Space Upper Bounds for $\alpha$-Perfect Hashing

TL;DR

This paper extends minimal perfect hashing to an approximate setting, introducing schemes that balance collision rates and space efficiency for large key sets.

Contribution

It proposes new randomized hashing schemes for $oldsymbol{ extit{ extalpha}}$-perfect hashing with improved space bounds over previous methods.

Findings

A simple baseline scheme combining perfect and zero-bit hashing.

A sampling-based scheme that reduces space requirements for all $ extit{ extalpha}$ values.

Abstract

In the problem of minimal perfect hashing, we are given a size $k$ subset $\mathcal{A}$ of a universe of keys $[n] = \{1,2, \cdots, n\}$, for which we wish to construct a hash function $h: [n] \to [k]$ such that $h(\cdot)$ maps $\mathcal{A}$ to $[k]$ with no collisions, i.e., the restriction of $h(\cdot)$ to $\mathcal{A}$ is injective. In this paper, we extend the study of minimal perfect hashing to the approximate setting. For an $\alpha \in [0, 1]$, we say that a randomized hashing scheme is $\alpha$-perfect if for any input $\mathcal{A}$ of size $k$, it outputs a hash function which exhibits at most $(1-\alpha)k$ collisions on $\mathcal{A}$ in expectation. One important performance consideration for any hashing scheme is the space required to store the hash functions. For minimal perfect hashing, it is well known that approximately $k\log(e)$ bits, or $\log(e)$ bits per key, is required to store the hash function. In this paper, we propose schemes for constructing minimal $\alpha$-perfect hash functions and analyze their space requirements. We begin by presenting a simple base-line scheme which randomizes between perfect hashing and zero-bit random hashing. We then present a more sophisticated hashing scheme based on sampling which significantly improves upon the space requirement of the aforementioned strategy for all values of $\alpha$.