I've recently been thinking about this problem.
Given an array $A$ containing $n$ integers and an index $i$, find the number of subgements of $A$ containing $A[i]$ as their minimum.
To better illustrate the problem, consider the array $A = [5, 1, 7]$ and we want to find the number of subsegments containing $A = 1$ as their minimum. Clearly the answer is 4 and we can easily enumerate them: $, [5, 1],[5, 1, 7], [1, 7]$.
What the fastest way to compute the number of subsegments?