LONG-TERM BALANCED ALLOCATION VIA THINNING

In the two-thinning balls-and-bins model, an overseer is provided with uniform random allocation of m balls into n bins in an on-line fashion. The overseer may reject the allocation of each ball, in which case it is placed into a new bin, drawn independently, uniformly at random. The purpose of the overseer is to reduce the maximum load, that is, the difference between the maximum number of balls in a single bin and the average number of balls among all bins. We provide tight estimates for three quantities: the lowest achievable maximum load at a given time m, the lowest achievable maximum load uniformly over the entire time interval [m]:= {1, 2, . . ., m} and the lowest achievable typical maximum load over the interval [m], that is, a load which upper-bounds 1 − o(1) portion of the times in [m]. In particular, for m polynomial in n and sufficiently large, we provide an explicit strategy, which achieves a typical maximum load of (log n)^1/2+o(1) , asymptotically the same as that can be achieved at a single time m. In contrast, we show that no strategy can achieve better than Θ(_log^log_logⁿ_n ) maximum load for all times up to time m.