Optimize algorithms and validate inputs

# 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 For a typical medium-difficulty coding task, walk through your approach: enumerate edge cases, justify your initial algorithm, analyze time and space complexity, and propose improvements. Describe production practices for handling invalid or unexpected inputs (validation, error reporting, logging/metrics, safe defaults, and failure modes). Provide example test cases that cover corner scenarios. ## Recommended Approach Start with a brute-force baseline to confirm correctness, then identify the repeated work or ordering property that enables a better data structure such as a hash map, heap, stack, queue, two pointers, prefix sums, BFS/DFS, or dynamic programming. Write the implementation around a small invariant and test that invariant directly. ## 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 State the baseline complexity and the optimized complexity. For most interview constraints, justify why the optimized approach meets the expected input size. ## Edge Cases and Tests Empty and singleton inputs, duplicates, ties, invalid inputs, boundary values, and tests that exercise the main invariant.