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DOI: 10.4230/LIPIcs.SoCG.2026.57

Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

Authors: Petar Hristov, Ingrid Hotz, and Talha Bin Masood

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework [Hristov et al., 2025]. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges [Tierny and Carr, 2017]. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

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Petar Hristov, Ingrid Hotz, and Talha Bin Masood. Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hristov_et_al:LIPIcs.SoCG.2026.57,
  author =	{Hristov, Petar and Hotz, Ingrid and Masood, Talha Bin},
  title =	{{Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.57},
  URN =		{urn:nbn:de:0030-drops-258644},
  doi =		{10.4230/LIPIcs.SoCG.2026.57},
  annote =	{Keywords: Computational topology, Reeb graph, Reeb space, Multivariate data, Multifield, Geometric arrangement}
}

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DOI: 10.4230/LIPIcs.SoCG.2020.17

Parallel Computation of Alpha Complexes for Biomolecules

Authors: Talha Bin Masood, Tathagata Ray, and Vijay Natarajan

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction of the Delaunay triangulation. The algorithm computes the alpha complex in two stages. The first stage proceeds in a bottom-up fashion and computes a superset of the edges, triangles, and tetrahedra belonging to the alpha complex. The false positives from this estimation stage are removed in a subsequent pruning stage to obtain the correct alpha complex. Computational experiments on several biomolecules demonstrate the superior performance of the algorithm, up to a factor of 50 when compared to existing methods that are optimized for biomolecules.

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Talha Bin Masood, Tathagata Ray, and Vijay Natarajan. Parallel Computation of Alpha Complexes for Biomolecules. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{masood_et_al:LIPIcs.SoCG.2020.17,
  author =	{Masood, Talha Bin and Ray, Tathagata and Natarajan, Vijay},
  title =	{{Parallel Computation of Alpha Complexes for Biomolecules}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{17:1--17:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.17},
  URN =		{urn:nbn:de:0030-drops-121758},
  doi =		{10.4230/LIPIcs.SoCG.2020.17},
  annote =	{Keywords: Delaunay triangulation, parallel algorithms, biomolecules, GPU}
}

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Documents authored by Masood, Talha Bin

Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

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Parallel Computation of Alpha Complexes for Biomolecules

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