The power of thinning in balanced allocation

Balls are sequentially allocated into n bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after Θ(n) balls have been allocated. We provide an asymptotically optimal strategy √ yielding a maximum load of (1 + o(1)) 8 log n log log n^balls..