What do you mean by Augmenting a Red black trees? Discuss in short.

Augmenting a Red-Black Tree refers to the process of enhancing the basic structure of a Red-Black Tree with additional information or metadata in each node, while maintaining the fundamental properties of the tree. This augmentation is typically done to support more complex operations or to improve the efficiency of certain queries.

A Red-Black Tree is a type of self-balancing binary search tree where each node contains an extra bit for denoting the color of the node, either red or black. The tree maintains the following properties:

1. Red/Black Property: Every node is colored, either red or black.
2. Root Property: The root is black.
3. Leaf Property: Every leaf (NIL node) is black.
4. Red Property: If a red node has children then, the children are always black.
5. Depth Property: For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.

Step-by-Step Approach to Augmenting a Red-Black Tree

1. Choose the Additional Information: Decide what extra information you want to store in each node. This could be the size of the subtree rooted at that node, the maximum or minimum value in the subtree, or any other data that is useful for the specific application.

2. Modify the Node Structure: Adjust the structure of each node in the tree to include space for the additional information.

3. Update Augmentation Information: Whenever the tree is modified (during insertions or deletions), ensure that the additional information is updated accordingly. This typically involves recalculating the information based on the node's children.

4. Maintain Red-Black Properties: Ensure that the basic operations of the Red-Black Tree (insertion, deletion, rotation) still maintain the Red-Black properties. The augmentation should not interfere with the self-balancing nature of the tree.

5. Utilize the Augmentation: Use the augmented information to perform operations or answer queries more efficiently than would be possible with a standard Red-Black Tree.

Example of Augmenting a Red-Black Tree

Let's consider an example where we augment a Red-Black Tree by storing the size of the subtree rooted at each node. This information can be useful for order-statistic queries, such as finding the k-th smallest element in the tree.

Node Structure

Attribute	Description
key	The key value of the node.
color	The color of the node (red or black).
left	Pointer to the left child.
right	Pointer to the right child.
parent	Pointer to the parent node.
subtree_size	The size of the subtree rooted at this node.

Updating Subtree Size

When a node is inserted or deleted, or a rotation is performed, the subtree_size attribute must be updated. Here's how you might update the subtree_size during an insertion:

1. Insert the new node as you would in a standard Red-Black Tree.
2. Recolor and perform rotations to maintain the Red-Black properties.
3. After each rotation or when moving up the tree, update the subtree_size of the affected nodes. The subtree_size of a node is the sum of the subtree_size of its left and right children, plus one (for the node itself):

   node.subtree_size = node.left.subtree_size + node.right.subtree_size + 1

Using Augmentation

To find the k-th smallest element, you can use the subtree_size information:

1. Start at the root of the tree.
2. If the subtree_size of the left child is greater than or equal to k, the k-th smallest element is in the left subtree.
3. If the subtree_size of the left child is k-1, the current node is the k-th smallest element.
4. If the subtree_size of the left child is less than k-1, subtract the size of the left subtree and the current node from k, and search for the (k - left_subtree_size - 1)-th smallest element in the right subtree.

By augmenting the Red-Black Tree with subtree_size, we can perform order-statistic queries in O(log n) time, where n is the number of nodes in the tree. Without augmentation, such queries would require O(n) time in the worst case.

In summary, augmenting a Red-Black Tree allows for additional functionality while preserving the balance and search properties of the tree. The key is to carefully maintain the augmented information during tree modifications to ensure that the additional benefits can be realized.