On the Relationship Between Several Variants of the Linear Hashing Conjecture
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In Linear Hashing (𝖫𝖧) with β bins on a size u universe 𝒰={0,1,…, u-1}, items {x_1,x_2,…, x_n}⊂𝒰 are placed in bins by the hash function x_i↦ (ax_i+b) p β for some prime p∈ [u,2u] and randomly chosen integers a,b ∈ [1,p]. The "maxload" of 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 𝖫𝖧 distributes items amongst the bins. Fix β=n. Despite 𝖫𝖧's simplicity, bounding 𝖫𝖧's worst-case maxload is extremely challenging. It is well-known that on random inputs 𝖫𝖧 achieves maxload Ω(log n/loglog n); this is currently the best lower bound for 𝖫𝖧's expected maxload. Recently Knudsen established an upper bound of O(n^1 / 3). The question "Is the worst-case expected maxload of 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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