HyperMinHash: MinHash in LogLog space

Abstract

In this extended abstract, we describe and analyze a lossy compression of MinHash from buckets of size $O(\log n)$ to buckets of size $O(\log \log n)$ by encoding using floating-point notation. This new compressed sketch, which we call HyperMinHash, as we build off a HyperLogLog scaffold, can be used as a drop-in replacement of MinHash. Unlike comparable Jaccard index fingerprinting algorithms in sub-logarithmic space (such as b-bit MinHash), HyperMinHash retains MinHash's features of streaming updates, unions, and cardinality estimation. For a multiplicative approximation error $1+\epsilon$ on a Jaccard index $t$, given a random oracle, HyperMinHash needs $O\left({\epsilon }^{-2}\left(\log \log n+\log \frac{1}{t\epsilon }\right)\right)$ space. HyperMinHash allows estimating Jaccard indices of 0.01 for set cardinalities on the order of $10^{19}$ with relative error of around 10\% using 64KiB of memory; MinHash can only estimate Jaccard indices for cardinalities of $10^{10}$ with the same memory consumption.

12 pages, 6 figures