Computing Homology: A Global Reduction Approach
A new algorithm to compute the homology of a finitely generated chain complex is proposed in this work. It is based on grouping algebraic reductions of the complex into structures that can be encoded as directed acyclic graphs. This leads to sequences of projection maps that reduce the number of generators in the complex while preserving its homology. This organization of reduction pairs allows to update the boundary information in a single step for a whole set of reductions which shows impressive gains in computational performance compared to existing methods. In addition, this method gives the homology generators for a small additional cost.
KeywordsHomology Computation Reduction Directed Acyclic Graphs Generators
- 1.Allili, M., Corriveau, D., Ziou, D.: Morse Homology Descriptor for Shape Characterization. Proceedings of the 17th ICPR 4, 27–30 (2004)Google Scholar
- 6.Storjohann, A.: Near Optimal Algorithms for Computing Smith Normal Forms of Integer Matrices. In: Proceedings of 1996 International Symposium on Symbolic and Algebraic Computation, ISSAC 1996, pp. 267–274 (1996)Google Scholar
- 10.Mrozek, M., Batko, B.: Coreduction Homology Algorithm. Discrete and Computational Geometry (in press)Google Scholar