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Efficient Computation of Simplicial Homology through Acyclic Matching

We consider the problem of efficiently computing homology with Z coefficients as well as homology generators for simplicial complexes of arbitrary dimension. We analyze, compare and discuss the equivalence of different methods based on combining reductions, co reductions and discrete Morse theory. We show that the combination of these methods produces theoretically sound approaches which are mutually equivalent. One of these methods has been implemented for simplicial complexes by using a compact data structure for representing the complex and a compact encoding of the discrete Morse gradient. We present experimental results and discuss further developments.