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

As a brief announcement \cite{PARK11}, we proposed the following balls-into-bins strategy allocating $n$ balls into $n$ bins, denoted by $(k,d)$-choice process. The placement 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 present complete analysis for our preliminary results and provide tight upper and lower bounds for the maximum load that hold with high probability 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 new results indicate that we can choose suitable parameters $k$ and $d$ to achieve the optimal tradeoff between the maximum bin load and message cost. For some $k$ and $d$, the $(k,d)$-choice process produces maximum load of O(1) whp at the expense of $(1+\epsilon)n$ messages (with certainty) for $0< \epsilon < 1$ constant. This shows that a non-adaptive balls-into-bins algorithm can outperform any sequential/parallel allocation algorithms including the ones using adaptive schemes, in the sense that any other algorithms achieving constant maximum load would require higher message cost than we have. Furthermore, we discuss some applications in which $(k,d)$-choice is more efficient than the standard multiple-choice algorithm; i.e. the cost of item insertion and search is as low as $(1+o(1))$ time on average.