How is the efficiency of inserting an element into a hash table evaluated under different conditions?

The efficiency of inserting an element into a hash table depends on several factors:

1. Load factor — the ratio of the number of elements to the size of the array. With a low load factor, insertion is usually performed in amortized O(1) time.

2. Hash function quality — uniform distribution of keys across buckets minimizes collisions.

3. Collision handling:

   • With chaining, insertion involves adding to a linked list or another container in the bucket. On average, O(1), but in the worst case, O(n) if all elements land in one bucket.
   • With open addressing (linear probing, quadratic probing), insertion may require searching for a free slot, which increases time at high load factors.

4. Rehashing — when the load factor threshold is reached, the table is expanded, requiring redistribution of elements and temporarily increasing insertion time.

In summary: with good hashing and moderate load, insertion is amortized O(1). At high load or with poor hashing, the time can degrade to O(n).