Homology Computations via Acyclic Subspace

  • Piotr Brendel
  • Paweł Dłotko
  • Marian Mrozek
  • Natalia Żelazna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7309)


Homology computations recently gain vivid attention in science. New methods, enabling fast and memory efficient computations are needed in order to process large simplicial complexes. In this paper we present the acyclic subspace reduction algorithm adapted to simplicial complexes. It provides fast and memory efficient preprocessing of the given data. A variant of the method for distributed computations is also presented. As a result, Betti numbers can be effectively computed.


Homology algorithms reduction algorithms acyclic subspace method 


  1. 1.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press and McGraw-Hill (1990)Google Scholar
  2. 2.
    Dłotko, P., Specogna, R.: Efficient Cohomology Computation for Electromagnetic Modeling. Computer Modeling in Engineering & Sciences 60(3), 247–277 (2010)Google Scholar
  3. 3.
    Dłotko, P.: Acyclic configurations for boundary elements of 3 and 4 dimensional simplices,
  4. 4.
    Gropp, W., Lusk, E., Skjellum, A.: Using MPI: Portable Parallel Programming with the Message-Passing Interface. MIT Press (1990)Google Scholar
  5. 5.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology, Appl. Math. Sci., vol. 157. Springer, New York (2004)Google Scholar
  6. 6.
    Kaczynski, T., Mrozek, M., Ślusarek, M.: Homology computation by reduction of chain complexes. Computers and Math. Appl. 35, 59–70 (1998)zbMATHCrossRefGoogle Scholar
  7. 7.
    Mrozek, M., Batko, B.: Coreduction Homology Algorithm. Discrete and Computational Geometry 41, 96–118 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Mrozek, M., Pilarczyk, P., Żelazna, N.: Homology algorithm based on acyclic subspace. Computers and Mathematics with Applications 55, 2395–2412 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Munkres, J.R.: Elements of Algebraic Topology. Perseus Publishing, Cambridge (1984)zbMATHGoogle Scholar
  10. 10.
    Computer Assisted Proofs in Dynamics,
  11. 11.
    The RedHom homology algorithms library,

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Piotr Brendel
    • 1
  • Paweł Dłotko
    • 1
  • Marian Mrozek
    • 1
  • Natalia Żelazna
    • 2
  1. 1.Institute of Computer ScienceJagiellonian UniversityPoland
  2. 2.Motorola SolutionsPoland

Personalised recommendations