



'''simple docstring'''






def    lowerCamelCase__ (		_A				,    _A       ):
		_enforce_args(_A				,    _A       )
		if n == 0:
				return 0
		a		:      Tuple       =  float('-inf'       )
		for i in range(1				,    n + 1       ):
				a		:      Tuple       =  max(
				    _A				,    prices[i - 1] + naive_cut_rod_recursive(n - i				,    _A       )       )

		return max_revue
def    lowerCamelCase__ (		_A				,    _A       ):
		_enforce_args(_A				,    _A       )
		a		:      Optional[Any]       =  [float('-inf'       ) for _ in range(n + 1       )]
		return _top_down_cut_rod_recursive(_A				,    _A				,    _A       )
def    lowerCamelCase__ (		_A				,    _A				,    _A       ):
		if max_rev[n] >= 0:
				return max_rev[n]
		elif n == 0:
				return 0
		else:
				a		:      Dict       =  float('-inf'       )
				for i in range(1				,    n + 1       ):
						a		:      Optional[Any]       =  max(
						    _A				,    prices[i - 1] + _top_down_cut_rod_recursive(n - i				,    _A				,    _A       )				,    )

				a		:      List[Any]       =  max_revenue

		return max_rev[n]
def    lowerCamelCase__ (		_A				,    _A       ):
		_enforce_args(_A				,    _A       )

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

		for i in range(1				,    n + 1       ):
				a		:      int       =  max_rev[i]
				for j in range(1				,    i + 1       ):
						a		:      Optional[Any]       =  max(_A				,    prices[j - 1] + max_rev[i - j]       )

				a		:      Optional[int]       =  max_revenue_i

		return max_rev[n]
def    lowerCamelCase__ (		_A				,    _A       ):
		if n < 0:
				a		:      Dict       =  f"""n must be greater than or equal to 0. Got n = {n}"""
				raise ValueError(_A       )

		if n > len(_A       ):
				a		:      Union[str, Any]       =  (
				    'Each integral piece of rod must have a corresponding price. '
				    f"""Got n = {n} but length of prices = {len(_A       )}"""
				)
				raise ValueError(_A       )
def    lowerCamelCase__ (		):
		a		:      Union[str, Any]       =  [6, 10, 12, 15, 20, 23]
		a		:      List[str]       =  len(_A       )

		# the best revenue comes from cutting the rod into 6 pieces, each
		# of length 1 resulting in a revenue of 6 * 6 = 36.
		a		:      List[Any]       =  36

		a		:      int       =  top_down_cut_rod(_A				,    _A       )
		a		:      int       =  bottom_up_cut_rod(_A				,    _A       )
		a		:      Optional[Any]       =  naive_cut_rod_recursive(_A				,    _A       )

		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()




'''simple docstring'''






from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

lowerCAmelCase:      List[str]		 =		TypeVar('T')
def    lowerCamelCase__ (		_A       ):
  return (position - 1) // 2
def    lowerCamelCase__ (		_A       ):
  return (2 * position) + 1
def    lowerCamelCase__ (		_A       ):
  return (2 * position) + 2

class 							a__(      Generic[T]			):







    def __init__(   self				:						Optional[Any]			):
      a		:      list[tuple[T, int]]       =  []
      a		:      dict[T, int]       =  {}
      a		:      int       =  0







    def __len__(   self				:						Union[str, Any]			):
      return self.elements







    def __repr__(   self				:						Any			):
      return str(self.heap			)







    def     lowercase_ (   self				:						Optional[int]			):
      # Check if the priority queue is empty
      return self.elements == 0







    def     lowercase_ (   self				:						Union[str, Any]		,    __snake_case				:						T		,    __snake_case				:						int			):
      # Add an element with given priority to the queue
      self.heap.append((elem, weight)			)
      a		:      int       =  self.elements
      self.elements += 1
      self._bubble_up(__snake_case			)







    def     lowercase_ (   self				:						Union[str, Any]			):
      # Remove and return the element with lowest weight (highest priority)
      if self.elements > 1:
        self._swap_nodes(0		,    self.elements - 1			)
      a     ,			a		:      str       =  self.heap.pop()
      del self.position_map[elem]
      self.elements -= 1
      if self.elements > 0:
        a     ,			a		:      List[Any]       =  self.heap[0]
        self._bubble_down(__snake_case			)
      return elem







    def     lowercase_ (   self				:						List[str]		,    __snake_case				:						T		,    __snake_case				:						int			):
      # Update the weight of the given key
      a		:      Any       =  self.position_map[elem]
      a		:      Union[str, Any]       =  (elem, weight)
      if position > 0:
        a		:      Optional[Any]       =  get_parent_position(__snake_case			)
        a     ,			a		:      Union[str, Any]       =  self.heap[parent_position]
        if parent_weight > weight:
          self._bubble_up(__snake_case			)
        else:
          self._bubble_down(__snake_case			)
      else:
        self._bubble_down(__snake_case			)







    def     lowercase_ (   self				:						int		,    __snake_case				:						T			):
      # Place a node at the proper position (upward movement) [to be used internally
      # only]
      a		:      Union[str, Any]       =  self.position_map[elem]
      if curr_pos == 0:
        return None
      a		:      Union[str, Any]       =  get_parent_position(__snake_case			)
      a     ,			a		:      Tuple       =  self.heap[curr_pos]
      a     ,			a		:      List[Any]       =  self.heap[parent_position]
      if parent_weight > weight:
        self._swap_nodes(__snake_case		,    __snake_case			)
        return self._bubble_up(__snake_case			)
      return None







    def     lowercase_ (   self				:						Tuple		,    __snake_case				:						T			):
      # Place a node at the proper position (downward movement) [to be used
      # internally only]
      a		:      str       =  self.position_map[elem]
      a     ,			a		:      List[Any]       =  self.heap[curr_pos]
      a		:      int       =  get_child_left_position(__snake_case			)
      a		:      Dict       =  get_child_right_position(__snake_case			)
      if child_left_position < self.elements and child_right_position < self.elements:
        a     ,			a		:      int       =  self.heap[child_left_position]
        a     ,			a		:      Optional[Any]       =  self.heap[child_right_position]
        if child_right_weight < child_left_weight and child_right_weight < weight:
          self._swap_nodes(__snake_case		,    __snake_case			)
          return self._bubble_down(__snake_case			)
      if child_left_position < self.elements:
        a     ,			a		:      Dict       =  self.heap[child_left_position]
        if child_left_weight < weight:
          self._swap_nodes(__snake_case		,    __snake_case			)
          return self._bubble_down(__snake_case			)
      else:
        return None
      if child_right_position < self.elements:
        a     ,			a		:      List[Any]       =  self.heap[child_right_position]
        if child_right_weight < weight:
          self._swap_nodes(__snake_case		,    __snake_case			)
          return self._bubble_down(__snake_case			)
      return None







    def     lowercase_ (   self				:						List[Any]		,    __snake_case				:						int		,    __snake_case				:						int			):
      # Swap the nodes at the given positions
      a		:      Optional[int]       =  self.heap[nodea_pos][0]
      a		:      int       =  self.heap[nodea_pos][0]
      a     ,			a		:      List[str]       =  (
          self.heap[nodea_pos],
          self.heap[nodea_pos],
      )
      a		:      Tuple       =  nodea_pos
      a		:      Any       =  nodea_pos

class 							a__(      Generic[T]			):







    def __init__(   self				:						Optional[int]			):
      a		:      dict[T, dict[T, int]]       =  {}
      a		:      int       =  0







    def __repr__(   self				:						List[str]			):
      return str(self.connections			)







    def __len__(   self				:						str			):
      return self.nodes







    def     lowercase_ (   self				:						int		,    __snake_case				:						T			):
      # Add a node in the graph if it is not in the graph
      if node not in self.connections:
        a		:      Optional[int]       =  {}
        self.nodes += 1







    def     lowercase_ (   self				:						Dict		,    __snake_case				:						T		,    __snake_case				:						T		,    __snake_case				:						int			):
      # Add an edge between 2 nodes in the graph
      self.add_node(__snake_case			)
      self.add_node(__snake_case			)
      a		:      Any       =  weight
      a		:      Dict       =  weight
def    lowerCamelCase__ (		_A				,    ):
  a		:      dict[T, int]       =  {node: maxsize for node in graph.connections}
  a		:      dict[T, T | None]       =  {node: None for node in graph.connections}

  a		:      MinPriorityQueue[T]       =  MinPriorityQueue()
  for node, weight in dist.items():
    priority_queue.push(_A				,    _A       )

  if priority_queue.is_empty():
    return dist, parent

  # initialization
  a		:      int       =  priority_queue.extract_min()
  a		:      Dict       =  0
  for neighbour in graph.connections[node]:
    if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
      a		:      Tuple       =  dist[node] + graph.connections[node][neighbour]
      priority_queue.update_key(_A				,    dist[neighbour]       )
      a		:      Dict       =  node

    # running prim's algorithm
  while not priority_queue.is_empty():
    a		:      List[str]       =  priority_queue.extract_min()
    for neighbour in graph.connections[node]:
      if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
        a		:      int       =  dist[node] + graph.connections[node][neighbour]
        priority_queue.update_key(_A				,    dist[neighbour]       )
        a		:      Optional[int]       =  node
  return dist, parent