{"paper":{"title":"Rectilinear Matching to the Integer Grid in Nearly-Linear Time","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Yu Gao","submitted_at":"2026-07-12T10:54:54Z","abstract_excerpt":"Rectilinear matching to the integer grid asks to assign each of $n$ points in $\\mathbb R^2$ to a distinct point of $\\mathbb Z^2$, minimizing total $\\ell_1$ movement. The main difficulty is that the target set is infinite: one must first identify a finite set of relevant grid points without losing optimality.\n  We prove a geometric compression theorem for this infinite-target problem. In $O(n\\log^2 n)$ time, we construct a set $\\mathcal{C}$ of asymptotically optimal size $O(n)$ such that, simultaneously for every $p\\in[1,\\infty]$, some optimal $\\ell_p$ assignment uses only points of $\\mathcal{C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10703","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.10703/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}