# 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 and implement a least-recently-used (LRU) cache with a fixed capacity. Support get(key) -> value and put(key, value) in average O( 1) time. When the cache is full, evict the least recently used entry. Explain your choice of data structures, detail how you update recency on reads and writes, and describe how you handle updates to existing keys, missing keys, and capacity edge cases (e.g., capacity = 0). Analyze time and space complexity. ## Recommended Approach Use a hash map from key to doubly linked-list node plus a doubly linked list ordered by recency. get moves the node to the front. put updates and moves an existing node, or inserts a new node at the front and evicts the tail when capacity is exceeded. ## 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 get and put are O(1) average time. Space is O(capacity). ## Edge Cases and Tests Capacity 0 or 1, updating an existing key, eviction order after get, repeated puts, and missing-key gets.