E - Expected Rain

There are $N$ buildings on a $1$-D line. The $i$-th building spans from $x$-coordinate $(i - 1)$ to $x$-coordinate $i$, and has a height of $H_i$. When rain falls, water gets collected between the buildings. There are no buildings to the left of $x = 0$ or to the right of $x = N$, and so no water gets collected there. Let $f(H)$ denote the amount of rain collected. ------- Unfortunately, yo