{"paper":{"title":"Optimal Private Halfspace Counting via Discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.DS","authors_text":"Aleksandar Nikolov, S. Muthukrishnan","submitted_at":"2012-03-24T22:25:12Z","abstract_excerpt":"A range counting problem is specified by a set $P$ of size $|P| = n$ of points in $\\mathbb{R}^d$, an integer weight $x_p$ associated to each point $p \\in P$, and a range space ${\\cal R} \\subseteq 2^{P}$. Given a query range $R \\in {\\cal R}$, the target output is $R(\\vec{x}) = \\sum_{p \\in R}{x_p}$. Range counting for different range spaces is a central problem in Computational Geometry.\n  We study $(\\epsilon, \\delta)$-differentially private algorithms for range counting. Our main results are for the range space given by hyperplanes, that is, the halfspace counting problem. We present an $(\\epsi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5453","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"}