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      "statement": "# Alon, Matias and Szegedy: frequency moments with streaming memory\n\nAlon, Matias and Szegedy developed a randomized small-memory estimator for the second frequency moment and a framework relating one-pass frequency-moment approximation to space complexity.\n\nSuppose a database receives items one by one and cannot keep the full history. Counting every item type needs a large table, while the second frequency moment measures how concentrated or repetitive the stream is. The method assigns a reproducible random sign to each type, adds those signs as items arrive, and squares the resulting counter. Its expectation is the desired second moment. Four-wise independence controls the variance; several independent counters and a median of averages reduce relative error and failure probability. Keeping the seed and counter can require only logarithmic memory for fixed accuracy. The work also gives a sampled-position estimator for general integer moments and studies which other estimates necessarily need more space. These are probabilistic approximation guarantees for a stated stream model, rather than exact answers or protection against every adaptive adversary. The contribution appeared in 1996 and a journal version in 1999; the complete scientific text inspected here is the authors’ separately dated 2002 public manuscript.\n\n## Scientific scope\n\nThe F2 streaming estimator uses four-wise independent signs and independent averaging/amplification. Pairwise independence suffices for its mean, but not the stated variance bound. Memory depends on accuracy, failure probability and the size of the stream/universe; this is neither exact nor deterministic sublinear counting. The source read is a 2002 author manuscript representing the 1996/1999 publication lineage, whose complete revision equivalence is unresolved. Adaptive adversarial-query guarantees and later sketch results are separate.\n\n## People and systems\n\nNoga Alon; Yossi Matias; Mario Szegedy\n\n\n\n## Original sources\n\n- [Alon, Matias and Szegedy: frequency moments with streaming memory — author manuscript dated 2002](https://web.math.princeton.edu/~nalon/PDFS/amsz4.pdf)\n- [Alon, Matias and Szegedy: frequency moments with streaming memory — 1996 conference publisher metadata](https://api.crossref.org/works/10.1145%2F237814.237823)\n- [Alon, Matias and Szegedy: frequency moments with streaming memory — 1999 journal publisher metadata](https://api.crossref.org/works/10.1006%2Fjcss.1997.1545)\n",
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