A Space and Time Efficient Implementation for Computing Persistent Homology
Résumé
The persistent homology with $\Z_2$-coefficients coincides with the same for cohomology because of duality. Recently it has been observed that the cohomology based algorithms perform much better in practice than the orginally proposed homology based persistence algorithm. We implement a cohomology based algorithm that attaches binary labels called annotations with the simplices. This algorithm fits very naturally with a recently developed data structure called simplex tree to represent simplicial complexes. By taking advantages of several practical tricks such as representing annotations compactly with memory words, using a union-find structure that eliminates duplicate annotation vectors, and a lazy evaluation, we save both space and time cost for computations. The complexity of the procedure, in practice, depends almost linearly on the size of the simplicial complex and to the variables related to the maximal dimension of the local homology groups we maintain during the computation, which remain small in practice. We provide a theoretical analysis as well as a detailed experimental study of our implementation. Experimental results show that our implementation performs several times better than the existing state-of-the-art software for computing persistent homology in terms of both time and memory requirements and can handle very large complexes efficiently (several hundred million simplices in high-dimension).
Domaines
Géométrie algorithmique [cs.CG]
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