{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6MZT6RRCROPYLCGLP7ZOUC6Y3L","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":"4f9478cf9281dcc2d0c0396697fcc4817ad2bcc7d2e7a5ff98474ca1b5681894","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-12-19T18:02:46Z","title_canon_sha256":"98e08fdfa1c840e12a4707f2bf716453118ac39246b1547a3484571b7b11b68d"},"schema_version":"1.0","source":{"id":"1412.6466","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6466","created_at":"2026-05-18T00:22:16Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6466v2","created_at":"2026-05-18T00:22:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6466","created_at":"2026-05-18T00:22:16Z"},{"alias_kind":"pith_short_12","alias_value":"6MZT6RRCROPY","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6MZT6RRCROPYLCGL","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6MZT6RRC","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:3b9d9ad29013ec49cb47d6a1f719b88c106c7450b5528b073033bd4686a90a08","target":"graph","created_at":"2026-05-18T00:22:16Z","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 faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an","authors_text":"Monika Henzinger, Sebastian Krinninger, Veronika Loitzenbauer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-12-19T18:02:46Z","title":"Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6466","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:d29384f0b154de558b42dad0feec193ea459975d0091bca7b795575b263cfb9b","target":"record","created_at":"2026-05-18T00:22:16Z","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":"4f9478cf9281dcc2d0c0396697fcc4817ad2bcc7d2e7a5ff98474ca1b5681894","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-12-19T18:02:46Z","title_canon_sha256":"98e08fdfa1c840e12a4707f2bf716453118ac39246b1547a3484571b7b11b68d"},"schema_version":"1.0","source":{"id":"1412.6466","kind":"arxiv","version":2}},"canonical_sha256":"f3333f46228b9f8588cb7ff2ea0bd8daea29bb7ae74302660a5d0c586bd218db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3333f46228b9f8588cb7ff2ea0bd8daea29bb7ae74302660a5d0c586bd218db","first_computed_at":"2026-05-18T00:22:16.637545Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:16.637545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5/cA0kNSkABCgMSHcL96VYhX5eI/k0/cesnThp7UVzi9NvrPw3EN8u4k3tbr8ZJEotKnMKFfjpZYCIhpkfi4Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:16.638134Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.6466","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d29384f0b154de558b42dad0feec193ea459975d0091bca7b795575b263cfb9b","sha256:3b9d9ad29013ec49cb47d6a1f719b88c106c7450b5528b073033bd4686a90a08"],"state_sha256":"ae880670295040d196fecdbe16f39432a387f1138023d935e4549900a823ab89"}