Non-empty Bins with Simple Tabulation Hashing

Abstract

We consider the hashing of a set X ⊆ U with |X| = m using a simple tabulation hash function h : U → [n] = {0, . . . , n − 1} and analyse the number of non-empty bins, that is, the size of h(X). We show that the expected size of h(X) matches that with fully random hashing to within low-order terms. We also provide concentration bounds. The number of non-empty bins is a fundamental measure in the balls and bins paradigm, and it is critical in applications such as Bloom filters and Filter hashing. For example, normally Bloom filters are proportioned for a desired low false-positive probability assuming fully random hashing (see en.wikipedia.org/wiki/Bloom_filter). Our results imply that if we implement the hashing with simple tabulation, we obtain the same low false-positive probability for any possible input.