Abstract
We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation.
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Notes
- 1.
The running time of the third step could be lowered to g ω, where ω is the matrix-multiplication exponent, using the method of [17].
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Acknowledgements
The authors thank Chao Chen, Herbert Edelsbrunner, and Hubert Wagner for helpful discussions. This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP and the Max Planck Center for Visual Computing and Communication.
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Bauer, U., Kerber, M., Reininghaus, J. (2014). Clear and Compress: Computing Persistent Homology in Chunks. In: Bremer, PT., Hotz, I., Pascucci, V., Peikert, R. (eds) Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-04099-8_7
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DOI: https://doi.org/10.1007/978-3-319-04099-8_7
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