A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing

2026-05-18

AI Summary

This work investigates the expected value and tail behavior of the maximum load under two-choice linear hashing, revealing its near-matching performance with fully independent hashing up to both leading and second-order correction terms. By refining the exponential potential function method and integrating subspace constructions with combinatorial estimates, the paper establishes the first asymptotically tight tail bound for the load of a single bin. The main contributions include proving an upper bound on the expected maximum load of $(1 + (1+o(1))\frac{\log\log\log n}{\log\log n})\frac{\log n}{\log\log n}$ and deriving refined tail probability bounds that significantly improve upon existing results, thereby narrowing the theoretical gap with ideal fully independent hashing.

Abstract

Let $S\subseteq F_2^u$ have size $n=2^\ell$, and let $h:F_2^u\to F_2^\ell$ be a uniformly random linear map. For $y\in F_2^\ell$, write $Load_h(y):=|h^{-1}(y)\cap S|$, and let $M(S,h):=\max_{y\in F_2^\ell} Load_h(y)$ be the maximum load. Jaber, Kumar and Zuckerman (STOC 2025) proved that the expected maximum load of $h$ on $S$ is at most $16\log n/\log\log n$, matching the fully independent keys-into-bins scale up to constants. Their proof also gives the tail estimate \[ \Pr\left[ M(S,h)\ge R\frac{\log n}{\log\log n} \right] \le O\left(\frac{1}{R^{2}}\right). \] We record a base optimization in their exponential-potential method showing that binary linear hashing nearly matches fully independent hashing also at the level of the second-order maximum-load scale. For every $R>1$ satisfying $R\ell^{1-1/R}\ge D\ln\ell$, where $D$ is an absolute constant, we prove \[ \Pr\left[ M(S,h)\ge R\frac{\log n}{\log\log n} \right] \le O\left( \frac{(\log\log n)^2}{R^2(\log n)^{2-2/R}} \right). \] Integrating this tail yields \[ E[M(S,h)] \le \left( 1+ (1+o(1)) \frac{\log\log\log n}{\log\log n} \right) \frac{\log n}{\log\log n}. \] Thus binary linear hashing matches fully independent hashing in the leading term and matches the dominant second-order correction up to a $1+o(1)$ factor. We also prove, by an independent self-contained argument, a sharp tail bound for one prescribed bucket: for fixed $y\in F_2^\ell$, \[ \Pr[ Load_h(y)>2^a-2]\le γ^{-1}2^{-a^2}, \] where $ γ=\prod_{j\ge1}(1-2^{-j}) $. A subspace construction shows that this is asymptotically tight even in the leading constant as $ a\to\infty $. However, this controls only a fixed bucket; a direct union bound over all buckets loses a factor $ 2^\ell $.

Problem

• binary linear hashing
• maximum load
• second-order bounds
• expected load
• hashing

Innovation

• binary linear hashing
• maximum load
• second-order analysis
• tail bounds
• exponential-potential method