import random







def 			A__      (	snake_case_ :		int       ,				snake_case_ :		float       ,				snake_case_ :		bool = False      ):
 SCREAMING_SNAKE_CASE__:   dict=      {i: [] for i in range(snake_case_      )}

 # if probability is greater or equal than 1, then generate a complete graph
 if probability >= 1:
  return complete_graph(snake_case_      )
 # if probability is lower or equal than 0, then return a graph without edges
 if probability <= 0:
  return graph

 # for each couple of nodes, add an edge from u to v
 # if the number randomly generated is greater than probability probability
 for i in range(snake_case_      ):
  for j in range(i + 1       ,				snake_case_      ):
   if random.random() < probability:
    graph[i].append(snake_case_      )
    if not directed:
     # if the graph is undirected, add an edge in from j to i, either
     graph[j].append(snake_case_      )
 return graph







def 			A__      (	snake_case_ :		int      ):
 return {
     i: [j for j in range(snake_case_      ) if i != j] for i in range(snake_case_      )
 }


if __name__ == "__main__":
      import doctest

      doctest.testmod()


def 			A__      (	snake_case_ :		str      ):
 if not head:
  return True
 # split the list to two parts
 SCREAMING_SNAKE_CASE__,	SCREAMING_SNAKE_CASE__:   Optional[int]=      head.next, head
 while fast and fast.next:
  SCREAMING_SNAKE_CASE__:   Dict=      fast.next.next
  SCREAMING_SNAKE_CASE__:   Union[str, Any]=      slow.next
 SCREAMING_SNAKE_CASE__:   Union[str, Any]=      slow.next
 SCREAMING_SNAKE_CASE__:   Union[str, Any]=      None  # Don't forget here! But forget still works!
 # reverse the second part
 SCREAMING_SNAKE_CASE__:   Optional[int]=      None
 while second:
  SCREAMING_SNAKE_CASE__:   Any=      second.next
  SCREAMING_SNAKE_CASE__:   int=      node
  SCREAMING_SNAKE_CASE__:   Optional[Any]=      second
  SCREAMING_SNAKE_CASE__:   Any=      nxt
 # compare two parts
 # second part has the same or one less node
 while node:
  if node.val != head.val:
   return False
  SCREAMING_SNAKE_CASE__:   Tuple=      node.next
  SCREAMING_SNAKE_CASE__:   Optional[int]=      head.next
 return True







def 			A__      (	snake_case_ :		Optional[Any]      ):
 if not head or not head.next:
  return True

 # 1. Get the midpoint (slow)
 SCREAMING_SNAKE_CASE__:   List[Any]=      head
 while fast and fast.next:
  SCREAMING_SNAKE_CASE__,	SCREAMING_SNAKE_CASE__:   Any=      fast.next.next, slow.next

 # 2. Push the second half into the stack
 SCREAMING_SNAKE_CASE__:   Optional[Any]=      [slow.val]
 while slow.next:
  SCREAMING_SNAKE_CASE__:   Optional[int]=      slow.next
  stack.append(slow.val      )

 # 3. Comparison
 while stack:
  if stack.pop() != cur.val:
   return False
  SCREAMING_SNAKE_CASE__:   Tuple=      cur.next

 return True







def 			A__      (	snake_case_ :		Any      ):
 if not head or not head.next:
  return True
 SCREAMING_SNAKE_CASE__:   Optional[int]=      {}
 SCREAMING_SNAKE_CASE__:   Union[str, Any]=      0
 while head:
  if head.val in d:
   d[head.val].append(snake_case_      )
  else:
   SCREAMING_SNAKE_CASE__:   Optional[int]=      [pos]
  SCREAMING_SNAKE_CASE__:   Dict=      head.next
  pos += 1
 SCREAMING_SNAKE_CASE__:   Dict=      pos - 1
 SCREAMING_SNAKE_CASE__:   str=      0
 for v in d.values():
  if len(snake_case_      ) % 2 != 0:
   middle += 1
  else:
   SCREAMING_SNAKE_CASE__:   List[Any]=      0
   for i in range(0       ,				len(snake_case_      )      ):
    if v[i] + v[len(snake_case_      ) - 1 - step] != checksum:
     return False
    step += 1
  if middle > 1:
   return False
 return True

