{
  "abstract": "Large-scale information retrieval systems, including retrieval-augmented generation (RAG) and recommendation engines, widely use multi-layered hierarchical data structures for ultra-fast approximate nearest-neighbor search in high-dimensional vector spaces. However, the geometric conditions that ensure accurate and efficient greedy navigation remain poorly understood. In this work, we study the efficiency of greedy navigation on a hierarchy of proximity graphs constructed from \\(n\\) data points on the \\(d\\)-dimensional torus~$\\mathbb{T}^d$. We identify a deterministic coverage condition under which, given any query $q\\in \\mathbb{T}^d$, greedy search returns a point within $(1+\\varepsilon)$-factor of the distance to the closest point. This coverage property holds with high probability when the data is distributed as a homogeneous Poisson process, a Hermitian determinantal process, or a bounded-density Cox process, as long as $d = o(\\log n/\\log \\log n)$. Under the same assumptions, the expected number of greedy hops for a fixed query is \\(O\\!\\left(\\exp\\!\\left(\\tfrac12 d\\log d+O(d)\\right)\\log n\\right)\\), yielding logarithmic expected hop count in fixed dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12312",
  "author_total": 4,
  "authors": [
    {
      "name": "Shankar Bhamidi"
    },
    {
      "name": "Souvik Dhara"
    },
    {
      "name": "Lars Schroeder"
    },
    {
      "name": "Clara Stegehuis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evvj684ffst65.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12312",
  "fulltext_url": null,
  "id": "evvj684ffst65",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12312",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "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"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1711|Data Management and Algorithms"
    ]
  ],
  "openalex": "W7221156190",
  "pdf_url": "https://arxiv.org/pdf/2610.12312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evvj684ffst65.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Geometry of Hierarchical Navigation: Accuracy and Query Cost for Point Process Input",
  "title_text": "The Geometry of Hierarchical Navigation: Accuracy and Query Cost for Point Process Input",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/evvj684ffst65",
  "venue": "arXiv",
  "versions": []
}