{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KDDESUWSSOE6BB5CW6AV62DBYT","short_pith_number":"pith:KDDESUWS","canonical_record":{"source":{"id":"1510.07462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-10-26T12:59:22Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"ac6a94a31ebd2c839138b05136b17b5854d19590fafe9f80840a9bc412af6f9d","abstract_canon_sha256":"e3447ca4493de0ec43ca39a206d880218a0b7489a972ff06049a79f4e6f0dd8f"},"schema_version":"1.0"},"canonical_sha256":"50c64952d29389e087a2b7815f6861c4c2d2f50e80f62fb385d97f3624c491b9","source":{"kind":"arxiv","id":"1510.07462","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07462","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07462v2","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07462","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"pith_short_12","alias_value":"KDDESUWSSOE6","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDDESUWSSOE6BB5C","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDDESUWS","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KDDESUWSSOE6BB5CW6AV62DBYT","target":"record","payload":{"canonical_record":{"source":{"id":"1510.07462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-10-26T12:59:22Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"ac6a94a31ebd2c839138b05136b17b5854d19590fafe9f80840a9bc412af6f9d","abstract_canon_sha256":"e3447ca4493de0ec43ca39a206d880218a0b7489a972ff06049a79f4e6f0dd8f"},"schema_version":"1.0"},"canonical_sha256":"50c64952d29389e087a2b7815f6861c4c2d2f50e80f62fb385d97f3624c491b9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:21.080659Z","signature_b64":"YGDUo0FLdXvKn8UcK+SXwVw/Fr3Ko02i8y0aS554GHux92CkIxzYF9dmYHkTRJZzw1+nIUgoR+IEIhmQ7qGqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50c64952d29389e087a2b7815f6861c4c2d2f50e80f62fb385d97f3624c491b9","last_reissued_at":"2026-05-18T00:53:21.080207Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:21.080207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.07462","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":"2026-05-18T00:53:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zw0Tlwjq0pkgVSi3l33ZHj7uk4HSBaMDSSWIIfFVuyfU2vke43NM5Lc/30TNBgEQ7+PORzt8C7u4Yy1gNeBiCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:03:15.338777Z"},"content_sha256":"5c19643c0a23bd8ace735d536bd32933c8877fba2922a47e4865d6f6594856e6","schema_version":"1.0","event_id":"sha256:5c19643c0a23bd8ace735d536bd32933c8877fba2922a47e4865d6f6594856e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KDDESUWSSOE6BB5CW6AV62DBYT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recognizing Union-Find trees is NP-complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CC","authors_text":"Kitti Gelle, Szabolcs Ivan","submitted_at":"2015-10-26T12:59:22Z","abstract_excerpt":"Disjoint-Set forests, consisting of Union-Find trees are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression). In this paper we provide such a characterization and show that the decision problem whether a given tree is a Union-Find tree is $\\NP$-complete."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07462","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":"2026-05-18T00:53:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lYV3ijZwOKr0PzlbbxmtZkvqe5no1sflKEAReu0oL5W5KFoTSDOirV8Dg3wAuGT0DGFUjJCXWbrpqlmOn9ldCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:03:15.339231Z"},"content_sha256":"6b6add92f2d401f80445284089c1a3d4652b004e4d8a7bc751fdc099984fe0cf","schema_version":"1.0","event_id":"sha256:6b6add92f2d401f80445284089c1a3d4652b004e4d8a7bc751fdc099984fe0cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDDESUWSSOE6BB5CW6AV62DBYT/bundle.json","state_url":"https://pith.science/pith/KDDESUWSSOE6BB5CW6AV62DBYT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDDESUWSSOE6BB5CW6AV62DBYT/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":"2026-06-24T06:03:15Z","links":{"resolver":"https://pith.science/pith/KDDESUWSSOE6BB5CW6AV62DBYT","bundle":"https://pith.science/pith/KDDESUWSSOE6BB5CW6AV62DBYT/bundle.json","state":"https://pith.science/pith/KDDESUWSSOE6BB5CW6AV62DBYT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDDESUWSSOE6BB5CW6AV62DBYT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KDDESUWSSOE6BB5CW6AV62DBYT","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":"e3447ca4493de0ec43ca39a206d880218a0b7489a972ff06049a79f4e6f0dd8f","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-10-26T12:59:22Z","title_canon_sha256":"ac6a94a31ebd2c839138b05136b17b5854d19590fafe9f80840a9bc412af6f9d"},"schema_version":"1.0","source":{"id":"1510.07462","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07462","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07462v2","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07462","created_at":"2026-05-18T00:53:21Z"},{"alias_kind":"pith_short_12","alias_value":"KDDESUWSSOE6","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KDDESUWSSOE6BB5C","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KDDESUWS","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:6b6add92f2d401f80445284089c1a3d4652b004e4d8a7bc751fdc099984fe0cf","target":"graph","created_at":"2026-05-18T00:53:21Z","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":"Disjoint-Set forests, consisting of Union-Find trees are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression). In this paper we provide such a characterization and show that the decision problem whether a given tree is a Union-Find tree is $\\NP$-complete.","authors_text":"Kitti Gelle, Szabolcs Ivan","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-10-26T12:59:22Z","title":"Recognizing Union-Find trees is NP-complete"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07462","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":"sha256:5c19643c0a23bd8ace735d536bd32933c8877fba2922a47e4865d6f6594856e6","target":"record","created_at":"2026-05-18T00:53:21Z","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":"e3447ca4493de0ec43ca39a206d880218a0b7489a972ff06049a79f4e6f0dd8f","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-10-26T12:59:22Z","title_canon_sha256":"ac6a94a31ebd2c839138b05136b17b5854d19590fafe9f80840a9bc412af6f9d"},"schema_version":"1.0","source":{"id":"1510.07462","kind":"arxiv","version":2}},"canonical_sha256":"50c64952d29389e087a2b7815f6861c4c2d2f50e80f62fb385d97f3624c491b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50c64952d29389e087a2b7815f6861c4c2d2f50e80f62fb385d97f3624c491b9","first_computed_at":"2026-05-18T00:53:21.080207Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:21.080207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YGDUo0FLdXvKn8UcK+SXwVw/Fr3Ko02i8y0aS554GHux92CkIxzYF9dmYHkTRJZzw1+nIUgoR+IEIhmQ7qGqCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:21.080659Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07462","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c19643c0a23bd8ace735d536bd32933c8877fba2922a47e4865d6f6594856e6","sha256:6b6add92f2d401f80445284089c1a3d4652b004e4d8a7bc751fdc099984fe0cf"],"state_sha256":"dc6248e176bf5a419d90993e1fa11317448c095887d9938638ba3b37208ce34b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5MuZt8q29W7XryEC8esGaqvLuD/QL/E9LOLX5mnaubqqY0xALtjoudwtm1Wrf7XytEzrULIQIfsXBLhBsoHmBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:03:15.341666Z","bundle_sha256":"7587bbdf1470f2032222d3b0fc6b099244064a34c47462d70fe978a17d5f97d5"}}