{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:AICOZLEVLG577HOQOTXHBJSX7N","short_pith_number":"pith:AICOZLEV","schema_version":"1.0","canonical_sha256":"0204ecac9559bbff9dd074ee70a657fb4ebac8774109ceb9d058a8bc46a18db2","source":{"kind":"arxiv","id":"1712.02913","version":1},"attestation_state":"computed","paper":{"title":"Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Esther Ezra, Haim Kaplan, Micha Sharir, Sariel Har-Peled","submitted_at":"2017-12-08T02:16:40Z","abstract_excerpt":"$\\renewcommand{\\Re}{\\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension $d$, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension.\n  For vertical decomposition, the combinatorial dimension is only $2d$, the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is at least $1 + d(d+1)/2$ and at most $O(d"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.02913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2017-12-08T02:16:40Z","cross_cats_sorted":[],"title_canon_sha256":"3668b744bfa9e48a07e6b3a0b7884fd4d8ca2d59ae89292de44a5c0414dbf8af","abstract_canon_sha256":"e28f17a47b7e9e2832a531b8a3802a059c7980f02c9818f31a320ba8b47d81d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:28.914237Z","signature_b64":"CFRb22ZozdvG10P98lfJz5JX7MQbTOTj1771ARDMFIbzEYiYnynV/7yI1T9VfK2GIZmomlM/Z/9Z+FKeP7FYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0204ecac9559bbff9dd074ee70a657fb4ebac8774109ceb9d058a8bc46a18db2","last_reissued_at":"2026-05-18T00:28:28.913569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:28.913569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Esther Ezra, Haim Kaplan, Micha Sharir, Sariel Har-Peled","submitted_at":"2017-12-08T02:16:40Z","abstract_excerpt":"$\\renewcommand{\\Re}{\\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex triangulation, and vertical decomposition, including their explicit dependence on the dimension $d$, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension.\n  For vertical decomposition, the combinatorial dimension is only $2d$, the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is at least $1 + d(d+1)/2$ and at most $O(d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02913","kind":"arxiv","version":1},"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":"1712.02913","created_at":"2026-05-18T00:28:28.913681+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.02913v1","created_at":"2026-05-18T00:28:28.913681+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02913","created_at":"2026-05-18T00:28:28.913681+00:00"},{"alias_kind":"pith_short_12","alias_value":"AICOZLEVLG57","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AICOZLEVLG577HOQ","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AICOZLEV","created_at":"2026-05-18T12:31:05.417338+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/AICOZLEVLG577HOQOTXHBJSX7N","json":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N.json","graph_json":"https://pith.science/api/pith-number/AICOZLEVLG577HOQOTXHBJSX7N/graph.json","events_json":"https://pith.science/api/pith-number/AICOZLEVLG577HOQOTXHBJSX7N/events.json","paper":"https://pith.science/paper/AICOZLEV"},"agent_actions":{"view_html":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N","download_json":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N.json","view_paper":"https://pith.science/paper/AICOZLEV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.02913&json=true","fetch_graph":"https://pith.science/api/pith-number/AICOZLEVLG577HOQOTXHBJSX7N/graph.json","fetch_events":"https://pith.science/api/pith-number/AICOZLEVLG577HOQOTXHBJSX7N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N/action/storage_attestation","attest_author":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N/action/author_attestation","sign_citation":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N/action/citation_signature","submit_replication":"https://pith.science/pith/AICOZLEVLG577HOQOTXHBJSX7N/action/replication_record"}},"created_at":"2026-05-18T00:28:28.913681+00:00","updated_at":"2026-05-18T00:28:28.913681+00:00"}