{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3UTMTMJL26PEOOHJQDGACUL7W3","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":"2a5afb401be1c0fb819b7b6c6834c1db753668dd0d9edbab87959f52b3df448e","cross_cats_sorted":["cs.DS","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-19T00:42:48Z","title_canon_sha256":"46b3893bfc640f7d5a28a2dc822a434746f72ac024df10c180f118d162e4e587"},"schema_version":"1.0","source":{"id":"1105.3770","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.3770","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"1105.3770v1","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.3770","created_at":"2026-05-18T04:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"3UTMTMJL26PE","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"3UTMTMJL26PEOOHJ","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"3UTMTMJL","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:7488d0df08358a7c802ca55f735fdc3fca2e5399d8ffba770c8e63d9d09ca8f3","target":"graph","created_at":"2026-05-18T04:21:51Z","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":"We present an all-pairs shortest path algorithm whose running time on a complete directed graph on $n$ vertices whose edge weights are chosen independently and uniformly at random from $[0,1]$ is $O(n^2)$, in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of \\emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability. We also present a dynamic version o","authors_text":"Benny Sudakov, Dimitry Sotnikov, Uri Zwick, Yuval Peres","cross_cats":["cs.DS","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-19T00:42:48Z","title":"All-Pairs Shortest Paths in $O(n^2)$ time with high probability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3770","kind":"arxiv","version":1},"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:b2260c04007136ff39873300736fb6a7e9b91dd966989b234148f402e53fc461","target":"record","created_at":"2026-05-18T04:21:51Z","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":"2a5afb401be1c0fb819b7b6c6834c1db753668dd0d9edbab87959f52b3df448e","cross_cats_sorted":["cs.DS","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-19T00:42:48Z","title_canon_sha256":"46b3893bfc640f7d5a28a2dc822a434746f72ac024df10c180f118d162e4e587"},"schema_version":"1.0","source":{"id":"1105.3770","kind":"arxiv","version":1}},"canonical_sha256":"dd26c9b12bd79e4738e980cc01517fb6eb4846cf375df2cdbd23fc9369675eae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd26c9b12bd79e4738e980cc01517fb6eb4846cf375df2cdbd23fc9369675eae","first_computed_at":"2026-05-18T04:21:51.354034Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:21:51.354034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P7hrmc+KzfHgU3cx4fmKlHH2D3hCLVXnVnk1mr+9s/+xFUqo9rflHhF2QCdrcI0GJzPXm5Taik7Qpj3BQ+6RDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:21:51.354644Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.3770","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2260c04007136ff39873300736fb6a7e9b91dd966989b234148f402e53fc461","sha256:7488d0df08358a7c802ca55f735fdc3fca2e5399d8ffba770c8e63d9d09ca8f3"],"state_sha256":"a594494dcb620d5d0bad69dd44c4717bbeb28699907fea3d83f0d9d06b5f5697"}