Lecture 4 : BST Introduction and Treaps

1 Binary Search Tree 1.1 Operations We want to design a data structure, an ordered set S that supports the following operations: 1. Search(k, S) 2. Insert(k, S) 3. Delete(k, S) 4. Range(k, k ′ , S) ≡ |{k ∈ S|k ≤ k ≤ k}| If only (1), (2), (3) are required, we could use a hash table. However, if we also need (4), then we will need a BST. Definition 1.1. T is a BST for keys S if: 1. T is an ordered binary tree with |S| nodes. 2. Each node stores a unique key, i.e., there is a 1-1 correspondence between keys and nodes. 3. With in-order traversal, we would gain an ordered sequence, i.e., the keys are stored in inorder. Definition 1.2. T is a balanced BST if max-depth(T ) = O(log n), and can be further categorized: 1. Always balanced: AVL, 2-3-4, Red-Black, B-Trees 2. Randomized balanced: Skip-lists, Treap 3. Amortized balanced: Splay tree