Common Mistakes - Walkthrough

$1.$ Wrong heap type. For min function (convex), use max-heap for left, min-heap for right.

$2.$ Forgetting to balance heaps. After adding a breakpoint, the heaps might violate the invariant $L.top \leq R.top$.

$3.$ Not tracking the base value. The function's minimum value changes with operations. Track it separately.

$4.$ Off-by-one in slopes. The slope change at breakpoint $x$ is +$1$ (or +$2$ for double insertion). Count carefully.