 341_550_071_728_321

     assert not miller_rabin(1_27_50_41_01_88_48_80_43_51			)
     assert miller_rabin(1_27_50_41_01_88_48_80_43_91			)
     # 3_825_123_056_546_413_051

     assert not miller_rabin(7_96_66_46_44_58_50_77_87_79_18_67			)
     assert miller_rabin(7_96_66_46_44_58_50_77_87_79_19_51			)
     # 318_665_857_834_031_151_167_461

     assert not miller_rabin(55_28_40_67_74_46_64_78_97_66_03_33			)
     assert miller_rabin(55_28_40_67_74_46_64_78_97_66_03_59			)
     # 3_317_044_064_679_887_385_961_981
     # upper limit for probabilistic test


if __name__ == "__main__":
      test_miller_rabin()
