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You need a monotonic increasing stack of indices.
Start by clarifying the problem and discussing a brute-force O(n^2) approach, then optimize using a monotonic stack to achieve O(n) time. Walk through the algorithm step-by-step, emphasizing how the stack maintains indices of bars in increasing height order to efficiently compute the maximum area.
Pro tip: Mention that this problem is a classic example of using a monotonic stack to solve 'next smaller element' problems, and that the same technique applies to similar problems like trapping rain water or maximal rectangle in a binary matrix. Also, discuss edge cases like empty input or all equal heights to show thoroughness.
Restate the problem in your own words and ask clarifying questions about input constraints, expected output, and edge cases.
Explain 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.
Describe how a stack can maintain indices of bars in increasing order of height, allowing O(n) computation by finding left and right boundaries for each bar.
Trace the algorithm on a small example (e.g., [2,1,5,6,2,3]) to demonstrate how the stack updates and areas are calculated.
State time and space complexity (O(n) time, O(n) space) and discuss handling of edge cases like empty array, single bar, or strictly increasing/decreasing heights.
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