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Re:Show HN: An improved version of HyperLogLog by zintinio5 on Wednesday June 28, 2017 @03:31PM (#14657455)
Neustar Research has a good article on this which may be of interest to you. However, I think the approach they took was still based on the inclusion-exclusion principle (still a good read though).

* https://research.neustar.biz/2012/12/17/hll-intersections-2/

Another approach if you have the flexibility of changing your implementation, is to use a different type of data-sketch that is more amenable to set expressions. There is some discussion (and references) here:

* https://datasketches.github.io/docs/Theta/ThetaSketchFramewo...

* https://datasketches.github.io/docs/Tuple/TupleOverview.html

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