{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:THNME6WDG4FZFVXLBWOE43D5JB","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":"4a7e2c992bbab5f037963a55c6ffb5c3228a53029678c9130d200bb3a67d572b","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-22T22:32:07Z","title_canon_sha256":"00f367d880bb68ed2c3e12c0dc97ef0e304222a82e4e7c6f74f98656a5d39e5b"},"schema_version":"1.0","source":{"id":"1211.5389","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5389","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5389v2","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5389","created_at":"2026-05-18T03:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"THNME6WDG4FZ","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"THNME6WDG4FZFVXL","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"THNME6WD","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:a117a987e495f7e7441a83532f2f4886c59b5f29427ad381ac0aa5456ef9929d","target":"graph","created_at":"2026-05-18T03:05:35Z","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":"Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \\emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\\sigma$ can have $\\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \\times \\sigma)$ using $O(n \\times \\sigma)$ space. We present an off-line algorithm based on a $\\sel$ function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present on-line algorithms that also enable to compute all the ","authors_text":"Arnaud Lefebvre, Elise Prieur-Gaston, Gabriele Fici, Thierry Lecroq","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-22T22:32:07Z","title":"Algorithms for Computing Abelian Periods of Words"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5389","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":"sha256:8a75140692d6a990d4cc3c5440f75765f5f4e57944ab81dd9686a07ef5f6b699","target":"record","created_at":"2026-05-18T03:05:35Z","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":"4a7e2c992bbab5f037963a55c6ffb5c3228a53029678c9130d200bb3a67d572b","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-11-22T22:32:07Z","title_canon_sha256":"00f367d880bb68ed2c3e12c0dc97ef0e304222a82e4e7c6f74f98656a5d39e5b"},"schema_version":"1.0","source":{"id":"1211.5389","kind":"arxiv","version":2}},"canonical_sha256":"99dac27ac3370b92d6eb0d9c4e6c7d485d047532498e9df3d2accf062c1d8dc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99dac27ac3370b92d6eb0d9c4e6c7d485d047532498e9df3d2accf062c1d8dc2","first_computed_at":"2026-05-18T03:05:35.780333Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:35.780333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nvtE1yKfF6IPACY1eMFqGSn/dDyKhVyFkUU7fuNJR6hIqsbb6tO6kTiPlGXfDONMvxi24N2YYrpWcMIebJIjAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:35.780935Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5389","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a75140692d6a990d4cc3c5440f75765f5f4e57944ab81dd9686a07ef5f6b699","sha256:a117a987e495f7e7441a83532f2f4886c59b5f29427ad381ac0aa5456ef9929d"],"state_sha256":"25e2ad399a2611b1786a9cf390e25e64f9cf21f8c42d7cfddf0db070eaaa3f1e"}