{"paper":{"title":"A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Nader H. Bshouty","submitted_at":"2026-05-18T12:51:10Z","abstract_excerpt":"Let $S\\subseteq F_2^u$ have size $n=2^\\ell$, and let $h:F_2^u\\to F_2^\\ell$ be a uniformly random linear map. For $y\\in F_2^\\ell$, write $Load_h(y):=|h^{-1}(y)\\cap S|$, and let $M(S,h):=\\max_{y\\in F_2^\\ell} Load_h(y)$ be the maximum load. Jaber, Kumar and Zuckerman (STOC 2025) proved that the expected maximum load of $h$ on $S$ is at most $16\\log n/\\log\\log n$, matching the fully independent keys-into-bins scale up to constants. Their proof also gives the tail estimate \\[\n  \\Pr\\left[\n  M(S,h)\\ge R\\frac{\\log n}{\\log\\log n}\n  \\right]\n  \\le O\\left(\\frac{1}{R^{2}}\\right). \\] We record a base optimi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every R>1 satisfying R ℓ^{1-1/R} ≥ D ln ℓ, Pr[M(S,h) ≥ R log n / log log n] ≤ O( (log log n)^2 / (R^2 (log n)^{2-2/R}) ), which integrates to E[M(S,h)] ≤ (1 + (1+o(1)) log log log n / log log n) * log n / log log n.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The base optimization of the exponential-potential function in the proof from Jaber et al. (STOC 2025) can be carried through without introducing new error terms that invalidate the improved exponent 2-2/R in the tail bound (see the derivation of the tail estimate in the note).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Binary linear hashing matches fully independent hashing in the leading term and dominant second-order correction of expected maximum load up to a 1+o(1) factor.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ed779228ebec8b7b2cfe01f3709aeed2449161c70d29dab34ce5df0b0a92b31d"},"source":{"id":"2605.18335","kind":"arxiv","version":1},"verdict":{"id":"f129dc42-8e3e-44af-bcde-f84d4a9666b3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:47:58.962125Z","strongest_claim":"For every R>1 satisfying R ℓ^{1-1/R} ≥ D ln ℓ, Pr[M(S,h) ≥ R log n / log log n] ≤ O( (log log n)^2 / (R^2 (log n)^{2-2/R}) ), which integrates to E[M(S,h)] ≤ (1 + (1+o(1)) log log log n / log log n) * log n / log log n.","one_line_summary":"Binary linear hashing matches fully independent hashing in the leading term and dominant second-order correction of expected maximum load up to a 1+o(1) factor.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The base optimization of the exponential-potential function in the proof from Jaber et al. (STOC 2025) can be carried through without introducing new error terms that invalidate the improved exponent 2-2/R in the tail bound (see the derivation of the tail estimate in the note).","pith_extraction_headline":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18335/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T00:02:35.073119Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:20.436118Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.170964Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.838630Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"83b630963c9508869683cb583c72e3584ddd87cc63d61a0544dde548a4815be0"},"references":{"count":12,"sample":[{"doi":"","year":1997,"title":"Is linear hashing good? InProceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 465–474, 1997","work_id":"24ef6bfd-aa5d-4754-b830-195d111fb2fd","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"Linear hash functions.Journal of the ACM, 46(5):667–683, 1999","work_id":"d776c225-f8ef-45ea-a47f-84f863478abe","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Kumar, and David Zuckerman","work_id":"20173400-cf66-48f1-97ff-907b5b3af378","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Also available as arXiv:2505.14061","work_id":"56c62d82-14d1-4b73-b9fd-bc344172a536","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Balls into bins","work_id":"257c7425-0b23-47fa-8569-94696123dc91","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"66585bb495ba56411ae4703f55be75bd7040ed45bebbc1ed4b2cab7dd60a3867","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"582406f642fc484a6539a8677565c35ab17429cd0cd50b2a0c4b762999dc3f0d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}