# 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 1) Design a fixed-capacity in-memory cache that evicts the least recently used key when full. Support get(key) and put(key, value) in O( 1) average time. Describe the data structures you would use, how recency is updated on get/put, how evictions occur, and analyze time and space complexity. 2) Given an array of 2D points and an integer k, return the k points closest to the origin by Euclidean distance. First outline a straightforward approach using sorting. Then discuss improvements using Quickselect and a max-heap, including their time and space complexity and when you would choose each approach. ## 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.