Hash, Displace, and Compress

A hash function h, i.e., a function from the set U of all keys to the range range [m] = {0, . . . ,m− 1} is called a perfect hash function (PHF) for a subset S ⊆ U of size n ≤ m if h is 1–1 on S. The important performance parameters of a PHF are representation size, evaluation time and construction time. In this paper, we present an algorithm that permits to obtain PHFs with representation size very close to optimal while retaining O(n) construction time and O(1) evaluation time. For example in the case m = 2n we obtain a PHF that uses space 0.67 bits per key, and for m = 1.23n we obtain space 1.4 bits per key, which was not achievable with previously known methods. Our algorithm is inspired by several known algorithms; the main new feature is that we combine a modification of Pagh’s “hash-and-displace” approach with data compression on a sequence of hash function indices. That combination makes it possible to significantly reduce space usage while retaining linear construction time and constant query time. Our algorithm can also be used for k-perfect hashing, where at most k keys may be mapped to the same value. For the analysis we assume that fully random hash functions are given for free; such assumptions can be justified and were made in previous papers.