


import heapq





def __UpperCAmelCase       (		__A			)						->					set[int]:



       '''simple docstring'''





       UpperCAmelCase__										=     []

       # 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__										=     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__										=     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__										=     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()

    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)}")









from __future__ import annotations





def __UpperCAmelCase       (		__A			)						->					list[int]:



       '''simple docstring'''





       UpperCAmelCase__										=     [True] * limit
       UpperCAmelCase__										=     False
       UpperCAmelCase__										=     False
       UpperCAmelCase__										=     True

       for i in range(3 ,						int(limit**0.5 + 1			) ,						2			):
              UpperCAmelCase__										=     i * 2
              while index < limit:
                     UpperCAmelCase__										=     False
                     UpperCAmelCase__										=     index + i

       UpperCAmelCase__										=     [2]

       for i in range(3 ,						__A ,						2			):
              if is_prime[i]:
                     primes.append(__A			)

       return primes





def __UpperCAmelCase       (		__A = 1_0_0_0_0_0_0			)						->					int:



       '''simple docstring'''





       UpperCAmelCase__										=     prime_sieve(__A			)
       UpperCAmelCase__										=     0
       UpperCAmelCase__										=     0

       for i in range(len(__A			)			):
              for j in range(i + length ,						len(__A			)			):
                     UpperCAmelCase__										=     sum(primes[i:j]			)
                     if sol >= ceiling:
                            break

                     if sol in primes:
                            UpperCAmelCase__										=     j - i
                            UpperCAmelCase__										=     sol

       return largest


if __name__ == "__main__":
    print(f"{solution() = }")





