On the Relationship Between Several Variants of the Linear Hashing Conjecture

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.