






'''simple docstring'''





def      snake_case		(							UpperCAmelCase	:       int,						UpperCAmelCase	:       list   ):
				_enforce_args(UpperCAmelCase,						UpperCAmelCase   )
				if n == 0:
								return 0
				A	 =      float('-inf'   )
				for i in range(1,						n + 1   ):
								A	 =      max(
								    UpperCAmelCase,						prices[i - 1] + naive_cut_rod_recursive(n - i,						UpperCAmelCase   )   )

				return max_revue







def      snake_case		(							UpperCAmelCase	:       int,						UpperCAmelCase	:       list   ):
				_enforce_args(UpperCAmelCase,						UpperCAmelCase   )
				A	 =      [float('-inf'   ) for _ in range(n + 1   )]
				return _top_down_cut_rod_recursive(UpperCAmelCase,						UpperCAmelCase,						UpperCAmelCase   )







def      snake_case		(							UpperCAmelCase	:       int,						UpperCAmelCase	:       list,						UpperCAmelCase	:       list   ):
				if max_rev[n] >= 0:
								return max_rev[n]
				elif n == 0:
								return 0
				else:
								A	 =      float('-inf'   )
								for i in range(1,						n + 1   ):
												A	 =      max(
												    UpperCAmelCase,						prices[i - 1] + _top_down_cut_rod_recursive(n - i,						UpperCAmelCase,						UpperCAmelCase   ),						)

								A	 =      max_revenue

				return max_rev[n]







def      snake_case		(							UpperCAmelCase	:       int,						UpperCAmelCase	:       list   ):
				_enforce_args(UpperCAmelCase,						UpperCAmelCase   )

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

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

								A	 =      max_revenue_i

				return max_rev[n]







def      snake_case		(							UpperCAmelCase	:       int,						UpperCAmelCase	:       list   ):
				if n < 0:
								A	 =      f'n must be greater than or equal to 0. Got n = {n}'
								raise ValueError(UpperCAmelCase   )

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







def      snake_case		(							):
				A	 =      [6, 10, 12, 15, 20, 23]
				A	 =      len(UpperCAmelCase   )

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

				A	 =      top_down_cut_rod(UpperCAmelCase,						UpperCAmelCase   )
				A	 =      bottom_up_cut_rod(UpperCAmelCase,						UpperCAmelCase   )
				A	 =      naive_cut_rod_recursive(UpperCAmelCase,						UpperCAmelCase   )

				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()













from collections.abc import Callable
class      UpperCamelCase :







		"""simple docstring"""



		def __init__(  self		:       Tuple						,_SCREAMING_SNAKE_CASE		:       Callable | None = None		)    ->   None:


						'''simple docstring'''
						# Stores actual heap items.
						A	 =      []
						# Stores indexes of each item for supporting updates and deletion.
						A	 =      {}
						# Stores current size of heap.
						A	 =      0
						# Stores function used to evaluate the score of an item on which basis ordering
						# will be done.
						A	 =      key or (lambda _SCREAMING_SNAKE_CASE		: x)
		def       A(  self		:       Tuple						,_SCREAMING_SNAKE_CASE		:       int		)    ->   int | None:


						'''simple docstring'''
						return int((i - 1) / 2		) if i > 0 else None
		def       A(  self		:       Union[str, Any]						,_SCREAMING_SNAKE_CASE		:       int		)    ->   int | None:


						'''simple docstring'''
						A	 =      int(2 * i + 1		)
						return left if 0 < left < self.size else None
		def       A(  self		:       Union[str, Any]						,_SCREAMING_SNAKE_CASE		:       int		)    ->   int | None:


						'''simple docstring'''
						A	 =      int(2 * i + 2		)
						return right if 0 < right < self.size else None
		def       A(  self		:       List[str]						,_SCREAMING_SNAKE_CASE		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						A ,		A	 =      (
						    self.pos_map[self.arr[j][0]],
						    self.pos_map[self.arr[i][0]],
						)
						# Then swap the items in the list.
						A ,		A	 =      self.arr[j], self.arr[i]
		def       A(  self		:       List[str]						,_SCREAMING_SNAKE_CASE		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   bool:


						'''simple docstring'''
						return self.arr[i][1] < self.arr[j][1]
		def       A(  self		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   int:


						'''simple docstring'''
						A	 =      self._left(_SCREAMING_SNAKE_CASE		)
						A	 =      self._right(_SCREAMING_SNAKE_CASE		)
						A	 =      i

						if left is not None and not self._cmp(_SCREAMING_SNAKE_CASE						,_SCREAMING_SNAKE_CASE		):
										A	 =      left
						if right is not None and not self._cmp(_SCREAMING_SNAKE_CASE						,_SCREAMING_SNAKE_CASE		):
										A	 =      right

						return valid_parent
		def       A(  self		:       Any						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						A	 =      self._parent(_SCREAMING_SNAKE_CASE		)
						while parent is not None and not self._cmp(_SCREAMING_SNAKE_CASE						,_SCREAMING_SNAKE_CASE		):
										self._swap(_SCREAMING_SNAKE_CASE						,_SCREAMING_SNAKE_CASE		)
										A ,		A	 =      parent, self._parent(_SCREAMING_SNAKE_CASE		)
		def       A(  self		:       List[Any]						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						A	 =      self._get_valid_parent(_SCREAMING_SNAKE_CASE		)
						while valid_parent != index:
										self._swap(_SCREAMING_SNAKE_CASE						,_SCREAMING_SNAKE_CASE		)
										A ,		A	 =      valid_parent, self._get_valid_parent(_SCREAMING_SNAKE_CASE		)
		def       A(  self		:       Optional[Any]						,_SCREAMING_SNAKE_CASE		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						if item not in self.pos_map:
										return
						A	 =      self.pos_map[item]
						A	 =      [item, self.key(_SCREAMING_SNAKE_CASE		)]
						# Make sure heap is right in both up and down direction.
						# Ideally only one of them will make any change.
						self._heapify_up(_SCREAMING_SNAKE_CASE		)
						self._heapify_down(_SCREAMING_SNAKE_CASE		)
		def       A(  self		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						if item not in self.pos_map:
										return
						A	 =      self.pos_map[item]
						del self.pos_map[item]
						A	 =      self.arr[self.size - 1]
						A	 =      index
						self.size -= 1
						# Make sure heap is right in both up and down direction. Ideally only one
						# of them will make any change- so no performance loss in calling both.
						if self.size > index:
										self._heapify_up(_SCREAMING_SNAKE_CASE		)
										self._heapify_down(_SCREAMING_SNAKE_CASE		)
		def       A(  self		:       Optional[Any]						,_SCREAMING_SNAKE_CASE		:       int						,_SCREAMING_SNAKE_CASE		:       int		)    ->   None:


						'''simple docstring'''
						A	 =      len(self.arr		)
						if arr_len == self.size:
										self.arr.append([item, self.key(_SCREAMING_SNAKE_CASE		)]		)
						else:
										A	 =      [item, self.key(_SCREAMING_SNAKE_CASE		)]
						A	 =      self.size
						self.size += 1
						self._heapify_up(self.size - 1		)
		def       A(  self		:       str		)    ->   tuple | None:


						'''simple docstring'''
						return self.arr[0] if self.size else None







		def       A(  self		:       Any		)    ->   tuple | None:


						'''simple docstring'''
						A	 =      self.get_top()
						if top_item_tuple:
										self.delete_item(top_item_tuple[0]		)
						return top_item_tuple







def      snake_case		(							):
				pass


if __name__ == "__main__":
					import doctest

					doctest.testmod()





