On the Relationship Between Several Variants of the Linear Hashing Conjecture
In Linear Hashing (š«š§) with Ī² bins on a size u universe š°={0,1,ā¦, u-1}, items {x_1,x_2,ā¦, x_n}āš° are placed in bins by the hash function x_iā¦ (ax_i+b) p Ī² for some prime pā [u,2u] and randomly chosen integers a,b ā [1,p]. The "maxload" of š«š§ 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 š«š§ distributes items amongst the bins. Fix Ī²=n. Despite š«š§'s simplicity, bounding š«š§'s worst-case maxload is extremely challenging. It is well-known that on random inputs š«š§ achieves maxload Ī©(log n/loglog n); this is currently the best lower bound for š«š§'s expected maxload. Recently Knudsen established an upper bound of O(n^1 / 3). The question "Is the worst-case expected maxload of š«š§ 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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