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

  • Jean-Daniel Boissonnat
  • Tamal K. Dey
  • Clément Maria
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)


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 substancially 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.


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  1. 1.
    Bauer, U., Kerber, M., Reininghaus, J.: Clear and compress: Computing persistent homology in chunks. arXiv/1303.0477 (2013)Google Scholar
  2. 2.
    Boissonnat, J.-D., Maria, C.: The simplex tree: An efficient data structure for general simplicial complexes. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 731–742. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Busaryev, O., Cabello, S., Chen, C., Dey, T.K., Wang, Y.: Annotating simplices with a homology basis and its applications. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 189–200. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Chen, C., Kerber, M.: Persistent homology computation with a twist. In: Proceedings 27th European Workshop on Computational Geometry (2011)Google Scholar
  5. 5.
    Chen, C., Kerber, M.: An output-sensitive algorithm for persistent homology. Comput. Geom. 46(4), 435–447 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete & Computational Geometry 37(1), 103–120 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persistent (co)homology. Inverse Problems 27, 124003 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Persistent cohomology and circular coordinates. Discrete Comput. Geom. 45, 737–759 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. CoRR, abs/1208.5018 (2012)Google Scholar
  10. 10.
    Edelsbrunner, H., Harer, J.: Computational Topology - an Introduction. American Mathematical Society (2010)Google Scholar
  11. 11.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Milosavljevic, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Symposium on Comp. Geom. (2011)Google Scholar
  13. 13.
    Morozov, D.: Persistence algorithm takes cubic time in worst case. In: BioGeometry News, Dept. Comput. Sci., Duke Univ. (2005)Google Scholar
  14. 14.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete & Computational Geometry 33(2), 249–274 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jean-Daniel Boissonnat
    • 1
  • Tamal K. Dey
    • 2
  • Clément Maria
    • 1
  1. 1.INRIA Sophia Antipolis-MéditerranéeFrance
  2. 2.The Ohio State UniversityUSA

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