Universal Hashing and Perfect Hashing

Each of the key values x comes from a universe U , i.e. x ∈ U . In this document, we assume U = {1, 2, . . . N}. Observe that the set S is a dynamic set. Each of the Insert and Delete operations may modify the set. Hence the size of the set S changes with each operation. We bound the maximum size of the set to n (n << N). What are the data structures that can be used to store the set S? One option is to use a balanced binary search tree. But each of the operations would take O(log n) time. Moreover, a balanced binary serch tree is more difficult to implement than, say, an array, or a singly linked list. Could we store the set S in an array? Is there a data structure that would perform the above operations in constant time? Yes, there is such a data structure, hash table, which provides an easy way of storing such information.