{"paper":{"title":"Tight Bounds for Low-Error Frequency Moment Estimation and the Power of Multiple Passes","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Naomi Green-Maimon, Or Zamir","submitted_at":"2025-09-09T11:17:12Z","abstract_excerpt":"Estimating the second frequency moment $F_2$ of a data stream up to a $(1 \\pm \\varepsilon)$ factor is a central problem in the streaming literature. For errors $\\varepsilon > \\Omega(1/\\sqrt{n})$, the tight bound $\\Theta\\left(\\log(\\varepsilon^2 n)/\\varepsilon^2\\right)$ was recently established by Braverman and Zamir. In this work, we complete the picture by resolving the remaining regime of small error, $\\varepsilon < 1/\\sqrt{n}$, showing that the optimal space complexity is $\\Theta\\left( \\min\\left(n, \\frac{1}{\\varepsilon^2} \\right) \\cdot \\left(1 + \\left| \\log(\\varepsilon^2 n) \\right| \\right) \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07599","kind":"arxiv","version":1},"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/2509.07599/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"}