






'''simple docstring'''





class 							A_    :






 '''simple docstring'''




 def __init__(	self							,  _A) ->		None:

     """simple docstring"""



     _UpperCAmelCase				:      List[str]			  =				set_counts
     _UpperCAmelCase				:      Optional[int]			  =				max(_A)
     _UpperCAmelCase				:      Union[str, Any]			  =				len(_A)
     _UpperCAmelCase				:      Optional[int]			  =				[1] * num_sets
     _UpperCAmelCase				:      Optional[int]			  =				list(range(_A))


 def  snake_case__				(	self							,  _A							,  _A) ->		bool:

     """simple docstring"""



     _UpperCAmelCase				:      Optional[Any]			  =				self.get_parent(_A)
     _UpperCAmelCase				:      Dict			  =				self.get_parent(_A)

     if src_parent == dst_parent:
         return False

     if self.ranks[dst_parent] >= self.ranks[src_parent]:
         self.set_counts[dst_parent] += self.set_counts[src_parent]
         _UpperCAmelCase				:      Any			  =				0
         _UpperCAmelCase				:      Union[str, Any]			  =				dst_parent
         if self.ranks[dst_parent] == self.ranks[src_parent]:
             self.ranks[dst_parent] += 1
         _UpperCAmelCase				:      Dict			  =				self.set_counts[dst_parent]
     else:
         self.set_counts[src_parent] += self.set_counts[dst_parent]
         _UpperCAmelCase				:      Dict			  =				0
         _UpperCAmelCase				:      Dict			  =				src_parent
         _UpperCAmelCase				:      List[Any]			  =				self.set_counts[src_parent]

     _UpperCAmelCase				:      Optional[Any]			  =				max(self.max_set							,  _A)
     return True


 def  snake_case__				(	self							,  _A) ->		int:

     """simple docstring"""



     if self.parents[disj_set] == disj_set:
         return disj_set
     _UpperCAmelCase				:      Dict			  =				self.get_parent(self.parents[disj_set])
     return self.parents[disj_set]












import heapq

def      _lowerCamelCase	(						__A    :							dict )      ->					set[int]:
    _UpperCAmelCase				:      list[list]			  =				[]

    # for each node and his adjacency list add them and the rank of the node to queue
    # using heapq module the queue will be filled like a Priority Queue
    # heapq works with a min priority queue, so I used -1*len(v) to build it
    for key, value in graph.items():
        # O(log(n))
        heapq.heappush(__A		,		[-1 * len(__A ), (key, value)] )

    # chosen_vertices = set of chosen vertices
    _UpperCAmelCase				:      Union[str, Any]			  =				set()

    # while queue isn't empty and there are still edges
    #   (queue[0][0] is the rank of the node with max rank)
    while queue and queue[0][0] != 0:
        # extract vertex with max rank from queue and add it to chosen_vertices
        _UpperCAmelCase				:      Tuple			  =				heapq.heappop(__A )[1][0]
        chosen_vertices.add(__A )

        # Remove all arcs adjacent to argmax
        for elem in queue:
            # if v haven't adjacent node, skip
            if elem[0] == 0:
                continue
            # if argmax is reachable from elem
            # remove argmax from elem's adjacent list and update his rank
            if argmax in elem[1][1]:
                _UpperCAmelCase				:      Dict			  =				elem[1][1].index(__A )
                del elem[1][1][index]
                elem[0] += 1
        # re-order the queue
        heapq.heapify(__A )
    return chosen_vertices


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  SCREAMING_SNAKE_CASE 							=						{0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
  print(F'Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}')




