






'''simple docstring'''
from __future__ import annotations




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase						,		_UpperCamelCase = None						,		_UpperCamelCase = None							):


   """simple docstring"""




   if start is None:
      lowercase_	:    List[Any]           =     0

   if end is None:
      lowercase_	:    Dict           =     len(_UpperCamelCase							) - 1

   if start >= end:
      return

   lowercase_	:    str           =     (start + end) // 2

   slowsort(_UpperCamelCase						,		_UpperCamelCase						,		_UpperCamelCase							)
   slowsort(_UpperCamelCase						,		mid + 1						,		_UpperCamelCase							)

   if sequence[end] < sequence[mid]:
      lowercase_	:    List[str]           =     sequence[mid], sequence[end]

   slowsort(_UpperCamelCase						,		_UpperCamelCase						,		end - 1							)


if __name__ == "__main__":
 from doctest import testmod

 testmod()












'''simple docstring'''
import heapq




def 		__SCREAMING_SNAKE_CASE					(  _UpperCamelCase							):


   """simple docstring"""




   lowercase_	:    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(_UpperCamelCase						,		[-1 * len(_UpperCamelCase							), (key, value)]							)

   # chosen_vertices = set of chosen vertices
   lowercase_	:    Optional[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_	:    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_	:    str           =     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()

 UpperCamelCase__ =			{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)}""")




