{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:XCWZGHPVSUXOPT7YERDYADRUPW","short_pith_number":"pith:XCWZGHPV","schema_version":"1.0","canonical_sha256":"b8ad931df5952ee7cff82447800e347d8fbeffc8bfdc7a2e4bc8db9bfef9002d","source":{"kind":"arxiv","id":"1509.01221","version":3},"attestation_state":"computed","paper":{"title":"Optimal searching of gapped repeats in a word","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Gregory Kucherov, Maxime Crochemore, Roman Kolpakov","submitted_at":"2015-09-03T19:11:14Z","abstract_excerpt":"Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\\em $\\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\\leq \\alpha |u|$. Our main result is the $O(\\alpha n)$ bound on the number of {\\em maximal} $\\alpha$-gapped repeats in a string of length $n$, previously proved to be $O(\\alpha^2 n)$ in (Kolpakov et al., 2013). For a closely related notion of maximal $\\delta$-subrepetition (maximal factors of exponent between $1+\\delta$ and $2$), our result implies the $O(n/\\delta)$ bound on their number, which improves the bound of (Kolpakov et "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.01221","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2015-09-03T19:11:14Z","cross_cats_sorted":[],"title_canon_sha256":"af27056038fdabd6ff3ea8b351ab5aff370cd25ae34eb1efee4700112cff7391","abstract_canon_sha256":"9a5688601b812109e1ed06f2dd345959b4a16d1507f786aeb27d6b4fd66c32ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:13.352969Z","signature_b64":"eR9mtm/YcqT/Dne+0EMqpaTDG9//jmwkS1elxJkuR1hmsXxX4DBnNG3jwgUUT6bUO50LiS+gm2fyzodSs1PpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8ad931df5952ee7cff82447800e347d8fbeffc8bfdc7a2e4bc8db9bfef9002d","last_reissued_at":"2026-05-18T01:31:13.352271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:13.352271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal searching of gapped repeats in a word","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Gregory Kucherov, Maxime Crochemore, Roman Kolpakov","submitted_at":"2015-09-03T19:11:14Z","abstract_excerpt":"Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\\em $\\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\\leq \\alpha |u|$. Our main result is the $O(\\alpha n)$ bound on the number of {\\em maximal} $\\alpha$-gapped repeats in a string of length $n$, previously proved to be $O(\\alpha^2 n)$ in (Kolpakov et al., 2013). For a closely related notion of maximal $\\delta$-subrepetition (maximal factors of exponent between $1+\\delta$ and $2$), our result implies the $O(n/\\delta)$ bound on their number, which improves the bound of (Kolpakov et "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01221","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.01221","created_at":"2026-05-18T01:31:13.352394+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.01221v3","created_at":"2026-05-18T01:31:13.352394+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01221","created_at":"2026-05-18T01:31:13.352394+00:00"},{"alias_kind":"pith_short_12","alias_value":"XCWZGHPVSUXO","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"XCWZGHPVSUXOPT7Y","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"XCWZGHPV","created_at":"2026-05-18T12:29:50.041715+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW","json":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW.json","graph_json":"https://pith.science/api/pith-number/XCWZGHPVSUXOPT7YERDYADRUPW/graph.json","events_json":"https://pith.science/api/pith-number/XCWZGHPVSUXOPT7YERDYADRUPW/events.json","paper":"https://pith.science/paper/XCWZGHPV"},"agent_actions":{"view_html":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW","download_json":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW.json","view_paper":"https://pith.science/paper/XCWZGHPV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.01221&json=true","fetch_graph":"https://pith.science/api/pith-number/XCWZGHPVSUXOPT7YERDYADRUPW/graph.json","fetch_events":"https://pith.science/api/pith-number/XCWZGHPVSUXOPT7YERDYADRUPW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW/action/storage_attestation","attest_author":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW/action/author_attestation","sign_citation":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW/action/citation_signature","submit_replication":"https://pith.science/pith/XCWZGHPVSUXOPT7YERDYADRUPW/action/replication_record"}},"created_at":"2026-05-18T01:31:13.352394+00:00","updated_at":"2026-05-18T01:31:13.352394+00:00"}