{
  "abstract": "In adaptive bitrate (ABR) streaming, the delivery technology behind most Internet video, each title is encoded at several bitrates, forming an \\emph{encoding ladder} of \\emph{renditions}. Each client plays the highest rendition that its network bandwidth can sustain. We show that choosing the ladder is a problem of \\emph{scalar quantization} of the bandwidth distribution. However, this quantization problem is of an unusual kind: the client's logic pins each quantization cell's reproduction value to the cell's \\emph{left edge}, and the distortion measure is a one-sided quality loss rather than a squared error. The constraint reshapes the classical theory. The \\emph{quality gap} of a ladder is the distance between its average delivered quality and the \\emph{quality limit} that infinitely many renditions would attain. For optimal $n$-rendition ladders the gap $\\mathcal{G}_n^*$ decays as $Θ(1/n)$, not the textbook $Θ(1/n^2)$. Moreover, with large $n$, $n\\,\\mathcal{G}_n^*\\to C_w=\\frac{1}{2}(\\int\\!\\sqrt{p\\,Q'}\\,\\,d B)^2$, with the limit attained when the ladder's rates follow the density $\\sqrt{p\\,Q'}$, which replaces Panter--Dite's $p^{1/3}$; here $p$ is the bandwidth density and $Q$ the quality--rate curve of the content. The law yields closed-form design rules: quantile rung placement, the rendition count $n(\\varepsilon)\\approx C_w/\\varepsilon$ needed to reach tolerance $\\varepsilon$, and the economic ladder size $n^*=\\sqrt{λC_w/κ}$ when a rendition costs $κ$ to operate and the quality gap is priced at $λ$. The same analysis extends to the design of ladders employing different video resolutions, codecs, perceptual quality metrics, and the two-dimensional adaptation logic of modern web players.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03745",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuriy A. Reznik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dx13vx1d5xfhd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03745",
  "fulltext_url": null,
  "id": "dx13vx1d5xfhd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03745",
  "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",
      "1707|Image and Video Quality Assessment"
    ]
  ],
  "openalex": "W7208723042",
  "pdf_url": "https://arxiv.org/pdf/2609.03745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dx13vx1d5xfhd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "A Quantization Problem Posed by Adaptive Streaming",
  "title_text": "A Quantization Problem Posed by Adaptive Streaming",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/dx13vx1d5xfhd",
  "venue": "arXiv",
  "versions": []
}