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Computing Zigzag Vineyard Efficiently Including Expansions and Contractions

Authors Tamal K. Dey, Tao Hou

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LIPIcs.SoCG.2024.49.pdf
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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

Funding

This research is partially supported by NSF grants CCF 2049010 and 2301360.

Cite AsGet BibTex

Tamal K. Dey and Tao Hou. Computing Zigzag Vineyard Efficiently Including Expansions and Contractions. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.49

Abstract

Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. [Cohen-Steiner et al., 2006]. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm [Tamal K. Dey and Tao Hou, 2022], expansions and contractions pose difficulty in breaking the barrier of cubic complexity [Dey and Hou, 2022]. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
Keywords
  • zigzag persistence
  • vines and vineyard
  • update operations

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References

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