# New cardinality estimation algorithms for HyperLogLog sketches

arXiv: https://arxiv.org/abs/1702.01284 · HTML: https://mlsanity.com/paper/1702.01284/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2017.

## Built on by (most cited first, 3 of 3)

- [Mergeable Model-Side Aggregation States for Long-Context Language Models](https://mlsanity.com/paper/2607.26448/) (2026)
- [Privacy-Preserving Record Linkage for Cardinality Counting](https://mlsanity.com/paper/2301.04000/) (2023)
- [Fast Gumbel-Max Sketch and its Applications](https://mlsanity.com/paper/2302.05176/) (2023)

## Abstract (arXiv)

This paper presents new methods to estimate the cardinalities of data sets recorded by HyperLogLog sketches. A theoretically motivated extension to the original estimator is presented that eliminates the bias for small and large cardinalities. Based on the maximum likelihood principle a second unbiased method is derived together with a robust and efficient numerical algorithm to calculate the estimate. The maximum likelihood approach can also be applied to more than a single HyperLogLog sketch. In particular, it is shown that it gives more precise cardinality estimates for union, intersection, or relative complements of two sets that are both represented by HyperLogLog sketches compared to the conventional technique using the inclusion-exclusion principle. All the new methods are demonstrated and verified by extensive simulations.