


"""simple docstring"""
from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

__SCREAMING_SNAKE_CASE							:    Tuple		=    TypeVar('T')







def 						_a				(      _SCREAMING_SNAKE_CASE				)    ->				int:
							return (position - 1) // 2







def 						_a				(      _SCREAMING_SNAKE_CASE				)    ->				int:
							return (2 * position) + 1







def 						_a				(      _SCREAMING_SNAKE_CASE				)    ->				int:
							return (2 * position) + 2





class      __A     (Generic[T]):





							'''simple docstring'''

							def __init__(				self							:      Optional[Any]       )					->None:



														"""simple docstring"""



														snake_case_           =		[]
														snake_case_           =		{}
														snake_case_           =		0



							def __len__(				self							:      Dict       )					->int:



														"""simple docstring"""



														return self.elements



							def __repr__(				self							:      Dict       )					->str:



														"""simple docstring"""



														return str(self.heap       )



							def 		lowerCAmelCase       (				self							:      Tuple       )					->bool:



														"""simple docstring"""



														return self.elements == 0



							def 		lowerCAmelCase       (				self							:      Any		,       UpperCAmelCase_							:      T		,       UpperCAmelCase_							:      int       )					->None:



														"""simple docstring"""



														self.heap.append((elem, weight)       )
														snake_case_           =		self.elements
														self.elements += 1
														self._bubble_up(UpperCAmelCase_       )



							def 		lowerCAmelCase       (				self							:      Optional[Any]       )					->T:



														"""simple docstring"""



														if self.elements > 1:
																					self._swap_nodes(0		,       self.elements - 1       )
														snake_case_       ,						snake_case_           =		self.heap.pop()
														del self.position_map[elem]
														self.elements -= 1
														if self.elements > 0:
																					snake_case_       ,						snake_case_           =		self.heap[0]
																					self._bubble_down(UpperCAmelCase_       )
														return elem



							def 		lowerCAmelCase       (				self							:      List[Any]		,       UpperCAmelCase_							:      T		,       UpperCAmelCase_							:      int       )					->None:



														"""simple docstring"""



														snake_case_           =		self.position_map[elem]
														snake_case_           =		(elem, weight)
														if position > 0:
																					snake_case_           =		get_parent_position(UpperCAmelCase_       )
																					snake_case_       ,						snake_case_           =		self.heap[parent_position]
																					if parent_weight > weight:
																												self._bubble_up(UpperCAmelCase_       )
																					else:
																												self._bubble_down(UpperCAmelCase_       )
														else:
																					self._bubble_down(UpperCAmelCase_       )



							def 		lowerCAmelCase       (				self							:      Optional[Any]		,       UpperCAmelCase_							:      T       )					->None:



														"""simple docstring"""



														snake_case_           =		self.position_map[elem]
														if curr_pos == 0:
																					return None
														snake_case_           =		get_parent_position(UpperCAmelCase_       )
														snake_case_       ,						snake_case_           =		self.heap[curr_pos]
														snake_case_       ,						snake_case_           =		self.heap[parent_position]
														if parent_weight > weight:
																					self._swap_nodes(UpperCAmelCase_		,       UpperCAmelCase_       )
																					return self._bubble_up(UpperCAmelCase_       )
														return None



							def 		lowerCAmelCase       (				self							:      Optional[Any]		,       UpperCAmelCase_							:      T       )					->None:



														"""simple docstring"""



														snake_case_           =		self.position_map[elem]
														snake_case_       ,						snake_case_           =		self.heap[curr_pos]
														snake_case_           =		get_child_left_position(UpperCAmelCase_       )
														snake_case_           =		get_child_right_position(UpperCAmelCase_       )
														if child_left_position < self.elements and child_right_position < self.elements:
																					snake_case_       ,						snake_case_           =		self.heap[child_left_position]
																					snake_case_       ,						snake_case_           =		self.heap[child_right_position]
																					if child_right_weight < child_left_weight and child_right_weight < weight:
																												self._swap_nodes(UpperCAmelCase_		,       UpperCAmelCase_       )
																												return self._bubble_down(UpperCAmelCase_       )
														if child_left_position < self.elements:
																					snake_case_       ,						snake_case_           =		self.heap[child_left_position]
																					if child_left_weight < weight:
																												self._swap_nodes(UpperCAmelCase_		,       UpperCAmelCase_       )
																												return self._bubble_down(UpperCAmelCase_       )
														else:
																					return None
														if child_right_position < self.elements:
																					snake_case_       ,						snake_case_           =		self.heap[child_right_position]
																					if child_right_weight < weight:
																												self._swap_nodes(UpperCAmelCase_		,       UpperCAmelCase_       )
																												return self._bubble_down(UpperCAmelCase_       )
														return None



							def 		lowerCAmelCase       (				self							:      List[Any]		,       UpperCAmelCase_							:      int		,       UpperCAmelCase_							:      int       )					->None:



														"""simple docstring"""



														snake_case_           =		self.heap[nodea_pos][0]
														snake_case_           =		self.heap[nodea_pos][0]
														snake_case_       ,						snake_case_           =		(
														    self.heap[nodea_pos],
														    self.heap[nodea_pos],
														)
														snake_case_           =		nodea_pos
														snake_case_           =		nodea_pos





class      __A     (Generic[T]):





							'''simple docstring'''

							def __init__(				self							:      Any       )					->None:



														"""simple docstring"""



														snake_case_           =		{}
														snake_case_           =		0



							def __repr__(				self							:      Tuple       )					->str:



														"""simple docstring"""



														return str(self.connections       )



							def __len__(				self							:      Dict       )					->int:



														"""simple docstring"""



														return self.nodes



							def 		lowerCAmelCase       (				self							:      int		,       UpperCAmelCase_							:      T       )					->None:



														"""simple docstring"""



														if node not in self.connections:
																					snake_case_           =		{}
																					self.nodes += 1



							def 		lowerCAmelCase       (				self							:      Any		,       UpperCAmelCase_							:      T		,       UpperCAmelCase_							:      T		,       UpperCAmelCase_							:      int       )					->None:



														"""simple docstring"""



														self.add_node(UpperCAmelCase_       )
														self.add_node(UpperCAmelCase_       )
														snake_case_           =		weight
														snake_case_           =		weight







def 						_a				(      _SCREAMING_SNAKE_CASE				, )    ->				tuple[dict[T, int], dict[T, T | None]]:
							snake_case_           =		{node: maxsize for node in graph.connections}
							snake_case_           =		{node: None for node in graph.connections}

							snake_case_           =		MinPriorityQueue()
							for node, weight in dist.items():
														priority_queue.push(_SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				)

							if priority_queue.is_empty():
														return dist, parent

							# initialization
							snake_case_           =		priority_queue.extract_min()
							snake_case_           =		0
							for neighbour in graph.connections[node]:
														if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
																					snake_case_           =		dist[node] + graph.connections[node][neighbour]
																					priority_queue.update_key(_SCREAMING_SNAKE_CASE				, dist[neighbour]				)
																					snake_case_           =		node

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







"""simple docstring"""
def 						_a				(      _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				)    ->				int:
							if index == number_of_items:
														return 0
							snake_case_           =		0
							snake_case_           =		0
							snake_case_           =		knapsack(_SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, index + 1				)
							if weights[index] <= max_weight:
														snake_case_           =		values[index] + knapsack(
														    _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				, max_weight - weights[index]				, index + 1				)
							return max(_SCREAMING_SNAKE_CASE				, _SCREAMING_SNAKE_CASE				)


if __name__ == "__main__":
				import doctest

				doctest.testmod()



