How is the access time to an element via a key in a HashMap data structure determined?

In the HashMap data structure, the average access time to an element by key is O(1) — constant time. This is achieved by using a hash function that converts the key into an array index (bucket), where the value is stored.

How it works:

1. The key is passed to the hash function, which computes a hash code.
2. The hash code determines the index of the bucket in the array.
3. The bucket stores key-value pairs. If multiple keys hash to the same bucket (collision), a traversal of elements in that bucket occurs.

In the worst case (many collisions), access time can degrade to O(n), but with a good hash function and uniform key distribution, this is unlikely.

Thus, HashMap provides very fast access to data by key thanks to hashing and efficient collision resolution.