{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:PNHVT6YJ2GAJM2FVEH4RBHLNB7","short_pith_number":"pith:PNHVT6YJ","canonical_record":{"source":{"id":"1007.1260","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-07-07T23:22:53Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"ce0054e52ddd7c3f101ab7f2643852770a1d65a417160d27acf6fb33105e0652","abstract_canon_sha256":"39e0d5925c5dd436856eb403a4943b1f46e057a42cb2ff9deef9ecfe4d1f62fd"},"schema_version":"1.0"},"canonical_sha256":"7b4f59fb09d1809668b521f9109d6d0fc4abcd18622dcd755d99d348c644d2cd","source":{"kind":"arxiv","id":"1007.1260","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1260","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1260v3","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1260","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"pith_short_12","alias_value":"PNHVT6YJ2GAJ","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PNHVT6YJ2GAJM2FV","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PNHVT6YJ","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:PNHVT6YJ2GAJM2FVEH4RBHLNB7","target":"record","payload":{"canonical_record":{"source":{"id":"1007.1260","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-07-07T23:22:53Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"ce0054e52ddd7c3f101ab7f2643852770a1d65a417160d27acf6fb33105e0652","abstract_canon_sha256":"39e0d5925c5dd436856eb403a4943b1f46e057a42cb2ff9deef9ecfe4d1f62fd"},"schema_version":"1.0"},"canonical_sha256":"7b4f59fb09d1809668b521f9109d6d0fc4abcd18622dcd755d99d348c644d2cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:58.152107Z","signature_b64":"8yAqkjhiPUgnTzsgmNzGoTSUkb9/vQ/fQ62a64qMRJDlK9GoIpbttYTsdQu4McHvVlGsN+ci/Bw7zu2n3YmqCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b4f59fb09d1809668b521f9109d6d0fc4abcd18622dcd755d99d348c644d2cd","last_reissued_at":"2026-05-18T04:27:58.151473Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:58.151473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.1260","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ki+LabmVVp7ik+8DhdSfekTCwTom8JubJqEpaVHkiXJ9ugnYyHUo9tKc/6NEzvE4yIjCUFC+GzvjtsDNL+T7Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:28:40.383777Z"},"content_sha256":"34d2b465cbe832e5ab71b976bba81e8864e8872730f6082ec1f9adc3b755cb9d","schema_version":"1.0","event_id":"sha256:34d2b465cbe832e5ab71b976bba81e8864e8872730f6082ec1f9adc3b755cb9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:PNHVT6YJ2GAJM2FVEH4RBHLNB7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"Bin Fu, Richard Beigel","submitted_at":"2010-07-07T23:22:53Z","abstract_excerpt":"The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes $a_1,..., a_n$ in $(0,1]$. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized $O({n(\\log n)(\\log\\log n)\\over \\sum_{i=1}^n a_i}+({1\\over \\epsilon})^{O({1\\over\\epsilon})})$ time $(1+\\epsilon)$-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs $\\Omega({n\\over \\sum_{i=1}^n a_i})$ time to give an $(1+\\epsilon)$-approximation. For each function $s(n): N\\rightar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1260","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xBU13c+ngfNr4WXh+5XFvQdevzMwjVQgRK3Cysb/+T7jtildtUJTSqmXA8FyBHpIfF8ews599cGOQBE8bFVZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:28:40.384409Z"},"content_sha256":"a1051f0271521c46fc54cda643c9a0c7f58a023b4eb8a9e4651d173af732f31b","schema_version":"1.0","event_id":"sha256:a1051f0271521c46fc54cda643c9a0c7f58a023b4eb8a9e4651d173af732f31b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/bundle.json","state_url":"https://pith.science/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T02:28:40Z","links":{"resolver":"https://pith.science/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7","bundle":"https://pith.science/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/bundle.json","state":"https://pith.science/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PNHVT6YJ2GAJM2FVEH4RBHLNB7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:PNHVT6YJ2GAJM2FVEH4RBHLNB7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"39e0d5925c5dd436856eb403a4943b1f46e057a42cb2ff9deef9ecfe4d1f62fd","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-07-07T23:22:53Z","title_canon_sha256":"ce0054e52ddd7c3f101ab7f2643852770a1d65a417160d27acf6fb33105e0652"},"schema_version":"1.0","source":{"id":"1007.1260","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1260","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1260v3","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1260","created_at":"2026-05-18T04:27:58Z"},{"alias_kind":"pith_short_12","alias_value":"PNHVT6YJ2GAJ","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PNHVT6YJ2GAJM2FV","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PNHVT6YJ","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:a1051f0271521c46fc54cda643c9a0c7f58a023b4eb8a9e4651d173af732f31b","target":"graph","created_at":"2026-05-18T04:27:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The bin packing problem is to find the minimum number of bins of size one to pack a list of items with sizes $a_1,..., a_n$ in $(0,1]$. Using uniform sampling, which selects a random element from the input list each time, we develop a randomized $O({n(\\log n)(\\log\\log n)\\over \\sum_{i=1}^n a_i}+({1\\over \\epsilon})^{O({1\\over\\epsilon})})$ time $(1+\\epsilon)$-approximation scheme for the bin packing problem. We show that every randomized algorithm with uniform random sampling needs $\\Omega({n\\over \\sum_{i=1}^n a_i})$ time to give an $(1+\\epsilon)$-approximation. For each function $s(n): N\\rightar","authors_text":"Bin Fu, Richard Beigel","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-07-07T23:22:53Z","title":"A Dense Hierarchy of Sublinear Time Approximation Schemes for Bin Packing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1260","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:34d2b465cbe832e5ab71b976bba81e8864e8872730f6082ec1f9adc3b755cb9d","target":"record","created_at":"2026-05-18T04:27:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"39e0d5925c5dd436856eb403a4943b1f46e057a42cb2ff9deef9ecfe4d1f62fd","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-07-07T23:22:53Z","title_canon_sha256":"ce0054e52ddd7c3f101ab7f2643852770a1d65a417160d27acf6fb33105e0652"},"schema_version":"1.0","source":{"id":"1007.1260","kind":"arxiv","version":3}},"canonical_sha256":"7b4f59fb09d1809668b521f9109d6d0fc4abcd18622dcd755d99d348c644d2cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b4f59fb09d1809668b521f9109d6d0fc4abcd18622dcd755d99d348c644d2cd","first_computed_at":"2026-05-18T04:27:58.151473Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:58.151473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8yAqkjhiPUgnTzsgmNzGoTSUkb9/vQ/fQ62a64qMRJDlK9GoIpbttYTsdQu4McHvVlGsN+ci/Bw7zu2n3YmqCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:58.152107Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.1260","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34d2b465cbe832e5ab71b976bba81e8864e8872730f6082ec1f9adc3b755cb9d","sha256:a1051f0271521c46fc54cda643c9a0c7f58a023b4eb8a9e4651d173af732f31b"],"state_sha256":"2702282f7a074b036451e25940a3c6a8a9b1071cd7ee9c934194af039e1787e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LU/1wGvwglPXNY23x/qdPHA2nAYoD/znngL2/CcOvPRpknv7HtM5HdLQmqkyj0SSv7x0wt4lT+oOMuXPAvRVDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:28:40.387809Z","bundle_sha256":"5584381791808d6bd261f5821135c09e5e11e6a9586366432f1c6bd26f9d4690"}}