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HyperLogLog cardinality estimate: harmonic sum skips empty registers · case 02

Raw estimates explode for sparse sketches because empty registers no longer contribute 1 each.

Member previewVariant 2 · 3 implementations · 10 checks per implementation

Case contract

Input {regs} with 16, 32, 64 or 128 registers. alpha is 0.673, 0.697, 0.709 for m = 16, 32, 64 and 0.7213/(1 + 1.079/m) otherwise. Raw E = alpha m^2 / sum 2^-r. If E <= 2.5m and V (zero registers) > 0, use linear counting m ln(m/V); otherwise if E > 2^32/30 apply the large-range correction -2^32 ln(1 - E/2^32). Return [round(E), V].

Why this case matters

Distinct-count dashboards and query planners rely on the bias corrections of the HyperLogLog estimator at both small and large cardinalities.

One recorded failure

Sample boundary fixture

This sample comes from the broken implementation of a controlled reproducer.

Boundary fixture	Actual	Expected	Outcome
sparse m=16	`[81, 10]`	`[8, 10]`	Failed

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