Long-term balanced allocation via thinning

Abstract

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 $[\mathit{m}]:=\{1,2,\dots ,\mathit{m}\}$ and the lowest achievable typical maximum load over the interval $[\mathit{m}]$, that is, a load which upper-bounds $1-\mathit{o}(1)$ portion of the times in $[\mathit{m}]$.

In particular, for m polynomial in n and sufficiently large, we provide an explicit strategy, which achieves a typical maximum load of ${(log\mathit{n})}^{1/2\mathbf{+}\mathit{o}(1)}$, asymptotically the same as that can be achieved at a single time m. In contrast, we show that no strategy can achieve better than $\mathrm{\Theta }(\frac{log\mathit{n}}{loglog\mathit{n}})$ maximum load for all times up to time m.