This is the classic monotonic stack problem and I knew I'd seen it before, which somehow made it worse because I second-guessed myself the whole time.
Start by clarifying the problem and edge cases, then propose a brute-force solution to establish a baseline. Introduce the optimal O(n) monotonic stack approach, explaining how it efficiently computes the largest rectangle by maintaining increasing bar heights and calculating areas when a smaller bar is encountered.
Pro tip: Emphasize the importance of handling edge cases like empty arrays and duplicate heights, and mention that the stack approach can be adapted to return the rectangle's boundaries if needed. This shows attention to detail and real-world applicability.
Ask questions to confirm the input format, expected output (area only or also boundaries), and constraints (e.g., array size, height range).
Outline a simple O(n^2) solution: for each bar, expand left and right to find the maximum width with height at least that bar's height, and track the maximum area.
Explain the monotonic stack method: iterate through bars, maintain a stack of indices with increasing heights, and when a lower bar is found, pop and calculate areas using the popped bar's height and the width between the current index and the new stack top.
Trace the algorithm on a small example (e.g., [2,1,5,6,2,3]) to demonstrate how the stack evolves and how the maximum area is computed.
State that the optimal solution runs in O(n) time and O(n) space, and mention handling of empty input, single bar, and all equal heights.
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