Falling Squares Variation

Start by restating the contract for Falling Squares Variation and explicitly clarify any difference between this candidate-reported label and the desired input/output behavior. Choose the representation that makes the central invariant visible: after each operation, state what is known to be correct and why it remains true. Begin with a simple baseline if it helps establish correctness, then replace its bottleneck with the appropriate data structure or recurrence. Walk through a small edge-case example aloud, including empty or minimal input, repeated values where relevant, and boundary indexes. Do not invent missing constraints: name the assumption and ask for confirmation. The final implementation should account for every stored state when reporting time and space complexity, not just the outer loop. This produces a useful prep answer even when the scraped source has incomplete wording. Complexity: coordinate compression plus range-max updates takes O(n log n) time and O(n) space for n squares.