{"paper":{"title":"Near-Optimal (Euclidean) Metric Compression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Piotr Indyk, Tal Wagner","submitted_at":"2016-09-20T19:18:08Z","abstract_excerpt":"The metric sketching problem is defined as follows. Given a metric on $n$ points, and $\\epsilon>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a $1+\\epsilon$ distortion. In this paper we consider metrics induced by $\\ell_2$ and $\\ell_1$ norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by $\\Phi>0$. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size $O(\\epsilon^{-2} \\log (\\Phi n) n\\log n)$, i.e., $O(\\epsilon^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06295","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"}