Homology algorithm based on acyclic subspace
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TL;DR: A new reduction algorithm is presented based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology, which proves itself to be significantly more efficient than other available cubical homology algorithms.
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Abstract: We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.
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Citations
Zigzag persistent homology in matrix multiplication time
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- 13 Jun 2011
TL;DR: A new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions, which takes O(n3) time in the worst case.
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TL;DR: A new Morse theoretic preprocessing framework for deriving chain maps from set-valued maps is introduced, and hence an effective scheme for computing the morphism induced on homology by the approximated continuous function is provided.
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Multivalued maps as a tool in modeling and rigorous numerics
TL;DR: In this paper, the fixed point theory of multivalued maps can be classified into several areas: (1) game theory and mathematical economics; (2) Discontinuous differential equations, differential inclusions, and optimal control; (3) Computing homology of maps; (4) Computer assisted proofs in dynamics; (5) Digital imaging.
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