# Solution Alignment The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests. ## Problem Restatement Design a data structure that tracks the top-K most frequent keys in a stream. Support operations increment(key), decrement(key), and topK(k) -> list of k keys with highest frequencies. Achieve O( 1) amortized updates by combining a hash map with a doubly linked list of frequency buckets. Specify tie-breaking, deletion behavior when counts drop to zero, memory usage, and complexity. ## Recommended Approach For one-time top-K, use a size-K min-heap or quickselect plus sorting the selected K. For streaming windows, maintain counts in a hash map plus a heap with lazy deletion or bucketed frequency structures when updates must be near O(1). Define deterministic tie-breaking. ## Correctness The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call. ## Complexity One-time heap: O(n log k) time and O(k) space. Quickselect: expected O(n) plus O(k log k) to order output. Streaming complexity depends on window eviction and tie-breaking. ## Edge Cases and Tests k = 0, k > n, duplicate values, ties, negative values, stale heap entries, and deterministic output ordering.