Question

You are given a red-black tree T that could contain repeated elements (i.e., duplicates).

(a) Design a linear time algorithm that returns a red-black tree that contains the elements in T, with repeated elements removed. In other words, if any duplicates are found in T, only one instance of that element will be kept in the tree.

(b) Prove that this algorithm is optimal.

Answer 1

To remove duplicates from a red-black tree, perform an inorder traversal and insert each element into a new tree while checking for duplicates. The linear time algorithm is proven to be optimal.

Step-by-step explanation:

(a) To design a linear time algorithm that returns a red-black tree with duplicates removed, we can perform an inorder traversal of the original red-black tree and insert each element into a new red-black tree. Before inserting each element, we can check if it already exists in the new tree. If it does, we skip the insertion. This algorithm has a time complexity of O(n), where n is the number of elements in the original tree.

(b) To prove the optimality of this algorithm, we can consider the worst-case scenario. In the worst case, all elements in the original tree are duplicates, resulting in a tree with only one unique element. In this case, no algorithm can remove duplicates in less than linear time. Therefore, the linear time complexity of this algorithm is optimal for the problem at hand.