{"paper":{"title":"Smoothed Analysis of Order Types","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","cs.DS","math.CO"],"primary_cat":"cs.CG","authors_text":"Ivor van der Hoog, Martijn van Schaik, Tillmann Miltzow","submitted_at":"2019-07-10T12:10:09Z","abstract_excerpt":"Consider an ordered point set $P = (p_1,\\ldots,p_n)$, its order type (denoted by $\\chi_P$) is a map which assigns to every triple of points a value in $\\{+,-,0\\}$ based on whether the points are collinear(0), oriented clockwise(-) or counter-clockwise(+). An abstract order type is a map $\\chi : \\left[\\substack{n\\\\3}\\right] \\rightarrow \\{+,-,0\\}$ (where $\\left[\\substack{n\\\\3}\\right]$ is the collection of all triples of a set of $n$ elements) that satisfies the following condition: for every set of five elements $S\\subset [n]$ its induced order type $\\chi_{|S}$ is realizable by a point set. To b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04645","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"}