{"paper":{"title":"The Price of Order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Andr\\'e van Renssen, Pat Morin, Prosenjit Bose","submitted_at":"2016-02-01T06:19:18Z","abstract_excerpt":"We present tight bounds on the spanning ratio of a large family of ordered $\\theta$-graphs. A $\\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\\theta = 2 \\pi/m$. An ordered $\\theta$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \\geq 1$, ordered $\\theta$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \\sin(\\theta/2) / (\\cos(\\theta/2) - \\sin(\\theta/2))$. We also show that for any integer $k \\geq 2$, or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00399","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"}