Balanced Allocations: The Heavily Loaded Case

We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of $d \ge 2$ randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when $m \approx n$. In this paper we present the first tight analysis in the heavily loaded case, that is, when $m \gg n$ rather than $m \approx n$.The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of $m/n + {\mbox{$\cal O$}}(\sqrt{m \ln n \,/\, n})$ with high probability. We show, however, that the multiple-choice...