# 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 ##### Question System failure troubleshooting: You can only SSH into the machine and the log file is huge. How would you locate the problem quickly? Flip a bit: Given an integer num and a bit position, flip that bit. Concurrency bug: Given multithreaded code that uses locks, find and fix the deadlock. Word frequency top-K: Given a directory that may contain nested sub-directories and files, count word frequencies across all files and return the top-K words. Follow-ups: handle too many files to fit in memory (MapReduce vs Count-Min Sketch + Space-Saving). ## 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.