Perfect Hash Families: Probabilistic Methods and Explicit Constructions

An (n, q, t)-perfect hash family of size s consists of a set V of order n, a set F of order q, and a sequence φ1, φ2, …, φs of functions from V to F with the following property. For all t-subsets X⊆V, there exists i∈{1, 2, …, s} such that φi is injective when restricted to X. An (n, q, t)-perfect hash family of minimal size is known as optimal. The paper presents a probabilistic existence result for perfect hash families which improves on the well known result of Mehlhorn for many parameter sets. The probabilistic methods are strong enough to establish the size of an optimal perfect hash family in many cases. The paper also gives several explicit constructions of classes of perfect hash families.