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  "path": "/abs/2605.21636v1",
  "publishedAt": "2026-05-22T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Ángel Javier Alonso",
    "Michael Kerber",
    "Tung Lam",
    "Michael Lesnick",
    "Abhishek Rathod"
  ],
  "textContent": "**Authors:** Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick, Abhishek Rathod\n\nA key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \\subset \\mathbb{R}^d$, our trifiltration has size $O\\bigl(|X|^{\\lceil(d+1)/2\\rceil+1}\\bigr)$. We present an algorithm that computes this trifiltration in time $O\\bigl(|X|^{\\lceil d/2\\rceil+2}\\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\\mathbb{R}^3$, with memory growth that is nearly linear.",
  "title": "Bifunction and Interlevel Delaunay Trifiltrations"
}