How is the time complexity determined when inserting new elements into a hash table?

The amortized time complexity of inserting an element into a hash table is usually considered to be O(1), meaning that insertion takes constant time on average.

This is achieved through:

• Fast computation of the hash function to determine the index.
• Fast access to the array cell.

However, in the worst case (for example, in case of collisions where multiple keys hash to the same bucket), insertion can degrade to O(n) if a linked list is used to store collisions.

Modern implementations (such as in Java HashMap) switch to balanced trees when there are many collisions, reducing the worst case to O(log n).

Thus, the time complexity depends on the efficiency of the hash function, the degree of collisions, and the data structure used for collision resolution.