# Derandomized Balanced Allocation

## TL;DR

This paper constructs a hash family with low randomness complexity that achieves near-optimal maximum load bounds in $d$-choice ball allocation schemes, matching those of fully random hash functions.

## Contribution

It introduces a hash family with $O( ext{log} n ext{ log log} n)$ random bits that attains optimal maximum loads in $d$-choice schemes, improving over previous $O( ext{log}^2 n)$-wise independent hash functions.

## Key findings

- Maximum load of $rac{ ext{log} ext{log} n}{ ext{log} d} + O(1)$ in Uniform-Greedy scheme.
- Maximum load of $rac{ ext{log} ext{log} n}{d ext{log} extphi_d} + O(1)$ in Always-Go-Left scheme.
- Hash family matches maximum loads of perfect randomness with significantly fewer random bits.

## Abstract

In this paper, we study the maximum loads of explicit hash families in the $d$-choice schemes when allocating sequentially $n$ balls into $n$ bins. We consider the \emph{Uniform-Greedy} scheme, which provides $d$ independent bins for each ball and places the ball into the bin with the least load, and its non-uniform variant --- the \emph{Always-Go-Left} scheme introduced by V\"ocking. We construct a hash family with $O(\log n \log \log n)$ random bits based on the previous work of Celis et al. and show the following results. 1. With high probability, this hash family has a maximum load of $\frac{\log \log n}{\log d} + O(1)$ in the \emph{Uniform-Greedy} scheme. 2. With high probability, it has a maximum load of $\frac{\log \log n}{d \log \phi_d} + O(1)$ in the \emph{Always-Go-Left} scheme for a constant $\phi_d>1.61$. The maximum loads of our hash family match the maximum loads of a perfectly random hash function in the \emph{Uniform-Greedy} and \emph{Always-Go-Left} scheme separately, up to the low order term of constants. Previously, the best known hash families matching the same maximum loads of a perfectly random hash function in $d$-choice schemes were $O(\log n)$-wise independent functions, which needs $\Theta(\log^2 n)$ random bits.