{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NN3OTDWR6P6TAB7HPP3BWEU4GI","short_pith_number":"pith:NN3OTDWR","canonical_record":{"source":{"id":"1611.00721","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-11-02T18:40:54Z","cross_cats_sorted":[],"title_canon_sha256":"707ed8d9558bf8deecf7d3e2873e0f62f34f9f936e55b8eca036620d97baaabb","abstract_canon_sha256":"7b86388a65f2bafa06e5823f5620ae2989f292d643608c7acea4987b91ec2112"},"schema_version":"1.0"},"canonical_sha256":"6b76e98ed1f3fd3007e77bf61b129c322ef4f59c8e3ff2435ec69e1c9f975a5d","source":{"kind":"arxiv","id":"1611.00721","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.00721","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"arxiv_version","alias_value":"1611.00721v2","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.00721","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"pith_short_12","alias_value":"NN3OTDWR6P6T","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NN3OTDWR6P6TAB7H","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NN3OTDWR","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NN3OTDWR6P6TAB7HPP3BWEU4GI","target":"record","payload":{"canonical_record":{"source":{"id":"1611.00721","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-11-02T18:40:54Z","cross_cats_sorted":[],"title_canon_sha256":"707ed8d9558bf8deecf7d3e2873e0f62f34f9f936e55b8eca036620d97baaabb","abstract_canon_sha256":"7b86388a65f2bafa06e5823f5620ae2989f292d643608c7acea4987b91ec2112"},"schema_version":"1.0"},"canonical_sha256":"6b76e98ed1f3fd3007e77bf61b129c322ef4f59c8e3ff2435ec69e1c9f975a5d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:23.925609Z","signature_b64":"zLxkIYpOpgyJOYT+WJCCzH4uTivVhZQ/qZkf6BvU6PbgRUK7ItzHHQRuljCbQhzza/VfJ3+1RIF/BdZMe3+xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b76e98ed1f3fd3007e77bf61b129c322ef4f59c8e3ff2435ec69e1c9f975a5d","last_reissued_at":"2026-05-18T00:08:23.925018Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:23.925018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.00721","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:08:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v54ZqNZrxKK16XNg4+mo9rYcxfnIVEOal/7LOuOHrHg2GZ2uvai0r1pKuxWLe/jCtzfU0TlRiKhR7d4iRLQgDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T11:19:11.352557Z"},"content_sha256":"9b4d477aa1fb2dbb208801fc1da5e05ed7ba6fdc1783285ca31ddf585c14bfed","schema_version":"1.0","event_id":"sha256:9b4d477aa1fb2dbb208801fc1da5e05ed7ba6fdc1783285ca31ddf585c14bfed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NN3OTDWR6P6TAB7HPP3BWEU4GI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Jakub Pachocki, Liam Roditty, Roei Tov, Virginia Vassilevska Williams","submitted_at":"2016-11-02T18:40:54Z","abstract_excerpt":"The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $\\Omega(\\min\\{n^{\\omega}, mn\\})$ time (for $2\\leq\\omega<2.373$). In this paper, we drastically improve these runtimes as follows:\n  * Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $\\widetilde{O}(m)$ time computes an $\\widetilde{O}(1)$-multiplicative approximation of the girth as well as an $\\widetilde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00721","kind":"arxiv","version":2},"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":"2026-05-18T00:08:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yxGCzj+8tJXYbMzDh+hH/PoTnvdGbEW9h0+hUVWDxFKfCyuyfBBfoSqxY0wzsU3qhoCphgGVZ2zKSQR3i6GBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T11:19:11.352914Z"},"content_sha256":"b9c682f47ff764fc2b35c9c50391ce864a92936ac9dea058220096aad62b80c0","schema_version":"1.0","event_id":"sha256:b9c682f47ff764fc2b35c9c50391ce864a92936ac9dea058220096aad62b80c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/bundle.json","state_url":"https://pith.science/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/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":"2026-06-11T11:19:11Z","links":{"resolver":"https://pith.science/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI","bundle":"https://pith.science/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/bundle.json","state":"https://pith.science/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NN3OTDWR6P6TAB7HPP3BWEU4GI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NN3OTDWR6P6TAB7HPP3BWEU4GI","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":"7b86388a65f2bafa06e5823f5620ae2989f292d643608c7acea4987b91ec2112","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-11-02T18:40:54Z","title_canon_sha256":"707ed8d9558bf8deecf7d3e2873e0f62f34f9f936e55b8eca036620d97baaabb"},"schema_version":"1.0","source":{"id":"1611.00721","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.00721","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"arxiv_version","alias_value":"1611.00721v2","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.00721","created_at":"2026-05-18T00:08:23Z"},{"alias_kind":"pith_short_12","alias_value":"NN3OTDWR6P6T","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NN3OTDWR6P6TAB7H","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NN3OTDWR","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:b9c682f47ff764fc2b35c9c50391ce864a92936ac9dea058220096aad62b80c0","target":"graph","created_at":"2026-05-18T00:08:23Z","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 girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $\\Omega(\\min\\{n^{\\omega}, mn\\})$ time (for $2\\leq\\omega<2.373$). In this paper, we drastically improve these runtimes as follows:\n  * Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $\\widetilde{O}(m)$ time computes an $\\widetilde{O}(1)$-multiplicative approximation of the girth as well as an $\\widetilde","authors_text":"Aaron Sidford, Jakub Pachocki, Liam Roditty, Roei Tov, Virginia Vassilevska Williams","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-11-02T18:40:54Z","title":"Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00721","kind":"arxiv","version":2},"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":"sha256:9b4d477aa1fb2dbb208801fc1da5e05ed7ba6fdc1783285ca31ddf585c14bfed","target":"record","created_at":"2026-05-18T00:08:23Z","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":"7b86388a65f2bafa06e5823f5620ae2989f292d643608c7acea4987b91ec2112","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-11-02T18:40:54Z","title_canon_sha256":"707ed8d9558bf8deecf7d3e2873e0f62f34f9f936e55b8eca036620d97baaabb"},"schema_version":"1.0","source":{"id":"1611.00721","kind":"arxiv","version":2}},"canonical_sha256":"6b76e98ed1f3fd3007e77bf61b129c322ef4f59c8e3ff2435ec69e1c9f975a5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6b76e98ed1f3fd3007e77bf61b129c322ef4f59c8e3ff2435ec69e1c9f975a5d","first_computed_at":"2026-05-18T00:08:23.925018Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:23.925018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zLxkIYpOpgyJOYT+WJCCzH4uTivVhZQ/qZkf6BvU6PbgRUK7ItzHHQRuljCbQhzza/VfJ3+1RIF/BdZMe3+xCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:23.925609Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.00721","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b4d477aa1fb2dbb208801fc1da5e05ed7ba6fdc1783285ca31ddf585c14bfed","sha256:b9c682f47ff764fc2b35c9c50391ce864a92936ac9dea058220096aad62b80c0"],"state_sha256":"585887be7899706e5e18c3c8ff746d148eaf7207ae25c9b312b8c892fc18ec5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jm/jFl7Z5aPRWqJZ4Occn6py+m3I+hdLpNez+sLjht82TKyWzyn6v0LUjQgpS8/07cHAFPlWHZ6AUUvs65PkDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T11:19:11.354916Z","bundle_sha256":"9912acd9c3b12de61ae3cf3bc1153cbb5ce98e90520600ec98520690f18b58c8"}}