{
  "abstract": "We introduce and study the adaptive covering complexity C adapt S (s), defined as the minimum number of contiguous factor blocks required to partition a string s when each block is either a factor of a fixed external dictionary S or a factor of the target's already decoded prefix. This hybrid admissibility model formalizes the structural interaction between static dictionary matching and dynamic nonoverlapping self-referencing. Our main theoretical result is the exact logarithmic law which is tight and proves that dynamic self-reference can reduce phrase count by at most an exponential factor over static dictionary covering alone. We establish structural equivalences showing that the measure bridges word-segmented Relative Lempel-Ziv and non-overlapping LZ77, and we prove exact preset-history LZ77 sandwich bounds. Furthermore, we establish the optimality of the longest-match greedy parse via an exchange lemma and present an optimal implementation based on dual suffix automata running in O(M + n) time and space for a constant alphabet. These analytical results are complemented by counting bounds, stationary-ergodic asymptotic invariance, a dictionaryselection NP-hardness result, and a concise empirical illustration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26021",
  "author_total": 1,
  "authors": [
    {
      "name": "Pierre-Fran{\\c c}ois Marteau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08nf53jmdaay6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26021",
  "fulltext_url": null,
  "id": "08nf53jmdaay6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26021",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.26021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08nf53jmdaay6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "On the Adaptive Covering Complexity of Strings: Unifying Relative and Self-Referential Compression",
  "title_text": "On the Adaptive Covering Complexity of Strings: Unifying Relative and Self-Referential Compression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/08nf53jmdaay6",
  "venue": "arXiv",
  "versions": []
}