Challenge: Online Queries

Standard Slope Trick gives the minimum. What if we need $f(x)$ for arbitrary $x$? Count breakpoints to the left of $x$. Each adds $1$ to slope.

Compute $f(x)$ from base + slope integral. Efficient query: use ordered set (like C++ set) instead of heaps. $O(\log n)$ per query. Trade-off: heaps are simpler and faster for operations, but don't support arbitrary queries. For arbitrary point queries, replace heaps with ordered sets. This gives $O(log n)$ query at the cost of complexity.