





"""simple docstring"""




def 					lowerCAmelCase_			(	__a		)					->		set:







  """simple docstring"""





  lowerCamelCase__:				Optional[Any]        =set()
  # edges = list of graph's edges
  lowerCamelCase__:				Tuple        =get_edges(__a		)

  # While there are still elements in edges list, take an arbitrary edge
  # (from_node, to_node) and add his extremity to chosen_vertices and then
  # remove all arcs adjacent to the from_node and to_node
  while edges:
    lowerCamelCase__:				Union[str, Any]        =edges.pop()
    chosen_vertices.add(__a		)
    chosen_vertices.add(__a		)
    for edge in edges.copy():
      if from_node in edge or to_node in edge:
        edges.discard(__a		)
  return chosen_vertices





def 					lowerCAmelCase_			(	__a		)					->		set:







  """simple docstring"""





  lowerCamelCase__:				str        =set()
  for from_node, to_nodes in graph.items():
    for to_node in to_nodes:
      edges.add((from_node, to_node)		)
  return edges


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     # graph = {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
     # print(f"Matching vertex cover:\n{matching_min_vertex_cover(graph)}")








from __future__ import annotations

from typing import Any



class        _SCREAMING_SNAKE_CASE						:







 '''simple docstring'''

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



   '''simple docstring'''



   lowerCamelCase__:				int        =num_of_nodes
   lowerCamelCase__:				list[list[int]]        =[]
   lowerCamelCase__:				dict[int, int]        ={}

 def 							SCREAMING_SNAKE_CASE_ (self    :       Tuple	,			UpperCAmelCase_    :       int	,			UpperCAmelCase_    :       int	,			UpperCAmelCase_    :       int)	->None:



   '''simple docstring'''



   self.m_edges.append([u_node, v_node, weight])

 def 							SCREAMING_SNAKE_CASE_ (self    :       List[str]	,			UpperCAmelCase_    :       int)	->int:



   '''simple docstring'''



   if self.m_component[u_node] == u_node:
     return u_node
   return self.find_component(self.m_component[u_node])

 def 							SCREAMING_SNAKE_CASE_ (self    :       Union[str, Any]	,			UpperCAmelCase_    :       int)	->None:



   '''simple docstring'''



   if self.m_component[u_node] != u_node:
     for k in self.m_component:
       lowerCamelCase__:				Dict        =self.find_component(UpperCAmelCase_)

 def 							SCREAMING_SNAKE_CASE_ (self    :       Union[str, Any]	,			UpperCAmelCase_    :       list[int]	,			UpperCAmelCase_    :       int	,			UpperCAmelCase_    :       int)	->None:



   '''simple docstring'''



   if component_size[u_node] <= component_size[v_node]:
     lowerCamelCase__:				Optional[int]        =v_node
     component_size[v_node] += component_size[u_node]
     self.set_component(UpperCAmelCase_)

   elif component_size[u_node] >= component_size[v_node]:
     lowerCamelCase__:				Tuple        =self.find_component(UpperCAmelCase_)
     component_size[u_node] += component_size[v_node]
     self.set_component(UpperCAmelCase_)







 def 							SCREAMING_SNAKE_CASE_ (self    :       List[str])	->None:



   '''simple docstring'''



   lowerCamelCase__:				List[Any]        =[]
   lowerCamelCase__:				List[str]        =0

   lowerCamelCase__:				list[Any]        =[-1] * self.m_num_of_nodes

   # A list of components (initialized to all of the nodes)
   for node in range(self.m_num_of_nodes):
     self.m_component.update({node: node})
     component_size.append(1)

   lowerCamelCase__:				List[str]        =self.m_num_of_nodes

   while num_of_components > 1:
     for edge in self.m_edges:
       lowerCamelCase__  , lowerCamelCase__  , lowerCamelCase__:				Tuple        =edge

       lowerCamelCase__:				List[Any]        =self.m_component[u]
       lowerCamelCase__:				str        =self.m_component[v]

       if u_component != v_component:

         for component in (u_component, v_component):
           if (
               minimum_weight_edge[component] == -1
               or minimum_weight_edge[component][2] > w
           ):
             lowerCamelCase__:				Union[str, Any]        =[u, v, w]

     for edge in minimum_weight_edge:
       if isinstance(UpperCAmelCase_	,			UpperCAmelCase_):
         lowerCamelCase__  , lowerCamelCase__  , lowerCamelCase__:				Tuple        =edge

         lowerCamelCase__:				str        =self.m_component[u]
         lowerCamelCase__:				Any        =self.m_component[v]

         if u_component != v_component:
           mst_weight += w
           self.union(UpperCAmelCase_	,			UpperCAmelCase_	,			UpperCAmelCase_)
           print(F"""Added edge [{u} - {v}]\nAdded weight: {w}\n""")
           num_of_components -= 1

     lowerCamelCase__:				Tuple        =[-1] * self.m_num_of_nodes
   print(F"""The total weight of the minimal spanning tree is: {mst_weight}""")





def 					lowerCAmelCase_			(	)					->		None:







  """simple docstring"""







if __name__ == "__main__":
     import doctest

     doctest.testmod()

