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arXiv · 2505.14061

Linear Hashing Is Optimal

Michael Jaber·Vinayak M. Kumar·David Zuckerman
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Abstract

We prove that hashing $n$ balls into $n$ bins via a random matrix over $\mathbf{F}_2$ yields expected maximum load $O(\log n / \log \log n)$. This matches the expected maximum load of a fully random function and resolves an open question posed by Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (STOC '97, JACM '99). More generally, we show that the maximum load exceeds $r\cdot\log n/\log\log n$ with probability at most $O(1/r^2)$.

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Michael Jaber, Vinayak M. Kumar, David Zuckerman. 2025-05-20. Linear Hashing Is Optimal. https://arxiv.org/abs/2505.14061

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