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A Global Reduction Based Algorithm for Computing Homology of Chain Complexes

A Global Reduction Based Algorithm for Computing Homology of Chain Complexes
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摘要 In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graphs. The organized reduction pairs lead to sequences of projection maps that reduce the number of generators while preserving the homology groups of the original chain complex. This sequencing of reduction pairs allows updating 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, our method gives the homology generators for a small additional cost. In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graphs. The organized reduction pairs lead to sequences of projection maps that reduce the number of generators while preserving the homology groups of the original chain complex. This sequencing of reduction pairs allows updating 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, our method gives the homology generators for a small additional cost.
作者 Madjid Allili David Corriveau Madjid Allili;David Corriveau(Department of Mathematics, Bishop’s University, Sherbrooke, Canada)
出处 《Advances in Pure Mathematics》 2016年第3期113-137,共25页 理论数学进展(英文)
关键词 Homology Algorithm Chain Complex Homology Generators Homology Algorithm Chain Complex Homology Generators
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