{"paper":{"title":"Covering many points with a small-area box","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Christian Knauer, David Eppstein, Mark de Berg, Otfried Cheong, Sergio Cabello","submitted_at":"2016-12-07T08:41:46Z","abstract_excerpt":"Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \\log n+ n\\log^2 n)$ time. We also consider the problem of, given a value $\\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\\alpha$. For this problem we give a probabilistic $(1-\\varepsilon)$-approximation that works in near-linear time: In $O((n/\\varepsilon^4)\\log^3 n \\log (1/\\varepsilon))$ time we find an axis-parallel rectangle of area at most $\\alpha$ that, with hi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.02149","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"}