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# NP-Incompleteness

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## Max Area Under a Histogram

09 Jan 2021

This is a classic programming puzzle. Suppose we’re given an array of $n$ values reprenting the height $h_i \ge 0$ of bars in a histogram, for $i = 0, \cdots, n-1$. We want to find the largest rectangle that fits “inside” that histogram.

More formally, we’d like to find indices $l$ and $r$, $l \le r$ such that $(l - r + 1) \bar h$ is maximum, where $\bar h$ is the smallest $h_i$ for $i \in \curly{l, \cdots, r}$.

In this post we’ll describe an $O(n)$ algorithm that solves this problem.