Binary Search - Walkthrough

We want exactly $K$ elements. Define $f(\lambda)$ = optimal count when each element costs $+\lambda$. $f$ is monotonically non-increasing in $\lambda$.

More $penalty = fewer$ elements used. Binary search: find smallest $\lambda$ where $f(\lambda) \leq K$. Or find $\lambda$ where $f(\lambda) = K$ exactly. Subtlety: $f$ might jump (not hit every value). Handle ties carefully. Often we binary search and check both boundaries.