Maximize the rectangular area under Histogram

Question

I have a histogram with integer heights and constant width 1. I want to maximize the rectangular area under a histogram. e.g.:

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Answer 1

I don't understand the other entries, but I think I know how to do it in O(n) as follows.

A) for each index find the largest rectangle inside the histogram ending at that index where the index column touches the top of the rectangle and remember where the rectangle starts. This can be done in O(n) using a stack based algorithm.

B) Similarly for each index find the largest rectangle starting at that index where the index column touches the top of the rectangle and remember where the rectangle ends. Also O(n) using the same method as (A) but scanning the histogram backwards.

C) For each index combine the results of (A) and (B) to determine the largest rectangle where the column at that index touches the top of the rectangle. O(n) like (A).

D) Since the largest rectangle must be touched by some column of the histogram the largest rectangle is the largest rectangle found in step (C).

The hard part is implementing (A) and (B), which I think is what JF Sebastian may have solved rather than the general problem stated.