Linear Hashing Is Optimal

TL;DR

This paper proves that linear hashing using a random matrix over GF(2) achieves an optimal maximum load comparable to fully random hashing, resolving a longstanding open problem in the field.

Contribution

It establishes that linear hashing is asymptotically optimal in terms of maximum load, matching the performance of fully random functions.

Findings

Expected maximum load is O(log n / log log n)

Maximum load exceeds r·log n / log log n with probability at most O(1/r^2)

Resolves an open question from prior research

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 $O(1/r^2)$.