class Solution:
  def loopExists(self, arr): # T O(N) | S O(1)
    p1 = self.getNext(arr, 0)
    p2 = self.getNext(arr, self.getNext(arr, 0))

    # step 1) find an element within the cycle
    while p1 != p2:
      p1 = self.getNext(arr, p1)
      p2 = self.getNext(arr, self.getNext(arr, p2))
    
    # step 2) make sure its a cycle that contains atleast 1 other element
    p2 = self.getNext(arr, p1)
    if p1 == p2:
      return False
    
    # step 3) make sure all the directions of the cycle are the same
    while p1 != p2:
      if (arr[p2] < 0 and arr[p1] > 0) or (arr[p2] > 0 and arr[p1] < 0):
        return False
      p2 = self.getNext(arr, p2)
    
    return True


  def getNext(self, arr, index): # T O(1) | S O(1)
    if index is None or len(arr) - 1 < index:
      return None

    nextIndex = index + arr[index]
    
    if nextIndex > len(arr) - 1:
      nextIndex -= len(arr)
    elif nextIndex < 0:
      nextIndex += len(arr) - 1
    return nextIndex