{
  "abstract": "We give the first polynomial-time constant-factor approximation of the Gromov-Hausdorff distance d_GH between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter and Wenk, 2024). Global alignment cannot succeed: the classical dimension drop defeats alignment by isometries, a multiplicity gap defeats alignment by bijections, and a reflection barrier defeats sorting under any single global reflection pattern. The algorithm is therefore local. Guessing the images of one diameter pair pins every point's longitudinal coordinate to within O(d_GH). Heights are read in windows whose height spread is at most a fixed multiple of their length, where a chain argument makes every compatible match local in the plane. One reflection sign per window is then chosen by 2-SAT; at the right frame and guess, any solution yields a correspondence of distortion O(d_GH). For the bijective relative of d_GH, half the least additive distortion over bijections, the same scheme reduces the planar problem to a matching question that we leave open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.17051",
  "author_total": 1,
  "authors": [
    {
      "name": "Sushovan Majhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4em5we3dhw1z0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.17051",
  "fulltext_url": null,
  "id": "4em5we3dhw1z0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.17051",
  "language": null,
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  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
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      "arxiv",
      "cs.DS"
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      "arxiv",
      "math.MG"
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      "openalex",
      "1704|Computational Geometry and Mesh Generation"
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  "openalex": "W7164926451",
  "pdf_url": "https://arxiv.org/pdf/2606.17051",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4em5we3dhw1z0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2606.17051",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.17051"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A constant-factor approximation of the Gromov-Hausdorff distance in the plane",
  "title_text": "A constant-factor approximation of the Gromov-Hausdorff distance in the plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/4em5we3dhw1z0",
  "venue": "arXiv",
  "versions": []
}