Non-empty Bins With Simple Tabulation Hashing

Aamand Anders, Thorup Mikkel. Arxiv 2018

[Paper]    
ARXIV Independent

We consider the hashing of a set \(X\subseteq U\) with \(|X|=m\) using a simple tabulation hash function \(h:U\to [n]=\{0,\dots,n-1\}\) 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 \url{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.

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