Space Upper Bounds for $α$-Perfect Hashing

2026-03-16

AI Summary

This work investigates the space complexity of constructing minimal α-perfect hash functions under an approximate setting that permits a bounded fraction of collisions. Two randomized constructions are proposed: a baseline hybrid strategy combining perfect hashing with zero-bit random hashing, and a novel sampling-based approach. Leveraging probabilistic analysis, information-theoretic arguments, and randomized algorithms, the study provides the first systematic characterization of the space requirements for α-perfect hashing. The analysis demonstrates that the proposed sampling scheme consistently outperforms the baseline method across the entire range of α ∈ [0,1], achieving substantial reductions in storage overhead.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In the problem of 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 $h: [n] \to [b]$ such that $h(\cdot)$ maps $\mathcal{A}$ to $[b]$ with no collisions, i.e., the restriction of $h(\cdot)$ to $\mathcal{A}$ is injective. When $b=k$, the problem is referred to as minimal perfect hashing. In this paper, we extend the study of minimal perfect hashing to the approximate setting. For some $α\in [0, 1]$, we say that a randomized hashing scheme is $α$-perfect if for any input $\mathcal{A}$ of size $k$, it outputs a hash function which exhibits at most $(1-α)k$ 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, i.e., $b = k$, 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 $α$-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 $α$.

Problem

• perfect hashing
• α-perfect hashing
• space complexity
• minimal hashing
• collision

Innovation

• alpha-perfect hashing
• space complexity
• minimal perfect hashing
• randomized hashing
• sampling-based construction