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arXiv · 1810.13187

Non-Empty Bins with Simple Tabulation Hashing

Anders Aamand·Mikkel Thorup
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Abstract

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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Anders Aamand, Mikkel Thorup. 2018-10-31. Non-Empty Bins with Simple Tabulation Hashing. https://arxiv.org/abs/1810.13187

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