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 

References

  1. 1.
    Bauer, U., Kerber, M., Reininghaus, J.: PHAT. https://code.google.com/p/phat/ (2013)
  2. 2.
    Bauer, U., Kerber, M., Reininghaus, J.: Clear and compress: computing persistent homology in chunks. In: Topological Methods in Data Analysis and Visualization III, pp. 103–117 (2014)Google Scholar
  3. 3.
    Bauer, U., Kerber, M., Reininghaus, J., Wagner, H.: PHAT—persistent homology algorithms toolbox. In: Mathematical Software—ICMS 2014—4th International Congress, Seoul, South Korea, August 5–9, 2014. Proceedings, pp. 137–143 (2014)Google Scholar
  4. 4.
    Boissonnat, J.-D., Maria, C.: The simplex tree: an efficient data structure for general simplicial complexes. Algorithmica 70(3), 406–427 (2014)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Busaryev, O., Cabello, S., Chen, C., Dey, T.K., Wang, Y.: Annotating simplices with a homology basis and its applications. In: SWAT, pp. 189–200 (2012)Google Scholar
  6. 6.
    Chen, C., Kerber, M.: Persistent homology computation with a twist. In: Proceedings 27th European Workshop on Computational Geometry (2011)Google Scholar
  7. 7.
    Chen, C., Kerber, M.: An output-sensitive algorithm for persistent homology. Comput. Geom. 46(4), 435–447 (2013)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persistent (co)homology. CoRR arXiv:1107.5665 (2011)
  10. 10.
    de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Persistent cohomology and circular coordinates. Discrete Comput. Geom. 45(4), 737–759 (2011)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. In: Symposium on Computational Geometry, p. 345 (2014)Google Scholar
  12. 12.
    Edelsbrunner, H., Harer, J.: Computational Topology—An Introduction. American Mathematical Society, Providence, RI (2010)MATHGoogle Scholar
  13. 13.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Maria, C.: Gudhi, Simplicial Complexes and Persistent Homology Packages. https://project.inria.fr/gudhi/software/
  15. 15.
    Maria, C., Boissonnat, J.-D., Glisse, M., Yvinec, M.: The Gudhi library: Simplicial complexes and persistent homology. In: International Congress on Mathematical Software, pp. 167–174 (2014)Google Scholar
  16. 16.
    Milosavljevic, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Symposium on Computational Geometry (2011)Google Scholar
  17. 17.
  18. 18.
    Morozov, D.: Persistence algorithm takes cubic time in worst case. In: BioGeometry News, Department of Computer Science, Duke University (2005)Google Scholar
  19. 19.
    Zomorodian, A., Carlsson, G.E.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)MATHMathSciNetCrossRefGoogle Scholar

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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