A Generalization of Multiple Choice Balls-into-Bins: Tight Bounds

We consider a general version of the multiple choice problem for assigning $n$ balls to $n$ bins, which we call $(k,d)$-choice process. The process consists of $n/k$ rounds, each of which $k<d$ balls are placed into $k$ least loaded bins out of $d$ bins chosen independently and uniformly at random. In this paper, we provide tight bounds for the maximum load for any $1 \leq k < d \leq n$. The entire spectrum of allocation schemes that lie between the boundaries of the single and multiple-choice algorithms is captured in this simple process. Our results enable one to choose suitable parameters $k$ and $d$ for which $(k,d)$-choice strategy achieves the optimal tradeoff between the maximum bin load and the message cost. While the standard multiple choice strategy requires $\Omega(n\log n)$ messages to achieve the maximum load of $O(1)$, the $(k,d)$-choice scheme produces constant maximum load at the expense of less than $2n$ messages with properly chosen $k$ and $d$. Our results indicate that a randomized allocation scheme can achieve the performance of the optimal scheme with very little overhead.