Heap Representation - Walkthrough

A piecewise linear function is defined by its slope changes. If slope goes from $0$ to $1$ at $x = 5$, we store $5$. observation: we only need the breakpoints, not the full function.

The slopes between breakpoints are implicit. Two heaps: $L$ (max-heap) for left breakpoints, $R$ (min-heap) for right breakpoints. The minimum is in the 'valley'. To add $|x - a|$: insert $a$ into both $L$ and $R$. Then balance: if $L.top > R.top$, swap and adjust the base value.