Linear Hashing is Not That Awesome
Consider the canonical universal hash family $h(x)= ((ax+b)\text{ mod } p)\text{ mod } m$, where $a,b$ are chosen uniformly from $\mathbb Z_p$, which we call linear hashing, being used to hash $n$ elements into $m=Θ(n)$ buckets. For any universal family, the expected size of the largest bucket is at least $Ω(\log n / \log\log n)$ and at most $O(\sqrt{n})$. The only improvement upon these trivial bounds for linear hashing is a 2019 upper bound of $\tilde{O}(n^{1/3})$ by Knudsen. We show that for any $p$ sufficiently larger than $n$, there is a set of $n$ keys whose expected maximum load is $n^{Ω(1/\log\log n)}$, proving linear hashing does not have a polylogarithmic maximum load. We extend the same bounds to the classical multiply-shift hash family of Dietzfelbinger, Hagerup, Katajainen, and Penttonen. Our main contribution is an equivalence between the maximum load problem to a density variant of arithmetic Kakeya sets. We then complete the lower bound using a construction of Green and Ruzsa of a small set containing long arithmetic progressions with every difference in a prescribed range. Surprisingly, our equivalence also implies that any substantial improvement over Knudsen's upper bound would imply new results about standard arithmetic Kakeya sets. More precisely, an $O(n^{1/3-\varepsilon})$ upper bound would improve known bounds for unions of complete integer arithmetic progressions, while an $n^{o(1)}$ upper bound would imply Bourgain's arithmetic-progression criterion.