"""simple docstring"""






import heapq



def SCREAMING_SNAKE_CASE__			(   __UpperCAmelCase )  ->  Union[str, Any]:
   lowercase__:   str					=      []

   # 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(__UpperCAmelCase       ,    [-1 * len(__UpperCAmelCase ), (key, value)] )

   # chosen_vertices = set of chosen vertices
   lowercase__:   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
      lowercase__:   List[Any]					=      heapq.heappop(__UpperCAmelCase )[1][0]
      chosen_vertices.add(__UpperCAmelCase )

      # 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]:
            lowercase__:   Any					=      elem[1][1].index(__UpperCAmelCase )
            del elem[1][1][index]
            elem[0] += 1
        # re-order the queue
      heapq.heapify(__UpperCAmelCase )
   return chosen_vertices


if __name__ == "__main__":
       import doctest

       doctest.testmod()

       __A			   = {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)}''')




'''simple docstring'''




def 	a_					( lowerCamelCase	:     Optional[Any]				):
   return [
       {
           0: [1, 2],
           1: [0, 2],
           2: [0, 1, 3, 5],
           3: [2, 4],
           4: [3],
           5: [2, 6, 8],
           6: [5, 7],
           7: [6, 8],
           8: [5, 7],
       },
       {
           0: [6],
           1: [9],
           2: [4, 5],
           3: [4],
           4: [2, 3],
           5: [2],
           6: [0, 7],
           7: [6],
           8: [],
           9: [1],
       },
       {
           0: [4],
           1: [6],
           2: [],
           3: [5, 6, 7],
           4: [0, 6],
           5: [3, 8, 9],
           6: [1, 3, 4, 7],
           7: [3, 6, 8, 9],
           8: [5, 7],
           9: [5, 7],
       },
       {
           0: [1, 3],
           1: [0, 2, 4],
           2: [1, 3, 4],
           3: [0, 2, 4],
           4: [1, 2, 3],
       },
   ][index]





def 	a_					( lowerCamelCase	:     dict[int, list[int]]				):
   lowerCAmelCase     =			0
   lowerCAmelCase     =			len(lowerCamelCase				)  # No of vertices in graph
   lowerCAmelCase     =			[0] * n
   lowerCAmelCase     =			[False] * n

   def dfs(lowerCamelCase	:     Tuple	, lowerCamelCase	:     str	, lowerCamelCase	:     Dict	, lowerCamelCase	:     str				):
      lowerCAmelCase     =			True
      lowerCAmelCase     =			id_
      id_ += 1
      for to in graph[at]:
         if to == parent:
            pass
         elif not visited[to]:
            dfs(lowerCamelCase	, lowerCamelCase	, lowerCamelCase	, id_				)
            lowerCAmelCase     =			min(low[at]	, low[to]				)
            if id_ <= low[to]:
               bridges.append((at, to) if at < to else (to, at)				)
         else:
            # This edge is a back edge and cannot be a bridge
            lowerCAmelCase     =			min(low[at]	, low[to]				)

   lowerCAmelCase     =			[]
   for i in range(lowerCamelCase				):
      if not visited[i]:
         dfs(lowerCamelCase	, -1	, lowerCamelCase	, id_				)
   return bridges


if __name__ == "__main__":
       import doctest

       doctest.testmod()





