HyperMinHash

Start from a plain HyperLogLog register: the first p bits of an item's hash select one of 2^p registers, and the register keeps ρ (rho), the maximum leading-zero-run length seen for that bucket. That alone gives cardinality, exactly like the HyperLogLog page.

HyperMinHash adds one more small field per register: aux, a b-bit hash (b is typically 4–8 bits) of whichever item most recently pushed that register to its current max ρ. When a new item reaches a strictly higher ρ, it becomes the register's new champion and aux is overwritten. When an item ties the current max ρ, the register keeps the smaller of the two aux values: the ordinary MinHash update rule, just scoped to "the items that are this register's current champion" instead of "every item in the set".

Two sketches that both saw the same item route it to the same register with the same ρ and the same aux deterministically. So comparing (ρ, aux) pairs register-by-register between sketch A and sketch B gives a MinHash-like collision signal: registers where both sketches agree are (probably) driven by a shared item, registers that disagree are (probably) not. Averaging that signal across every "touched" register (corrected for the small chance two unrelated items coincidentally produce the same (ρ, aux) pair) estimates the Jaccard similarity. From there, |A∩B| ≈ J × |A∪B|, and |A∪B| itself comes from the ordinary register-wise HLL merge (keep the max ρ per register). All of this stays in O(2^p) space, with no separate MinHash structure needed alongside the cardinality sketch.