Document Open Access Logo

Fast Computation of Zigzag Persistence

Authors Tamal K. Dey, Tao Hou

Thumbnail PDF


  • Filesize: 0.8 MB
  • 15 pages

Document Identifiers

Author Details

Tamal K. Dey
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Tao Hou
  • School of Computing, DePaul University, Chicago, IL, USA


We thank the Stanford Computer Graphics Laboratory and Ryan Holmes for providing the triangular meshes used in the experiment of this paper.

Cite AsGet BibTex

Tamal K. Dey and Tao Hou. Fast Computation of Zigzag Persistence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 43:1-43:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a Δ-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • zigzag persistence
  • persistent homology
  • fast computation


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 522-539. SIAM, 2021. Google Scholar
  2. Ulrich Bauer. Ripser: Efficient computation of vietoris-rips persistence barcodes. Journal of Applied and Computational Topology, 5(3):391-423, 2021. Google Scholar
  3. Ulrich Bauer, Michael Kerber, and Jan Reininghaus. Clear and compress: Computing persistent homology in chunks. In Topological methods in data analysis and visualization III, pages 103-117. Springer, 2014. Google Scholar
  4. Ulrich Bauer, Michael Kerber, Jan Reininghaus, and Hubert Wagner. Phat - persistent homology algorithms toolbox. Journal of Symbolic Computation, 78:76-90, 2017. Google Scholar
  5. Jean-Daniel Boissonnat, Tamal K. Dey, and Clément Maria. The compressed annotation matrix: An efficient data structure for computing persistent cohomology. Algorithmica, 73(3):607-619, 2015. URL:
  6. Gunnar Carlsson and Vin de Silva. Zigzag persistence. Foundations of Computational Mathematics, 10(4):367-405, 2010. Google Scholar
  7. Gunnar Carlsson, Vin de Silva, Sara Kališnik, and Dmitriy Morozov. Parametrized homology via zigzag persistence. Algebraic & Geometric Topology, 19(2):657-700, 2019. Google Scholar
  8. Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. In Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, pages 247-256, 2009. Google Scholar
  9. Gunnar Carlsson, Anjan Dwaraknath, and Bradley J. Nelson. Persistent and zigzag homology: A matrix factorization viewpoint. arXiv preprint, 2019. URL:
  10. Chao Chen and Michael Kerber. Persistent homology computation with a twist. In Proceedings 27th European Workshop on Computational Geometry, volume 11, pages 197-200, 2011. Google Scholar
  11. Chao Chen and Michael Kerber. An output-sensitive algorithm for persistent homology. Comput. Geom.: Theory and Applications, 46(4):435-447, 2013. URL:
  12. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics, 9(1):79-103, 2009. Google Scholar
  13. Vin de Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Dualities in persistent (co)homology. Inverse Problems, 27(12):124003, 2011. Google Scholar
  14. Harm Derksen and Jerzy Weyman. Quiver representations. Notices of the AMS, 52(2):200-206, 2005. Google Scholar
  15. Tamal K. Dey and Tao Hou. Computing zigzag persistence on graphs in near-linear time. In 37th International Symposium on Computational Geometry, SoCG 2021, volume 189 of LIPIcs, pages 30:1-30:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  16. Tamal K. Dey and Tao Hou. Updating zigzag persistence and maintaining representatives over changing filtrations. arXiv preprint, 2021. URL:
  17. Tamal K. Dey, Woojin Kim, and Facundo Mémoli. Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its applications. In 38th International Symposium on Computational Geometry, SoCG 2022, volume 224 of LIPIcs, pages 34:1-34:17, 2022. Google Scholar
  18. Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 454-463. IEEE, 2000. Google Scholar
  19. Peter Gabriel. Unzerlegbare Darstellungen I. Manuscripta Mathematica, 6(1):71-103, 1972. Google Scholar
  20. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Google Scholar
  21. Petter Holme and Jari Saramäki. Temporal networks. Physics Reports, 519(3):97-125, 2012. Google Scholar
  22. Clément Maria and Steve Y. Oudot. Zigzag persistence via reflections and transpositions. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 181-199. SIAM, 2014. Google Scholar
  23. Clément Maria and Steve Y. Oudot. Computing zigzag persistent cohomology. arXiv preprint, 2016. URL:
  24. Clément Maria and Hannah Schreiber. Discrete morse theory for computing zigzag persistence. In Workshop on Algorithms and Data Structures, pages 538-552. Springer, 2019. Google Scholar
  25. Nikola Milosavljević, Dmitriy Morozov, and Primoz Skraba. Zigzag persistent homology in matrix multiplication time. In Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry, pages 216-225, 2011. Google Scholar
  26. Dmitriy Morozov. Dionysus2. URL:
  27. Steve Y. Oudot and Donald R. Sheehy. Zigzag zoology: Rips zigzags for homology inference. Foundations of Computational Mathematics, 15(5):1151-1186, 2015. Google Scholar
  28. The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial Board, 2015. URL:
  29. Simon Zhang, Mengbai Xiao, and Hao Wang. GPU-accelerated computation of Vietoris-Rips persistence barcodes. In 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  30. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249-274, 2005. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail