Expected number of uniformly distributed balls in a most loaded bin using placement with simple linear functions

TL;DR

This paper analyzes the expected maximum load in bins when balls are placed using simple linear functions over finite fields, showing the load remains constant, unlike the logarithmic growth seen with fully random hash functions.

Contribution

It demonstrates that using simple linear functions for placement results in a constant expected maximum load, a significant improvement over traditional random hashing.

Findings

Expected maximum load is constant with linear functions.

Fully random hash functions lead to logarithmic maximum load.

The result holds for balls chosen from a transformed interval.

Abstract

We estimate the size of a most loaded bin in the setting when the balls are placed into the bins using a random linear function in a finite field. The balls are chosen from a transformed interval. We show that in this setting the expected load of the most loaded bins is constant. This is an interesting fact because using fully random hash functions with the same class of input sets leads to an expectation of $\Theta\left(\frac{\log m}{\log \log m}\right)$ balls in most loaded bins where $m$ is the number of balls and bins. Although the family of the functions is quite common the size of largest bins was not known even in this simple case.