{"paper":{"title":"Circle separability queries in logarithmic time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Greg Aloupis, Luis Barba, Stefan Langerman","submitted_at":"2012-03-28T13:37:56Z","abstract_excerpt":"Let $P$ be a set of $n$ points in the plane. In this paper we study a new variant of the circular separability problem in which a point set $P$ is preprocessed so that one can quickly answer queries of the following form: Given a geometric object $Q$, report the minimum circle containing $P$ and exluding $Q$. Our data structure can be constructed in $O(n\\log n)$ time using O(n) space, and can be used to answer the query when $Q$ is either a circle or a convex $m$-gon in $O(\\log n)$ or $O(\\log n + \\log m)$ time, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6266","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"}