An asymptotically optimal algorithm for generating bin cardinalities

Abstract

In the balls-into-bins setting, $n$ balls are thrown uniformly at random into $n$ bins. The naïve way to generate the final load vector takes $\Theta \left(n\right)$ time. However, it is well-known that this load vector has with high probability bin cardinalities of size $\Theta \left(\frac{logn}{loglogn}\right)$. Here, we present an algorithm in the RAM model that generates the bin cardinalities of the final load vector in the optimal $\Theta \left(\frac{logn}{loglogn}\right)$ time in expectation and with high probability.

Further, the algorithm that we present is still optimal for any $m\in [n,nlogn]$ balls and can also be used as a building block to efficiently simulate more involved load balancing algorithms. In particular, for the Two-Choice algorithm, which samples two bins in each step and allocates to the least-loaded of the two, we obtain roughly a quadratic speed-up over the naïve simulation.