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Documents authored by Russold, Florian

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}
}
Artifact
Software
DOI: 10.4230/artifacts.22452
TDA-Jyamiti/Algos-cplxs-pers-modules

Authors: Tamal K. Dey, Florian Russold, and Shreyas N. Samaga

Abstract
Cite as

Tamal K. Dey, Florian Russold, Shreyas N. Samaga. TDA-Jyamiti/Algos-cplxs-pers-modules (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@misc{dagstuhl-artifact-22452,
   title = {{TDA-Jyamiti/Algos-cplxs-pers-modules}}, 
   author = {Dey, Tamal K. and Russold, Florian and Samaga, Shreyas N.},
   note = {Software, NSF 2301360, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:6c13c2c3aeb94cc68377d695005250d1ab892cb7;origin=https://github.com/TDA-Jyamiti/Algos-cplxs-pers-modules;visit=swh:1:snp:cf1c636b5c687db1f9fd059ec1c433c6c280180b;anchor=swh:1:rev:5040010c80b997ed83eaee51c314ac2a48c9cfeb}{\texttt{swh:1:dir:6c13c2c3aeb94cc68377d695005250d1ab892cb7}} (visited on 2024-11-28)},
   url = {https://github.com/TDA-Jyamiti/Algos-cplxs-pers-modules/},
   doi = {10.4230/artifacts.22452},
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.44
Stability and Approximations for Decorated Reeb Spaces

Authors: Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.
Cite as

Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold. Stability and Approximations for Decorated Reeb Spaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{curry_et_al:LIPIcs.SoCG.2024.44,
  author =	{Curry, Justin and Mio, Washington and Needham, Tom and Okutan, Osman Berat and Russold, Florian},
  title =	{{Stability and Approximations for Decorated Reeb Spaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{44:1--44:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.44},
  URN =		{urn:nbn:de:0030-drops-199891},
  doi =		{10.4230/LIPIcs.SoCG.2024.44},
  annote =	{Keywords: Reeb spaces, Gromov-Hausdorff distance, Persistent homology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.51
Efficient Algorithms for Complexes of Persistence Modules with Applications

Authors: Tamal K. Dey, Florian Russold, and Shreyas N. Samaga

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.
Cite as

Tamal K. Dey, Florian Russold, and Shreyas N. Samaga. Efficient Algorithms for Complexes of Persistence Modules with Applications. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2024.51,
  author =	{Dey, Tamal K. and Russold, Florian and Samaga, Shreyas N.},
  title =	{{Efficient Algorithms for Complexes of Persistence Modules with Applications}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{51:1--51:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.51},
  URN =		{urn:nbn:de:0030-drops-199969},
  doi =		{10.4230/LIPIcs.SoCG.2024.51},
  annote =	{Keywords: Persistent (co)homology, Persistence modules, Sheaves, Presentations}
}
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