Design a data structure for dynamic top‑K frequency

Design a data structure that maintains a multiset of integers and supports the following operations efficiently: ### Operations - `insert(x)`: Add one occurrence of value `x`. - `remove(x)`: Remove one occurrence of value `x` if it exists (if `x` is not present, do nothing). - `topK(k) -> List[int]`: Return the **k most frequent** values currently in the multiset. ### Requirements / Clarifications - Frequency means the current count of each value in the multiset. - If fewer than `k` distinct values exist, return all distinct values. - If multiple values have the same frequency, you may return them in any order (or state and implement a deterministic tie-break such as smaller value first). - Discuss time and space complexity for each operation. - Explain why your chosen data structures are appropriate compared with alternatives (e.g., sorting on every query, balanced BST/TreeMap, etc.). - Consider edge cases (removing down to zero, repeated inserts/removes, large `k`, empty structure). - Bonus discussion: how would your approach change if the number of total updates is extremely large (e.g., up to billions)?