Discrete & Computational Geometry

, Volume 33, Issue 2, pp 249–274 | Cite as

Computing Persistent Homology

  • Afra Zomorodian
  • Gunnar Carlsson


We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.


Computational Mathematic Simplicial Complex Homology Group Polynomial Ring Arbitrary Dimension 
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Copyright information

© Springer Science + Business Media Inc. 2004

Authors and Affiliations

  1. 1.Department of Computer Science, Stanford University, Stanford, CA 94305USA
  2. 2.Department of Mathematics, Stanford University, Stanford, CA 94305USA

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