Question

Design a data structure to support the following operations for a dynamic multiset S of integers, which allows duplicate values,

INSERT(S, x), inserts x into S
DELETE-LARGER-HALF(S, x), deletes the largest [|S|/2] from S numbers
Explain how to implement this data structure so that any sequence of m INSERT and DELETE-LARGER-HALF operations runs in O(m) time. Your implementation should also include a way to output the elements of S in O(|S|) time.

Answer 1

To support the required operations, we can use a balanced binary search tree (BST) data structure such as AVL tree or Red-Black tree. These trees provide efficient insertion and deletion operations in O(log n) time complexity.

Step-by-step explanation:

To support the required operations, we can use a balanced binary search tree (BST) data structure such as AVL tree or Red-Black tree. These trees provide efficient insertion and deletion operations in O(log n) time complexity. The height of the tree (log n) ensures that any sequence of m operations runs in O(m log n) time.

To implement the DELETE-LARGER-HALF operation, we can perform an in-order traversal of the binary search tree, removing the largest half of the elements. Since we can visit each element once during the traversal, the time complexity of this operation is O(|S|).

To output the elements of S in O(|S|) time, we can perform an in-order traversal of the binary search tree and collect the elements in a list or an array.