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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 ∆-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.
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Apex Representatives
Given a zigzag filtration, we want to find its barcode representatives, i.e., a compatible choice of bases for the homology groups that diagonalize the linear maps in the zigzag. To achieve this, we convert the input zigzag to a levelset zigzag of a real-valued function. This function generates a Mayer-Vietoris pyramid of spaces, which generates an infinite strip of homology groups. We call the origins of indecomposable (diamond) summands of this strip their apexes and give an algorithm to find representative cycles in these apexes from ordinary persistence computation. The resulting representatives map back to the levelset zigzag and thus yield barcode representatives for the input zigzag. Our algorithm for lifting a p-dimensional cycle from ordinary persistence to an apex representative takes O(p ⋅ m log m) time. From this we can recover zigzag representatives in time O(log m + C), where C is the size of the output.
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- Award ID(s):
- 2437030
- PAR ID:
- 10651836
- Editor(s):
- Aichholzer, Oswin; Wang, Haitao
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 332
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 40:1-40:16
- Subject(s) / Keyword(s):
- zigzag persistent homology Mayer-Vietoris pyramid cycle representatives Theory of computation → Computational geometry Mathematics of computing → Algebraic topology
- Format(s):
- Medium: X Size: 16 pages; 1141016 bytes Other: application/pdf
- Size(s):
- 16 pages 1141016 bytes
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/lipics.socg.2025.40
