Under what conditions will the add operation in a HashMap have a time complexity proportional to logarithmic complexity?

Adding an element to a HashMap typically has an amortized constant time complexity of O(1). However, under certain conditions, the time can increase to O(log n):

• If a balanced tree (e.g., a red-black tree) is used in the HashMap to resolve collisions instead of linked lists.
• When many elements collide in a single bucket, and the data structure within the bucket changes from a list to a tree.

This approach is implemented, for example, in Java starting from version 8, where when the collision threshold in a bucket is exceeded, the linked list is transformed into a tree, improving search and insertion to logarithmic complexity.

Thus, adding an element reaches logarithmic complexity when a balanced tree is used within a bucket to store elements due to collisions.