50 Search Results for "Dey, Tamal K."

Document
DOI: 10.4230/LIPIcs.SoCG.2026.10
Fast Free Resolutions of Bifiltered Chain Complexes

Authors: Ulrich Bauer, Tamal K. Dey, Michael Kerber, Florian Russold, and Matthias Söls

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

Abstract
In a k-critical bifiltration, every simplex enters along a staircase with at most k steps. Examples with k > 1 include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a k-critical bifiltration into a 1-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a k-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.
Cite as

Ulrich Bauer, Tamal K. Dey, Michael Kerber, Florian Russold, and Matthias Söls. Fast Free Resolutions of Bifiltered Chain Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bauer_et_al:LIPIcs.SoCG.2026.10,
  author =	{Bauer, Ulrich and Dey, Tamal K. and Kerber, Michael and Russold, Florian and S\"{o}ls, Matthias},
  title =	{{Fast Free Resolutions of Bifiltered Chain Complexes}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{10:1--10:21},
  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.2026.10},
  URN =		{urn:nbn:de:0030-drops-258161},
  doi =		{10.4230/LIPIcs.SoCG.2026.10},
  annote =	{Keywords: Topological Data Analysis, Multi-Parameter Persistence, Multi-Critical Bifiltrations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.6
Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations

Authors: Ethan André, Jingyi Li, David Loiseaux, and Steve Oudot

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

Abstract
Given an unknown ℝⁿ-valued function f on a metric space X, can we approximate the persistent homology of f from a finite sampling of X with known pairwise distances and function values? This question has been answered in the case n = 1, assuming f is Lipschitz continuous and X is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary n, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results.
Cite as

Ethan André, Jingyi Li, David Loiseaux, and Steve Oudot. Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{andre_et_al:LIPIcs.SoCG.2026.6,
  author =	{Andr\'{e}, Ethan and Li, Jingyi and Loiseaux, David and Oudot, Steve},
  title =	{{Estimating the Persistent Homology of \mathbb{R}ⁿ-Valued Functions Using Function-Geometric Multifiltrations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{6:1--6:18},
  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.2026.6},
  URN =		{urn:nbn:de:0030-drops-258120},
  doi =		{10.4230/LIPIcs.SoCG.2026.6},
  annote =	{Keywords: Topological data analysis, multi-parameter persistent homology, function-Rips multifiltration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.26
Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice

Authors: Guangya Cai

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

Abstract
We study the problem of finding a fair linear scoring function over (numerical) attributes for top-k selection, ensuring fairness through a proportional representation constraint on the protected group. Existing algorithms do not scale efficiently, particularly in higher dimensions. Our hardness analysis shows that in more than two dimensions, no algorithm is likely to scale efficiently with respect to dataset size, and the computational complexity is likely to grow rapidly with dimensionality. However, the hardness results also provide key insights guiding algorithm design, leading to our two-pronged solution: (1) For small k, our analysis reveals a gap in the hardness barrier. By addressing various engineering challenges, including achieving efficient parallelism, we turn this potential of efficiency into an optimized geometry-based algorithm delivering substantial performance gains. (2) For large k, where the hardness is robust, we employ a practically efficient optimization-based algorithm which, despite being theoretically worse, achieves superior real-world performance. Experimental evaluations on real-world datasets then explore scenarios where worst-case behavior does not manifest, identifying areas critical to practical performance. Our solution achieves speedups of up to several orders of magnitude compared to the state of the art, an efficiency made possible through a tight integration of hardness analysis, algorithm design, practical engineering, and empirical evaluation.
Cite as

Guangya Cai. Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai:LIPIcs.SoCG.2026.26,
  author =	{Cai, Guangya},
  title =	{{Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{26:1--26:17},
  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.2026.26},
  URN =		{urn:nbn:de:0030-drops-258320},
  doi =		{10.4230/LIPIcs.SoCG.2026.26},
  annote =	{Keywords: Fairness, Top-k, Integration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.47
Computing the Skyscraper Invariant

Authors: Marc Fersztand and Jan Jendrysiak

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

Abstract
We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value. Cheng’s algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For d-parameter modules, this produces an FPT ε-approximate algorithm with runtime dominated by 𝒪(1/ε^d ⋅ T_dec), where T_dec is the time for decomposition, which we compute with aida [DJK25]. We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree d - 1 polynomials. This allows for an exact computation of the Skyscraper Invariant roughly in 𝒪(n^d ⋅ T_dec) time for n the size of the presentation and enables a fast hybrid algorithm. For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng’s algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.
Cite as

Marc Fersztand and Jan Jendrysiak. Computing the Skyscraper Invariant. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fersztand_et_al:LIPIcs.SoCG.2026.47,
  author =	{Fersztand, Marc and Jendrysiak, Jan},
  title =	{{Computing the Skyscraper Invariant}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{47:1--47:23},
  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.2026.47},
  URN =		{urn:nbn:de:0030-drops-258535},
  doi =		{10.4230/LIPIcs.SoCG.2026.47},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence, Persistence, Harder-Narasimhan Filtration, Skyscraper Invariant}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.62
Computing the Bottleneck Distance Between Persistent Homology Transforms

Authors: Michael Kerber and Elena Xinyi Wang

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

Abstract
The Persistent Homology Transform (PHT) summarizes a shape in ℝ^m by collecting persistence diagrams obtained from linear height filtrations in all directions on 𝕊^{m-1}. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact integral of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to Õ(n⁵) in place of the earlier Õ(n⁶) bound, where n denotes the number of simplices. For the max objective, we give an Õ(n³) expected-time algorithm in ℝ² and an Õ(n⁵) expected-time algorithm in ℝ³.
Cite as

Michael Kerber and Elena Xinyi Wang. Computing the Bottleneck Distance Between Persistent Homology Transforms. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2026.62,
  author =	{Kerber, Michael and Wang, Elena Xinyi},
  title =	{{Computing the Bottleneck Distance Between Persistent Homology Transforms}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{62:1--62:15},
  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.2026.62},
  URN =		{urn:nbn:de:0030-drops-258693},
  doi =		{10.4230/LIPIcs.SoCG.2026.62},
  annote =	{Keywords: Kinetic data structure, bottleneck distance, persistent homology transform, vineyards}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.72
Topological Simplification Guided by Forbidden Regions

Authors: Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov

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

Abstract
Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.
Cite as

Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov. Topological Simplification Guided by Forbidden Regions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{leskiewicz_et_al:LIPIcs.SoCG.2026.72,
  author =	{Le\'{s}kiewicz, Jakub and Furmanek, Bartosz and Lipi\'{n}ski, Micha{\l} and Morozov, Dmitriy},
  title =	{{Topological Simplification Guided by Forbidden Regions}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{72:1--72:17},
  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.2026.72},
  URN =		{urn:nbn:de:0030-drops-258797},
  doi =		{10.4230/LIPIcs.SoCG.2026.72},
  annote =	{Keywords: persistent homology, topological simplification, depth posets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.78
Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper

Authors: Dhruv Meduri, Chuan-Shen Hu, Cong Shen, Kelin Xia, and Bei Wang

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

Abstract
The vast chemical space, encompassing virtually innumerable molecules and materials, presents both immense opportunities and significant challenges. The design and discovery of novel drugs and functional materials may be viewed as a search within this space; however, the sheer scale of potential candidates renders exhaustive exploration infeasible. To address this, we introduce Chemical Mapper, a framework that integrates topological data analysis with deep learning to enable the visual exploration and analysis of chemical latent spaces. At its core, Chemical Mapper employs mapper, a widely used tool in topological data analysis, to investigate the organizational principles of chemical latent spaces defined by molecular representations learned by geometric deep learning models. In doing so, Chemical Mapper not only highlights groups of molecular representations but also uncovers the relationships among them through linkages and branching structures. Our results show that Chemical Mapper reveals intrinsic patterns associated with molecular scaffolds, functional groups, and chemical properties, as well as the structural and functional evolutions of the molecules.
Cite as

Dhruv Meduri, Chuan-Shen Hu, Cong Shen, Kelin Xia, and Bei Wang. Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{meduri_et_al:LIPIcs.SoCG.2026.78,
  author =	{Meduri, Dhruv and Hu, Chuan-Shen and Shen, Cong and Xia, Kelin and Wang, Bei},
  title =	{{Mapping Chemical Space: Topological Data Analysis of Chemical Latent Space with Mapper}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{78:1--78:20},
  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.2026.78},
  URN =		{urn:nbn:de:0030-drops-258854},
  doi =		{10.4230/LIPIcs.SoCG.2026.78},
  annote =	{Keywords: Practice of computational topology, topological data analysis, applications in chemistry, mapper algorithm, high-dimensional data analysis, chemical spaces, geometric deep learning, latent space geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.79
D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence

Authors: Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, and Tamal K. Dey

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

Abstract
End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called Gril. We establish a theory for gradient descent on Gril producing D-Gril. We show that D-Gril can be used to learn a bifiltration function on benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.
Cite as

Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, and Tamal K. Dey. D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 79:1-79:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{mukherjee_et_al:LIPIcs.SoCG.2026.79,
  author =	{Mukherjee, Soham and Samaga, Shreyas N. and Xin, Cheng and Oudot, Steve and Dey, Tamal K.},
  title =	{{D-GRIL: End-To-End Topological Learning with 2-Parameter Persistence}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{79:1--79:17},
  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.2026.79},
  URN =		{urn:nbn:de:0030-drops-258865},
  doi =		{10.4230/LIPIcs.SoCG.2026.79},
  annote =	{Keywords: Topological Data Analysis, Persistent Homology, Multiparameter Persistence, Graph Learning, Graph Neural Networks}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.82
A Persistent Version of Latschev’s Theorem

Authors: Steve Oudot and Lukas Waas

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

Abstract
Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex ℛ^δ(𝕄) of any nearby space 𝕄 in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f:M → ℝ^N) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex ℛ^δ(𝕄^•) of any nearby pair (𝕄, 𝕗). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
Cite as

Steve Oudot and Lukas Waas. A Persistent Version of Latschev’s Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{oudot_et_al:LIPIcs.SoCG.2026.82,
  author =	{Oudot, Steve and Waas, Lukas},
  title =	{{A Persistent Version of Latschev’s Theorem}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{82:1--82:17},
  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.2026.82},
  URN =		{urn:nbn:de:0030-drops-258891},
  doi =		{10.4230/LIPIcs.SoCG.2026.82},
  annote =	{Keywords: Topological data analysis (TDA), metric geometry, Vietoris-Rips complex, homotopy theory, multi-parameter persistent homology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.84
The Voronoi Diagram of Four Lines in ℝ³

Authors: Evanthia Papadopoulou and Zeyu Wang

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

Abstract
We consider the Voronoi diagram of lines in ℝ³ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a twist, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called full and partial twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in ℝ³ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.
Cite as

Evanthia Papadopoulou and Zeyu Wang. The Voronoi Diagram of Four Lines in ℝ³. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{papadopoulou_et_al:LIPIcs.SoCG.2026.84,
  author =	{Papadopoulou, Evanthia and Wang, Zeyu},
  title =	{{The Voronoi Diagram of Four Lines in \mathbb{R}³}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{84:1--84:17},
  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.2026.84},
  URN =		{urn:nbn:de:0030-drops-258916},
  doi =		{10.4230/LIPIcs.SoCG.2026.84},
  annote =	{Keywords: Voronoi diagram, lines, three dimensions, structural properties}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.87
Robustness of Persistent Topological Features and Minimum Homological Cuts

Authors: Pepijn Roos Hoefgeest and Lucas Slot

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

Abstract
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of adversarial robustness, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in 0-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-1 in n-dimensional complexes embedded in ℝⁿ. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.
Cite as

Pepijn Roos Hoefgeest and Lucas Slot. Robustness of Persistent Topological Features and Minimum Homological Cuts. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rooshoefgeest_et_al:LIPIcs.SoCG.2026.87,
  author =	{Roos Hoefgeest, Pepijn and Slot, Lucas},
  title =	{{Robustness of Persistent Topological Features and Minimum Homological Cuts}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{87:1--87:15},
  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.2026.87},
  URN =		{urn:nbn:de:0030-drops-258636},
  doi =		{10.4230/LIPIcs.SoCG.2026.87},
  annote =	{Keywords: Topological Data Analysis, Persistent Homology, Min-cut Max-flow, Robustness, Vietoris-Rips Filtration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.90
Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells

Authors: Jonathan Richard Shewchuk

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

Abstract
The restricted Delaunay triangulation of a closed surface Σ and a finite point set V ⊂ Σ is a subcomplex of the Delaunay tetrahedralization of V whose triangles approximate Σ. It is well known that if V is a sufficiently dense sample of a smooth Σ, then the union of the restricted Delaunay triangles is homeomorphic to Σ. We show that an ε-sample with ε ≤ 0.3245 suffices. By comparison, Dey proves it for a 0.18-sample; our improved sampling bound reduces the number of sample points required by a factor of 3.25. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of 21. The first step of our homeomorphism proof is particularly interesting: we show that for a 0.44-sample, the restricted Voronoi cell of each site v ∈ V is homeomorphic to a disk, and the orthogonal projection of the cell onto T_vΣ (the plane tangent to Σ at v) is star-shaped.
Cite as

Jonathan Richard Shewchuk. Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 90:1-90:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shewchuk:LIPIcs.SoCG.2026.90,
  author =	{Shewchuk, Jonathan Richard},
  title =	{{Better Sampling Bounds for Restricted Delaunay Triangulations and a Star-Shaped Property for Restricted Voronoi Cells}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{90:1--90:16},
  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.2026.90},
  URN =		{urn:nbn:de:0030-drops-258961},
  doi =		{10.4230/LIPIcs.SoCG.2026.90},
  annote =	{Keywords: Restricted Delaunay triangulation, restricted Voronoi diagram, surface sampling, surface mesh generation, surface reconstruction, \epsilon-sample, homeomorphism}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.93
Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations

Authors: Raphaël Tinarrage

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

Abstract
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.
Cite as

Raphaël Tinarrage. Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 93:1-93:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{tinarrage:LIPIcs.SoCG.2026.93,
  author =	{Tinarrage, Rapha\"{e}l},
  title =	{{Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{93:1--93:22},
  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.2026.93},
  URN =		{urn:nbn:de:0030-drops-258991},
  doi =		{10.4230/LIPIcs.SoCG.2026.93},
  annote =	{Keywords: Triangulation of manifolds, Simplicial approximation, CW complexes, Delaunay complexes, List homomorphism problem, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.5
Bifunction and Interlevel Delaunay Trifiltrations

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

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

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

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)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2026.5,
  author =	{Alonso, \'{A}ngel Javier and Kerber, Michael and Lam, Tung and Lesnick, Michael and Rathod, Abhishek},
  title =	{{Bifunction and Interlevel Delaunay Trifiltrations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{5:1--5:20},
  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.2026.5},
  URN =		{urn:nbn:de:0030-drops-258118},
  doi =		{10.4230/LIPIcs.SoCG.2026.5},
  annote =	{Keywords: Delaunay triangulation, Multiparameter persistent homology, Interlevel, Bowyer-Watson}
}
Artifact
Software
DOI: 10.4230/artifacts.26130
D-GRIL: End-to-End Topological Learning with 2-parameter Persistence

Authors: Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, and Tamal K. Dey

Abstract
Cite as

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@misc{dagpub-supp--paper-24468-url-github.com-TDA-Jyamiti-d-gril,
   title = {{D-GRIL: End-to-End Topological Learning with 2-parameter Persistence}}, 
   author = {Mukherjee, Soham and Samaga, Shreyas N. and Xin, Cheng and Oudot, Steve and Dey, Tamal K.},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:379b8266de5d11b54b8041b9a4af9b9dde1fa254;origin=https://github.com/TDA-Jyamiti/d-gril;visit=swh:1:snp:258c458a4305a717ca6956a7438350e1cda5b1f5;anchor=swh:1:rev:c5039986410fa9e715dd86c20b12834275b7b810}{\texttt{swh:1:dir:379b8266de5d11b54b8041b9a4af9b9dde1fa254}} (visited on 2026-05-27)},
   url = {https://github.com/TDA-Jyamiti/d-gril/},
}
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