Complexity Analysis - Walkthrough

Each element adds $O(1)$ breakpoints (sometimes $2$). Total breakpoints: $O(n)$. Each heap operation (insert, pop) is $O(\log n)$.

Each element does $O(1)$ operations. Total time: $O(n \log n)$. Space: $O(n)$ for the heaps. Compare to naive: $O(n \cdot \text{range})$ where range is the value domain. Slope Trick ignores the range. The observation: each element adds at most two breakpoints. Total breakpoints stay $O(n)$ throughout.