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        "description": "The estimator assumes a sufficiently uniform, shared hash function and fixed register configuration. Its approximately 1.04/sqrt(m) relative standard error is an asymptotic distributional statement, not a deterministic error bound for every stream. The analysis orders the large-cardinality and large-register limits. Small-range and hash-collision corrections have their own conditions. Registerwise maximum merges compatible streams but does not support deletion. The original extended abstract sketches some analytic steps. Later HyperLogLog++ improvements are separate. WRS has not reconstructed the full probability proof or replayed the published simulations.",
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      "statement": "# HyperLogLog: counting distinct items with a small sketch\n\nHyperLogLog estimates the number of distinct stream items by combining per-bucket bit-pattern registers through a normalized harmonic mean, reducing variance while keeping a small mergeable state.\n\nA stream may contain billions of entries but far fewer distinct items. Instead of storing every item, HyperLogLog hashes them into buckets and remembers an extreme bit pattern in each bucket. A harmonic combination turns those small registers into an approximate distinct count. Compatible sketches can be merged by taking registerwise maxima, making the approach useful for distributed data. The compactness comes from accepting probabilistic error and a hash-model assumption, rather than recovering the complete set.\n\n## Scientific scope\n\nThe estimator assumes a sufficiently uniform, shared hash function and fixed register configuration. Its approximately 1.04/sqrt(m) relative standard error is an asymptotic distributional statement, not a deterministic error bound for every stream. The analysis orders the large-cardinality and large-register limits. Small-range and hash-collision corrections have their own conditions. Registerwise maximum merges compatible streams but does not support deletion. The original extended abstract sketches some analytic steps. Later HyperLogLog++ improvements are separate. WRS has not reconstructed the full probability proof or replayed the published simulations.\n\n## People and systems\n\nPhilippe Flajolet; Éric Fusy; Olivier Gandouet; Frédéric Meunier\n\n\n\n## Original sources\n\n- [Flajolet, Fusy, Gandouet and Meunier, HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm (AofA07,DMTCS AH,127-146), original author copy](https://algo.inria.fr/flajolet/Publications/FlFuGaMe07.pdf)\n",
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