{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:IHARSQT5OSH57232UA5NAXWNIT","short_pith_number":"pith:IHARSQT5","schema_version":"1.0","canonical_sha256":"41c119427d748fdfeb7aa03ad05ecd44ed201d60467b59dbb139bc70d68fb37e","source":{"kind":"arxiv","id":"1512.07533","version":2},"attestation_state":"computed","paper":{"title":"Geometric k-Center Problems with Centers Constrained to Two Lines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Binay Bhattacharya, Naoki Katoh, Sandip Das, Tsunehiko Kameda, Yuya Higashikawa","submitted_at":"2015-12-23T16:33:11Z","abstract_excerpt":"We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(n\\log^2 n)$ time algorithm, which minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(n\\log^2 n)$ time also in this case."},"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":"1512.07533","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2015-12-23T16:33:11Z","cross_cats_sorted":[],"title_canon_sha256":"76570669e058e44cba494fc84064a0edadedbad525925ec5d73ed8367065826a","abstract_canon_sha256":"8b83ac8a011b2ebc33d20760594deddb9247e78f72ddea45ed401615dc95ab70"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:15.035151Z","signature_b64":"2UCr8Ye+7oip4EvJsrE43EcUMe1vIFaA+WXGMONIqBI2qbpNCE8yus9UUaIHeudt6lFGYxx7AktRZuI3aJ4bAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41c119427d748fdfeb7aa03ad05ecd44ed201d60467b59dbb139bc70d68fb37e","last_reissued_at":"2026-05-18T01:16:15.034514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:15.034514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric k-Center Problems with Centers Constrained to Two Lines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Binay Bhattacharya, Naoki Katoh, Sandip Das, Tsunehiko Kameda, Yuya Higashikawa","submitted_at":"2015-12-23T16:33:11Z","abstract_excerpt":"We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(n\\log^2 n)$ time algorithm, which minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(n\\log^2 n)$ time also in this case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07533","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.07533","created_at":"2026-05-18T01:16:15.034608+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.07533v2","created_at":"2026-05-18T01:16:15.034608+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07533","created_at":"2026-05-18T01:16:15.034608+00:00"},{"alias_kind":"pith_short_12","alias_value":"IHARSQT5OSH5","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"IHARSQT5OSH57232","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"IHARSQT5","created_at":"2026-05-18T12:29:25.134429+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/IHARSQT5OSH57232UA5NAXWNIT","json":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT.json","graph_json":"https://pith.science/api/pith-number/IHARSQT5OSH57232UA5NAXWNIT/graph.json","events_json":"https://pith.science/api/pith-number/IHARSQT5OSH57232UA5NAXWNIT/events.json","paper":"https://pith.science/paper/IHARSQT5"},"agent_actions":{"view_html":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT","download_json":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT.json","view_paper":"https://pith.science/paper/IHARSQT5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.07533&json=true","fetch_graph":"https://pith.science/api/pith-number/IHARSQT5OSH57232UA5NAXWNIT/graph.json","fetch_events":"https://pith.science/api/pith-number/IHARSQT5OSH57232UA5NAXWNIT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT/action/storage_attestation","attest_author":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT/action/author_attestation","sign_citation":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT/action/citation_signature","submit_replication":"https://pith.science/pith/IHARSQT5OSH57232UA5NAXWNIT/action/replication_record"}},"created_at":"2026-05-18T01:16:15.034608+00:00","updated_at":"2026-05-18T01:16:15.034608+00:00"}