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Bifunction and Interlevel Delaunay Trifiltrations

Authors Ángel Javier Alonso , Michael Kerber , Tung Lam , Michael Lesnick , Abhishek Rathod

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Topology
  • Theory of computation → Computational geometry
Keywords
  • Delaunay triangulation
  • Multiparameter persistent homology
  • Interlevel
  • Bowyer-Watson

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Abstract

A 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 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 ℝ²-valued function, also satisfying an analogous weak equivalence. For a point cloud X ⊂ ℝ^d, our trifiltration has size O(|X|^{⌈(d+1)/2⌉+1}). We present an algorithm that computes this trifiltration in time O(|X|^{⌈d/2⌉+2}), together with an implementation. Our experiments demonstrate that the implementation can handle thousands of points in ℝ³, with memory growth that is nearly linear.

Cite As Get BibTex

Ángel Javier Alonso, Michael Kerber, Tung Lam, Michael Lesnick, and Abhishek Rathod. Bifunction and Interlevel Delaunay Trifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.5

Author Details

Ángel Javier Alonso
  • CUNEF Universidad, Madrid, Spain
  • Institute of Geometry, Graz University of Technology, Austria
Michael Kerber
  • Institute of Geometry, Graz University of Technology, Austria
Tung Lam
  • University at Albany, SUNY, NY, USA
Michael Lesnick
  • University at Albany, SUNY, NY, USA
Abhishek Rathod
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel

Funding

  • Alonso, Ángel Javier: Austrian Science Fund (FWF) grants 10.55776/P33765 and 10.55776/ W1230.
  • Kerber, Michael: Austrian Science Fund (FWF) grant 10.55776/P33765.
  • Lam, Tung: Simons Foundation Award 96384.
  • Lesnick, Michael: Simons Foundation Award 963845.
  • Rathod, Abhishek: European Research Council (ERC) grant titled PARAPATH (101039913).

Supplementary Materials

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