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Robust Detection of Singularities in Vector Fields Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Clear and Compress: Computing Persistent Homology in Chunks

Authors : Ulrich Bauer, Michael Kerber, Jan Reininghaus

Published in: Topological Methods in Data Analysis and Visualization III

Publisher: Springer International Publishing

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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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previous chapter Distributed Contour Trees
next chapter Parallel Computation of Nearly Recurrent Components of Piecewise Constant Vector Fields
Footnotes
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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D. Günther, J. Reininghaus, H. Wagner, I. Hotz, Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theory. Vis. Comput. 1–11 (2012)
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N. Milosavljević, D. Morozov, P. Škraba, Zigzag persistent homology in matrix multiplication time, in Proceedings of the 27th Annual Symposium on Computational Geometry, Paris, 2011, pp. 216–225
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D. Morozov, Persistence algorithm takes cubic time in the worst case, in BioGeometry News (Duke Computer Science, Durham, 2005)
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Metadata
Title
Clear and Compress: Computing Persistent Homology in Chunks
Authors
Ulrich Bauer
Michael Kerber
Jan Reininghaus
Copyright Year
2014
Book
Topological Methods in Data Analysis and Visualization III

Print ISBN: 978-3-319-04098-1

Electronic ISBN: 978-3-319-04099-8

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-04099-8_7

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