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In this work our goal is to find a function $F:[n]\\rightarrow \\field{R}$ which optimizes the following objective function:\n  $$ \\min_{F} \\sum_{i=1}^n (P_i-F_i)^2 + C\\times |\\{i:F_i \\neq F_{i+1} \\} | $$\n  The above optimization problem reduces to solving the following recurrence, which can be done efficiently using dynamic programming in $O(n^2)$ time:\n  $$ OPT_i = \\min_{0 \\leq j \\leq i-1} [ OPT_j + \\sum_{k=j+1}^i (P_k - (\\sum_{m=j+1}^i P_m)/(i-j) )^2 ]+ C $$\n  The above recurrence arises na"},"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":"1003.4942","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2010-03-25T16:06:44Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"8741ca1dd69c0d7c6ac34d9649cde536a482c4a3cbcb9d63726da0da55d9d799","abstract_canon_sha256":"2a4e78b86b3e2bef2258d0e1ba5f24e2e1bf9b844191f7ffc8c77ffffcbc26c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:26.693639Z","signature_b64":"oB9uAiTOYG6zHmCYOPyl6U72bk/K2mQf8OIyYOiL58+werzUfHpfJa5xb4fJhPRylhFZR+n98Z1vBS7DkzAiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc52f1932ca9275ed7ff4295a24b5b6ea9aaa3e1e647de2b48550c3b6abc0930","last_reissued_at":"2026-05-18T02:24:26.692886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:26.692886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximate Dynamic Programming using Halfspace Queries and Multiscale Monge decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.DS","authors_text":"Charalampos E. Tsourakakis, Gary L. Miller, Richard Peng, Russell Schwartz","submitted_at":"2010-03-25T16:06:44Z","abstract_excerpt":"Let $P=(P_1, P_2, \\ldots, P_n)$, $P_i \\in \\field{R}$ for all $i$, be a signal and let $C$ be a constant. In this work our goal is to find a function $F:[n]\\rightarrow \\field{R}$ which optimizes the following objective function:\n  $$ \\min_{F} \\sum_{i=1}^n (P_i-F_i)^2 + C\\times |\\{i:F_i \\neq F_{i+1} \\} | $$\n  The above optimization problem reduces to solving the following recurrence, which can be done efficiently using dynamic programming in $O(n^2)$ time:\n  $$ OPT_i = \\min_{0 \\leq j \\leq i-1} [ OPT_j + \\sum_{k=j+1}^i (P_k - (\\sum_{m=j+1}^i P_m)/(i-j) )^2 ]+ C $$\n  The above recurrence arises na"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.4942","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":"1003.4942","created_at":"2026-05-18T02:24:26.693035+00:00"},{"alias_kind":"arxiv_version","alias_value":"1003.4942v2","created_at":"2026-05-18T02:24:26.693035+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.4942","created_at":"2026-05-18T02:24:26.693035+00:00"},{"alias_kind":"pith_short_12","alias_value":"3RJPDEZMVETV","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"3RJPDEZMVETV5V77","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"3RJPDEZM","created_at":"2026-05-18T12:26:03.138858+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/3RJPDEZMVETV5V77IKK2ES23N2","json":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2.json","graph_json":"https://pith.science/api/pith-number/3RJPDEZMVETV5V77IKK2ES23N2/graph.json","events_json":"https://pith.science/api/pith-number/3RJPDEZMVETV5V77IKK2ES23N2/events.json","paper":"https://pith.science/paper/3RJPDEZM"},"agent_actions":{"view_html":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2","download_json":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2.json","view_paper":"https://pith.science/paper/3RJPDEZM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1003.4942&json=true","fetch_graph":"https://pith.science/api/pith-number/3RJPDEZMVETV5V77IKK2ES23N2/graph.json","fetch_events":"https://pith.science/api/pith-number/3RJPDEZMVETV5V77IKK2ES23N2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2/action/storage_attestation","attest_author":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2/action/author_attestation","sign_citation":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2/action/citation_signature","submit_replication":"https://pith.science/pith/3RJPDEZMVETV5V77IKK2ES23N2/action/replication_record"}},"created_at":"2026-05-18T02:24:26.693035+00:00","updated_at":"2026-05-18T02:24:26.693035+00:00"}