Derandomized Asymmetrical Balanced Allocation

Abstract

Balls-in-bins model, in which n balls are sequentially placed into n bins according to some dispatching policy, is an important model with a wide range of applications despite its simplicity. The power-of-d choices (Pod) policy, in which each ball samples d independent uniform random bins and join the one with the least load (where ties are broken arbitrarily), can yield a maximum load of $\frac{\log\log n}{\log d} + \Theta(1)$ with high probability whenever $d\geq 2$. Vöking later proposed a variant of power-of-d scheme in which bins are divided into d groups, and d bins are sampled from each group respectively. One important feature of this scheme is that ties are broken asymmetrically based on groups. Comparing with Pod, this scheme reduces the maximum load to $\frac{\log\log n}{d\log\phi_{d}}+\Theta(1)$ where $1 \lt \phi_{d} \lt 2$. Our recent work shows that one can replace independent uniform sampling with random walk based sampling while having the same performance of Pod in terms of the maximum load of all bins. In this work, we propose multiple derandomized variants of Vöking’s asymmetrical schemes and we show that they can yield the same performance as the original scheme, i.e. the maximum load is bounded by $\frac{\log \log n}{d \log \phi_{d}}+\Theta(1)$