In the ideal case, the time complexity of a hash table insert operation is O(1), or constant time. This is under the assumption that the hash function distributes the elements evenly across the hash table buckets, thereby avoiding or minimizing collisions, and that the size of the hash table is sufficient to accommodate the elements without having to perform resizing operations frequently.
Practically, collisions can occur where two different elements are hashed to the same index. To handle collisions, various techniques can be used like chaining or open addressing which may add an overhead.
Chaining: Each slot in the hash table holds a linked list of elements that hash to the same slot. The time complexity would be O(n) in worst case scenario where all elements hash to the same slot, effectively creating a linked list.
Open Addressing: Elements are probed into next slots upon collision. In the worst case scenario (hash table almost full), it could potentially have to probe through much/most of the table, giving a time complexity of O(n).
However, even considering collision handling, hash table operations are remarkably efficient if we can assume the hash function is well-designed, and given a large enough hash table, the time complexity of an insert operation still approximates to O(1), on an average case.
Note, when the hash table reaches a certain load factor (ratio of number of items / table size), we may need to resize the hash table (usually doubling its size) and rehash all the items. This is an O(n) operation, but it is amortized over n insertions, so it doesn’t affect the average time complexity of an individual insertion operation. Its effect becomes constant when spread over the individual insertions that triggered the resizing, hence we can still say the time complexity of a hash table insert operation is generally regarded as O(1).
The time complexity for the operations can be expressed as:
- Average Case: O(1)
- Worst Case: O(n)
Where "n" is the number of entries in the hash table.