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)?