{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:U6DICJ6Y6DCT4PU5IRCMRI6OGO","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":"1ace76bec5a83e09056cad580ee0c540c125d1e86b214525e3eb3297952f9ba1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2026-07-21T05:11:13Z","title_canon_sha256":"5d09fb0040309f6c0d8ff6d31cdcd5cf3544f79560065e05ea5fbce8bdd9f8b2"},"schema_version":"1.0","source":{"id":"2607.18714","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.18714","created_at":"2026-07-22T01:23:07Z"},{"alias_kind":"arxiv_version","alias_value":"2607.18714v1","created_at":"2026-07-22T01:23:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.18714","created_at":"2026-07-22T01:23:07Z"},{"alias_kind":"pith_short_12","alias_value":"U6DICJ6Y6DCT","created_at":"2026-07-22T01:23:07Z"},{"alias_kind":"pith_short_16","alias_value":"U6DICJ6Y6DCT4PU5","created_at":"2026-07-22T01:23:07Z"},{"alias_kind":"pith_short_8","alias_value":"U6DICJ6Y","created_at":"2026-07-22T01:23:07Z"}],"graph_snapshots":[{"event_id":"sha256:b2b10ddfd349551d73eb53a014eec6e1c57208bae49edf5f5f3e38a2153bb773","target":"graph","created_at":"2026-07-22T01:23:07Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.18714/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given an undirected, unweighted graph $G$, we aim to compute a 2-approximation of all-pairs shortest paths (APSP). This problem admits a natural lower bound of $\\Omega(n^2)$ since the output size is $\\Theta(n^2)$. A central goal in this area is to achieve a running time of $O(n^2)$. Dor, Halperin, and Zwick (FOCS 1996, SICOMP 2001) designed an algorithm with a running time of $\\tilde{O}(n^2)$ that guarantees a 2-approximation only for pairs at a distance of at least $O(\\log n)$. Recently, Gupta (FOCS 2025) improved this bound, handling all pairs at a distance of at least $O(\\log \\log n)$. We n","authors_text":"Manoj Gupta, Mrigankashekhar Shandilya","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2026-07-21T05:11:13Z","title":"$\\tilde{O}$ptimal Algorithm for 2-Approximate All Pair Shortest Paths -- almost"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18714","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":"sha256:134e4c12cbaed737244aeb8fd09fc862e86d38df80df027bdb053332c5bcef41","target":"record","created_at":"2026-07-22T01:23:07Z","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":"1ace76bec5a83e09056cad580ee0c540c125d1e86b214525e3eb3297952f9ba1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2026-07-21T05:11:13Z","title_canon_sha256":"5d09fb0040309f6c0d8ff6d31cdcd5cf3544f79560065e05ea5fbce8bdd9f8b2"},"schema_version":"1.0","source":{"id":"2607.18714","kind":"arxiv","version":1}},"canonical_sha256":"a7868127d8f0c53e3e9d4444c8a3ce33af36594f069eb7bceb6db347527c2aba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7868127d8f0c53e3e9d4444c8a3ce33af36594f069eb7bceb6db347527c2aba","first_computed_at":"2026-07-22T01:23:07.953909Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-22T01:23:07.953909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gnR6kZhLFQOn9S3qWO7NdD08Oc+GX6LbG1APOkxYFm/BrjhtI/LZzb8CWeRtVDpsMO6j1AfsE4/QLcIFxmowCg==","signature_status":"signed_v1","signed_at":"2026-07-22T01:23:07.954723Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.18714","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:134e4c12cbaed737244aeb8fd09fc862e86d38df80df027bdb053332c5bcef41","sha256:b2b10ddfd349551d73eb53a014eec6e1c57208bae49edf5f5f3e38a2153bb773"],"state_sha256":"c67f6180fd50d2613f5a1e265fcff26f950aa1cccb226a3cc8c1ea48e6f9ef92"}