5 Search Results for "Robins, Vanessa"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.41
The Depth Poset Under Transpositions in the Filter

Authors: Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin

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

Abstract
The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.
Cite as

Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin. The Depth Poset Under Transpositions in the Filter. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2026.41,
  author =	{Edelsbrunner, Herbert and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Soriano-Trigueros, Manuel and Zimin, Fedor},
  title =	{{The Depth Poset Under Transpositions in the Filter}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{41:1--41:18},
  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.2026.41},
  URN =		{urn:nbn:de:0030-drops-258479},
  doi =		{10.4230/LIPIcs.SoCG.2026.41},
  annote =	{Keywords: Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions}
}
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.2026.85
ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes

Authors: Geevarghese Philip and Erlend Raa Vågset

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

Abstract
The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth k, OMM has long been known to be solvable on triangulations of 3-manifolds in 2^O(k²) n^O(1) time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on k has remained an open question. We resolve this by giving a new 2^O(k log k) n-time algorithm for any finite regular CW complex, and show that no 2^o(k log k) n^O(1)-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.
Cite as

Geevarghese Philip and Erlend Raa Vågset. ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{philip_et_al:LIPIcs.SoCG.2026.85,
  author =	{Philip, Geevarghese and V\r{a}gset, Erlend Raa},
  title =	{{ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{85:1--85:19},
  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.2026.85},
  URN =		{urn:nbn:de:0030-drops-258926},
  doi =		{10.4230/LIPIcs.SoCG.2026.85},
  annote =	{Keywords: Discrete Morse Theory, Simplicial Complexes, Optimal Morse Matching, Treewidth, Parameterized Algorithms, Computational Topology, Dynamic Programming, Exponential Time Hypothesis, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.60
Slice, Simplify and Stitch: Topology-Preserving Simplification Scheme for Massive Voxel Data

Authors: Hubert Wagner

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We focus on efficient computations of topological descriptors for voxel data. This type of data includes 2D greyscale images, 3D medical scans, but also higher-dimensional scalar fields arising from physical simulations. In recent years we have seen an increase in applications of topological methods for such data. However, computational issues remain an obstacle. We therefore propose a streaming scheme which simplifies large 3-dimensional voxel data - while provably retaining its persistent homology. We combine this scheme with an efficient boundary matrix reduction implementation, obtaining an end-to-end tool for persistent homology of large data. Computational experiments show its state-of-the-art performance. In particular, we are now able to robustly handle complex datasets with several billions voxels on a regular laptop. A software implementation called Cubicle is available as open-source: https://bitbucket.org/hubwag/cubicle.
Cite as

Hubert Wagner. Slice, Simplify and Stitch: Topology-Preserving Simplification Scheme for Massive Voxel Data. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{wagner:LIPIcs.SoCG.2023.60,
  author =	{Wagner, Hubert},
  title =	{{Slice, Simplify and Stitch: Topology-Preserving Simplification Scheme for Massive Voxel Data}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{60:1--60:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.60},
  URN =		{urn:nbn:de:0030-drops-179107},
  doi =		{10.4230/LIPIcs.SoCG.2023.60},
  annote =	{Keywords: Computational topology, topological data analysis, topological image analysis, persistent homology, persistence diagram, discrete Morse theory, algorithm engineering, implementation, voxel data, volume data, image data}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.SoCG.2017.1
The Geometry and Topology of Crystals: From Sphere-Packing to Tiling, Nets, and Knots (Invited Talk)

Authors: Vanessa Robins

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract
Crystal structures have inspired developments in geometry since the Ancient Greeks conceived of Platonic solids after observing tetrahedral, cubical and octahedral mineral forms in their local environment. The internal structure of crystals became accessible with the development of x-ray diffraction techniques just over 100 years ago, and a key step in developing this method was understanding the arrangement of atoms in the simplest crystals as close-packings of spheres. Determining a crystal structure via x-ray diffraction unavoidably requires prior models, and this has led to the intense study of sphere packing, atom-bond networks, and arrangements of polyhedra by crystallographers investigating ever more complex compounds. In the 21st century, chemists are exploring the possibilities of coordination polymers, a wide class of crystalline materials that self-assemble from metal cations and organic ligands into periodic framework materials. Longer organic ligands mean these compounds can form multi-component interwoven network structures where the "edges" are no longer constrained to join nearest-neighbour "nodes" as in simpler atom-bond networks. The challenge for geometers is to devise algorithms for enumerating relevant structures and to devise invariants that will distinguish between different modes of interweaving. This talk will survey various methods from computational geometry and topology that are currently used to describe crystalline structures and outline research directions to address some of the open questions suggested above.
Cite as

Vanessa Robins. The Geometry and Topology of Crystals: From Sphere-Packing to Tiling, Nets, and Knots (Invited Talk). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{robins:LIPIcs.SoCG.2017.1,
  author =	{Robins, Vanessa},
  title =	{{The Geometry and Topology of Crystals: From Sphere-Packing to Tiling, Nets, and Knots}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.1},
  URN =		{urn:nbn:de:0030-drops-72374},
  doi =		{10.4230/LIPIcs.SoCG.2017.1},
  annote =	{Keywords: Mathematical crystallography, Combinatorial tiling theory, Graphs and surfaces in the 3-torus}
}
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