基于算术Kakeya的线性哈希下界
Lower Bounds for Linear Hashing via Arithmetic Kakeya
浏览论文内容
中文总结 AI 辅助
该研究针对线性哈希的最大负载问题，通过两次归约结合算术Kakeya构造，证明了键空间为n^(1+o(1))时的指数级下界，还给出了上界的替代路径及Kakeya猜想的关联结论。
中文摘要 AI 辅助
仿射模线性哈希是最简单的经典哈希族之一。对于素数p > u，哈希函数通过从ℤₚ中均匀选取s、t，将每个键x ∈ {0,…,u-1}映射到n个桶之一，公式为h(x) = [(sx + t) mod p] mod n。尽管结构简单，线性哈希的最大负载仍未被充分理解：对于n个键哈希到n个桶，目前最佳上界为O((n log n)^(1/3))，而最佳下界仅为Ω(log n / log log n)。我们针对大小为n^(1+o(1))的键空间，证明了下界exp(Ω(log n / log log n))。令人惊讶的是，存在一个键集，其负载不仅在期望意义下成立，对每个随机种子均成立。证明基于两个简单归约：其一将实线性哈希的下界传递到模线性哈希，其二将算术Kakeya构造传递到实哈希。我们进一步证明，对于足够大的p，模与实场景下的期望最大负载本质相同，为获得n^(1/3+o(1))上界提供了另一条路径。最后，我们指出，任一场景下的均匀亚多项式上界，都将蕴含多项式长度的算术Kakeya猜想，进而蕴含上Minkowski维数的Kakeya猜想。
英文摘要
Affine modular linear hashing is one of the simplest classical hash families. For a prime $p > u$, the hash function is obtained by choosing $s,t$ uniformly from $\mathbb{Z}_p$ and mapping each key $x \in \{0,\ldots,u-1\}$ to one of $n$ bins by $h(x) = [(sx+t) \bmod p] \bmod n$. Despite its simplicity, the maximum load of linear hashing remains poorly understood. For $n$ keys hashed into $n$ bins, the best known upper bound is $O((n \log n)^{1/3})$, whereas the best known lower bound is only $Ω(\log n / \log\log n)$. We prove a lower bound of $\exp(Ω(\log n / \log\log n))$ for universes of size $n^{1+o(1)}$. Surprisingly, there is a key set for which this load holds not just in expectation, but for every random seed. The proof is driven by two simple reductions: one transfers lower bounds from a real version of linear hashing to modular linear hashing, and the other transfers arithmetic Kakeya constructions to real hashing. We further show that, for sufficiently large $p$, the expected maximum loads in the modular and real settings are essentially the same, giving an alternative route to an $n^{1/3+o(1)}$ upper bound. Finally, we show that any uniform subpolynomial upper bound for either setting would imply a polynomial-length arithmetic Kakeya conjecture and hence the Kakeya conjecture for upper Minkowski dimension.
