{"paper":{"title":"Optimal Separation and Strong Direct Sum for Randomized Query Complexity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Eric Blais, Joshua Brody","submitted_at":"2019-08-02T19:16:22Z","abstract_excerpt":"We establish two results regarding the query complexity of bounded-error randomized algorithms.\n  * Bounded-error separation theorem. There exists a total function $f : \\{0,1\\}^n \\to \\{0,1\\}$ whose $\\epsilon$-error randomized query complexity satisfies $\\overline{\\mathrm{R}}_\\epsilon(f) = \\Omega( \\mathrm{R}(f) \\cdot \\log\\frac1\\epsilon)$.\n  * Strong direct sum theorem. For every function $f$ and every $k \\ge 2$, the randomized query complexity of computing $k$ instances of $f$ simultaneously satisfies $\\overline{\\mathrm{R}}_\\epsilon(f^k) = \\Theta(k \\cdot \\overline{\\mathrm{R}}_{\\frac\\epsilon k}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01020","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/1908.01020/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"}