On the Relationship Between Several Variants of the Linear Hashing Conjecture

TL;DR

This paper explores the challenging problem of bounding the worst-case expected maximum load in linear hashing, proposing intermediate questions and partial progress to advance understanding of this fundamental open problem.

Contribution

It introduces intermediate open questions relating to linear hashing's maxload and provides partial results, advancing the theoretical understanding of this complex problem.

Findings

Established relationships between intermediate open questions

Provided partial progress on bounding maxload

Connected problem to broader discrete math challenges

Abstract

In Linear Hashing ($\mathsf{LH}$) with $\beta$ bins on a size $u$ universe ${\mathcal{U}=\{0,1,\ldots, u-1\}}$, items $\{x_1,x_2,\ldots, x_n\}\subset \mathcal{U}$ are placed in bins by the hash function $$x_i\mapsto (ax_i+b)\mod p \mod \beta$$ for some prime $p\in [u,2u]$ and randomly chosen integers $a,b \in [1,p]$. The "maxload" of $\mathsf{LH}$ is the number of items assigned to the fullest bin. Expected maxload for a worst-case set of items is a natural measure of how well $\mathsf{LH}$ distributes items amongst the bins. Fix $\beta=n$. Despite $\mathsf{LH}$'s simplicity, bounding $\mathsf{LH}$'s worst-case maxload is extremely challenging. It is well-known that on random inputs $\mathsf{LH}$ achieves maxload $\Omega\left(\frac{\log n}{\log\log n}\right)$; this is currently the best lower bound for $\mathsf{LH}$'s expected maxload. Recently Knudsen established an upper bound of $\widetilde{O}(n^{1 / 3})$. The question "Is the worst-case expected maxload of $\mathsf{LH}$ $n^{o(1)}$?" is one of the most basic open problems in discrete math. In this paper we propose a set of intermediate open questions to help researchers make progress on this problem. We establish the relationship between these intermediate open questions and make some partial progress on them.