On The Relationship Between Several Variants Of The Linear Hashing Conjecture

Westover Alek. Arxiv 2023

[Paper]    
ARXIV Independent

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}\( achieves maxload \)Ω\left(\frac{log n}{loglog 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.

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