




lowerCamelCase__			:       Optional[int]         =							{
    '''Pillow''': '''Pillow''',
    '''accelerate''': '''accelerate>=0.11.0''',
    '''compel''': '''compel==0.1.8''',
    '''black''': '''black~=23.1''',
    '''datasets''': '''datasets''',
    '''filelock''': '''filelock''',
    '''flax''': '''flax>=0.4.1''',
    '''hf-doc-builder''': '''hf-doc-builder>=0.3.0''',
    '''huggingface-hub''': '''huggingface-hub>=0.13.2''',
    '''requests-mock''': '''requests-mock==1.10.0''',
    '''importlib_metadata''': '''importlib_metadata''',
    '''invisible-watermark''': '''invisible-watermark''',
    '''isort''': '''isort>=5.5.4''',
    '''jax''': '''jax>=0.2.8,!=0.3.2''',
    '''jaxlib''': '''jaxlib>=0.1.65''',
    '''Jinja2''': '''Jinja2''',
    '''k-diffusion''': '''k-diffusion>=0.0.12''',
    '''torchsde''': '''torchsde''',
    '''note_seq''': '''note_seq''',
    '''librosa''': '''librosa''',
    '''numpy''': '''numpy''',
    '''omegaconf''': '''omegaconf''',
    '''parameterized''': '''parameterized''',
    '''protobuf''': '''protobuf>=3.20.3,<4''',
    '''pytest''': '''pytest''',
    '''pytest-timeout''': '''pytest-timeout''',
    '''pytest-xdist''': '''pytest-xdist''',
    '''ruff''': '''ruff>=0.0.241''',
    '''safetensors''': '''safetensors''',
    '''sentencepiece''': '''sentencepiece>=0.1.91,!=0.1.92''',
    '''scipy''': '''scipy''',
    '''onnx''': '''onnx''',
    '''regex''': '''regex!=2019.12.17''',
    '''requests''': '''requests''',
    '''tensorboard''': '''tensorboard''',
    '''torch''': '''torch>=1.4''',
    '''torchvision''': '''torchvision''',
    '''transformers''': '''transformers>=4.25.1''',
    '''urllib3''': '''urllib3<=2.0.0''',
}





from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

lowerCamelCase__			:       Dict         =							TypeVar('T')

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (position - 1) // 2

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (2 * position) + 1

def  UpperCAmelCase_							(				__UpperCAmelCase						:     int							) ->			int:
      return (2 * position) + 2





class 	lowerCamelCase_							(				Generic[T] ):


 '''simple docstring'''







 def __init__(							self  :		Optional[Any]						):
       SCREAMING_SNAKE_CASE_     						=      []
       SCREAMING_SNAKE_CASE_     						=      {}
       SCREAMING_SNAKE_CASE_     						=      0



 def __len__(							self  :		Optional[Any]						):
       return self.elements



 def __repr__(							self  :		Optional[int]						):
       return str(self.heap						)



 def 				lowerCAmelCase_  (							self  :		Union[str, Any]						):
       # Check if the priority queue is empty
       return self.elements == 0



 def 				lowerCAmelCase_  (							self  :		Tuple    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Add an element with given priority to the queue
       self.heap.append((elem, weight)						)
       SCREAMING_SNAKE_CASE_     						=      self.elements
       self.elements += 1
       self._bubble_up(_lowerCAmelCase						)



 def 				lowerCAmelCase_  (							self  :		Optional[int]						):
       # Remove and return the element with lowest weight (highest priority)
       if self.elements > 1:
             self._swap_nodes(0    ,    self.elements - 1						)
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap.pop()
       del self.position_map[elem]
       self.elements -= 1
       if self.elements > 0:
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[0]
             self._bubble_down(_lowerCAmelCase						)
       return elem



 def 				lowerCAmelCase_  (							self  :		Optional[int]    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Update the weight of the given key
       SCREAMING_SNAKE_CASE_     						=      self.position_map[elem]
       SCREAMING_SNAKE_CASE_     						=      (elem, weight)
       if position > 0:
             SCREAMING_SNAKE_CASE_     						=      get_parent_position(_lowerCAmelCase						)
             SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      self.heap[parent_position]
             if parent_weight > weight:
                   self._bubble_up(_lowerCAmelCase						)
             else:
                   self._bubble_down(_lowerCAmelCase						)
       else:
             self._bubble_down(_lowerCAmelCase						)



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



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




 def 				lowerCAmelCase_  (							self  :		Optional[Any]    ,    _lowerCAmelCase  :		int    ,    _lowerCAmelCase  :		int						):
       # Swap the nodes at the given positions
       SCREAMING_SNAKE_CASE_     						=      self.heap[nodea_pos][0]
       SCREAMING_SNAKE_CASE_     						=      self.heap[nodea_pos][0]
       SCREAMING_SNAKE_CASE_						,			SCREAMING_SNAKE_CASE_     						=      (
           self.heap[nodea_pos],
           self.heap[nodea_pos],
       )
       SCREAMING_SNAKE_CASE_     						=      nodea_pos
       SCREAMING_SNAKE_CASE_     						=      nodea_pos





class 	lowerCamelCase_							(				Generic[T] ):


 '''simple docstring'''







 def __init__(							self  :		Union[str, Any]						):
       SCREAMING_SNAKE_CASE_     						=      {}
       SCREAMING_SNAKE_CASE_     						=      0



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



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



 def 				lowerCAmelCase_  (							self  :		Dict    ,    _lowerCAmelCase  :		T						):
       # Add a node in the graph if it is not in the graph
       if node not in self.connections:
             SCREAMING_SNAKE_CASE_     						=      {}
             self.nodes += 1




 def 				lowerCAmelCase_  (							self  :		List[Any]    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		T    ,    _lowerCAmelCase  :		int						):
       # Add an edge between 2 nodes in the graph
       self.add_node(_lowerCAmelCase						)
       self.add_node(_lowerCAmelCase						)
       SCREAMING_SNAKE_CASE_     						=      weight
       SCREAMING_SNAKE_CASE_     						=      weight


def  UpperCAmelCase_							(				__UpperCAmelCase						:     GraphUndirectedWeighted[T]      ,      ) ->			tuple[dict[T, int], dict[T, T | None]]:
      SCREAMING_SNAKE_CASE_     						=      {node: maxsize for node in graph.connections}
      SCREAMING_SNAKE_CASE_     						=      {node: None for node in graph.connections}

      SCREAMING_SNAKE_CASE_     						=      MinPriorityQueue()
      for node, weight in dist.items():
            priority_queue.push(__UpperCAmelCase      ,      __UpperCAmelCase							)

      if priority_queue.is_empty():
            return dist, parent

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

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