Explanation:
- A **Scapegoat Tree** is a type of self-balancing binary search tree that uses an amortized **O(log n)** time complexity for insertion and deletion.
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- Unlike AVL and Red-Black Trees, it does not maintain a strict balance at every node but ensures that the tree does not become unbalanced by periodically rebuilding subtrees.
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Steps:
- Insertion: Insert elements as in a regular binary search tree. When a subtree becomes too unbalanced, a “scapegoat” node is found and the subtree is rebuilt to restore balance.
- Rebuilding: The tree periodically rebuilds a subtree when an insertion violates the balance property.
Time Complexity:
- Insertion: O(log n) amortized
- Rebuilding: O(n) for the worst case (rebuilding the tree)
class ScapegoatTreeNode:
def __init__(self, key):
self.key = key
self.left = None
self.right = None
self.parent = None
class ScapegoatTree:
def __init__(self):
self.root = None
def insert(self, key):
# Perform insertion as a regular binary search tree
self.root = self._insert(self.root, key)
# Check if the tree is balanced and rebuild if necessary
if self._is_unbalanced(self.root):
self._rebalance(self.root)
def _insert(self, node, key):
# Perform binary search tree insertion
if node is None:
return ScapegoatTreeNode(key)
if key < node.key:
node.left = self._insert(node.left, key)
else:
node.right = self._insert(node.right, key)
return node
def _is_unbalanced(self, node):
# Check if a subtree is unbalanced (e.g., height difference)
return False # Placeholder for balance checking logic
def _rebalance(self, node):
# Rebuild the tree or subtree for balance
pass
# Example usage
scapegoat_tree = ScapegoatTree()
scapegoat_tree.insert(10)
scapegoat_tree.insert(5)
scapegoat_tree.insert(20)