A real problem from the hospital.
Suppose we have $n$ blood samples to test HIV. We don't want test them one by one, since the cost is huge and most people don't have a positive result. So we try to merge them into groups, and then we just use the machine to test groups. Assume we have $k$ positive in $n$ samples, but we don't know $k$. Does a algorithm of $O(k\log_2n)$ tests (times of using the machine) exist?