{"paper":{"title":"Optimality of Frequency Moment Estimation","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.CC","cs.IT","math.IT"],"primary_cat":"cs.DS","authors_text":"Mark Braverman, Or Zamir","submitted_at":"2024-11-04T15:06:16Z","abstract_excerpt":"Estimating the second frequency moment of a stream up to $(1\\pm\\varepsilon)$ multiplicative error requires at most $O(\\log n / \\varepsilon^2)$ bits of space, due to a seminal result of Alon, Matias, and Szegedy. It is also known that at least $\\Omega(\\log n + 1/\\varepsilon^{2})$ space is needed. We prove an optimal lower bound of $\\Omega\\left(\\log \\left(n \\varepsilon^2 \\right) / \\varepsilon^2\\right)$ for all $\\varepsilon = \\Omega(1/\\sqrt{n})$. Note that when $\\varepsilon>n^{-1/2 + c}$, where $c>0$, our lower bound matches the classic upper bound of AMS. For smaller values of $\\varepsilon$ we a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.02148","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.02148/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}