Chef and Dragon Dens

Chef is on an adventure to find the tastiest exotic spices. Everyone knows that the best spices are guarded by several ancient dragons that dwell within mountains. The mountain range can be modeled by a polyline in the 2D $xz$ plane that passes through $N$ points $p_1 = (1,h_1), p_2 = (2, h_2), \ldots, p_N = (N, h_N)$, in this order, where $h_i$ is a positive integer for each valid $i$. This po