{"paper":{"title":"Approximating Majority Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Dan Chen, Pat Morin","submitted_at":"2012-05-07T20:05:01Z","abstract_excerpt":"We consider the problem of approximating the majority depth (Liu and Singh, 1993) of a point q with respect to an n-point set, S, by random sampling. At the heart of this problem is a data structures question: How can we preprocess a set of n lines so that we can quickly test whether a randomly selected vertex in the arrangement of these lines is above or below the median level. We describe a Monte-Carlo data structure for this problem that can be constructed in O(nlog n) time, can answer queries O((log n)^{4/3}) expected time, and answers correctly with high probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1524","kind":"arxiv","version":3},"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"}