{"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"}