# 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 Solve the following algorithmic tasks: 1) Given a singly linked list, return the k-th node from the end (1-indexed). Describe one-pass and two-pass approaches, edge cases (k <= 0, k > n), and time/space complexity. 2) Given an array of integers, return the top k most frequent numbers. Explain how to handle ties, streaming data, and large ranges; compare min-heap, bucket sort, and quickselect approaches. 3) Given a list of 2D points, return the k points closest to a target point (default origin). Specify the distance metric, constraints, and the complexity of heap-based vs divide-and-conquer solutions. 4) Given a string, repeatedly remove any maximal group of adjacent equal characters until n... ## 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.