






from __future__ import annotations






def     _snake_case    (__lowercase    ,      __lowercase):
			UpperCamelCase_									=     []
			UpperCamelCase_									=     []
			UpperCamelCase_									=     0
			UpperCamelCase_									=     sum(__lowercase)
			create_state_space_tree(__lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase)
			return result






def     _snake_case    (__lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase    ,      __lowercase    ,      ):
			if sum(__lowercase) > max_sum or (remaining_nums_sum + sum(__lowercase)) < max_sum:
						return
			if sum(__lowercase) == max_sum:
						result.append(__lowercase)
						return
			for index in range(__lowercase    ,      len(__lowercase)):
						create_state_space_tree(
						    __lowercase    ,      __lowercase    ,      index + 1    ,      [*path, nums[index]]    ,      __lowercase    ,      remaining_nums_sum - nums[index]    ,      )
snake_case__      :	Optional[int]       	=      [3, 3_4, 4, 1_2, 5, 2]
snake_case__      :	Union[str, Any]       	=      9
snake_case__      :	Dict       	=      generate_sum_of_subsets_soln(nums, max_sum)

print(*result)













def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)
   if n == 0:
      return 0
   UpperCamelCase_									=     float('-inf')
   for i in range(1    ,      n + 1):
      UpperCamelCase_									=     max(
          __lowercase    ,      prices[i - 1] + naive_cut_rod_recursive(n - i    ,      __lowercase))

   return max_revue






def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     [float('-inf') for _ in range(n + 1)]
   return _top_down_cut_rod_recursive(__lowercase    ,      __lowercase    ,      __lowercase)






def     _snake_case    (__lowercase    ,      __lowercase    ,      __lowercase):
   if max_rev[n] >= 0:
      return max_rev[n]
   elif n == 0:
      return 0
   else:
      UpperCamelCase_									=     float('-inf')
      for i in range(1    ,      n + 1):
         UpperCamelCase_									=     max(
             __lowercase    ,      prices[i - 1] + _top_down_cut_rod_recursive(n - i    ,      __lowercase    ,      __lowercase)    ,      )

      UpperCamelCase_									=     max_revenue

   return max_rev[n]






def     _snake_case    (__lowercase    ,      __lowercase):
   _enforce_args(__lowercase    ,      __lowercase)

   # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
   # length 0.
   UpperCamelCase_									=     [float('-inf') for _ in range(n + 1)]
   UpperCamelCase_									=     0

   for i in range(1    ,      n + 1):
      UpperCamelCase_									=     max_rev[i]
      for j in range(1    ,      i + 1):
         UpperCamelCase_									=     max(__lowercase    ,      prices[j - 1] + max_rev[i - j])

      UpperCamelCase_									=     max_revenue_i

   return max_rev[n]






def     _snake_case    (__lowercase    ,      __lowercase):
   if n < 0:
      UpperCamelCase_									=     f"""n must be greater than or equal to 0. Got n = {n}"""
      raise ValueError(__lowercase)

   if n > len(__lowercase):
      UpperCamelCase_									=     (
          'Each integral piece of rod must have a corresponding price. '
          f"""Got n = {n} but length of prices = {len(__lowercase)}"""
      )
      raise ValueError(__lowercase)






def     _snake_case    ():
   UpperCamelCase_									=     [6, 10, 12, 15, 20, 23]
   UpperCamelCase_									=     len(__lowercase)

   # the best revenue comes from cutting the rod into 6 pieces, each
   # of length 1 resulting in a revenue of 6 * 6 = 36.
   UpperCamelCase_									=     36

   UpperCamelCase_									=     top_down_cut_rod(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     bottom_up_cut_rod(__lowercase    ,      __lowercase)
   UpperCamelCase_									=     naive_cut_rod_recursive(__lowercase    ,      __lowercase)

   assert expected_max_revenue == max_rev_top_down
   assert max_rev_top_down == max_rev_bottom_up
   assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
 main()





