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Algorithmica

, Volume 73, Issue 3, pp 607–619 | Cite as

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

  • Jean-Daniel Boissonnat
  • Tamal K. Dey
  • Clément Maria
Article

Abstract

Persistent homology with coefficients in a field \(\mathbb {F}\) coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substantially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology.

Keywords

Persistent cohomology Annotation Data structure  Simplicial complex Algorithm Implementation 

Notes

Acknowledgments

This research is partially supported by the 7th Framework Programme for Research of the European Commission, under FET-Open Grant No. 255827 (CGL Computational Geometry Learning). This research is also partially supported by NSF (National Science Foundation, USA) Grants CCF-1048983 and CCF-1116258.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jean-Daniel Boissonnat
    • 1
  • Tamal K. Dey
    • 2
  • Clément Maria
    • 1
  1. 1.INRIA Sophia Antipolis-MéditerranéeSophia Antipolis CedexFrance
  2. 2.The Ohio State UniversityColumbusUSA

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