Each element is pushed to the heap at most twice: once for itself, once for slope adjustment after popping. Each push and pop operation is O(logn). Total operations: O(n) pushes and O(n) pops in the worst case.
Time complexity: O(nlogn). Space complexity: O(n) for the heap storing all breakpoints. Compare to naive DP: O(n⋅V2) with V=109 means 1018 operations. Impossible. Slope Trick reduces this to O(nlogn), making previously impossible problems solvable in milliseconds.