Abstract
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial complexes, and we define an efficient encoding for the discrete Morse gradient on the most compact of such representations. We theoretically compare methods based on reductions and coreductions for computing a discrete Morse gradient, proving that the combination of reductions and coreductions produces new mutually equivalent approaches. We design and implement a new algorithm for computing a discrete Morse complex on simplicial complexes. We show that our approach scales very well with the size and the dimension of the simplicial complex also through comparisons with the only existing public-domain algorithm for discrete Morse complex computation. We discuss applications to the computation of multi-parameter persistent homology and of extremum graphs for visualization of time-varying 3D scalar fields.
Highlights
In recent years, computational topology has become a fundamental tool for the analysis and visualization of scientific data
We evaluate the performances of the coreduction-based algorithm for Forman gradient computation and of the algorithm for computing the boundary maps that give a Morse complex, described in Section 2.3, which are based on the encoding of the original simplicial complex as an IA∗ data structure
We have studied different strategies to endow a simplicial complex with a Forman gradient through the use of homology-preserving operators and to extract the corresponding discrete Morse complex
Summary
Computational topology has become a fundamental tool for the analysis and visualization of scientific data. We propose a solution to compactly encode a Forman gradient attached to the IA∗ data structure Based on the latter encoding, we have defined and implemented an efficient, dimensionindependent, algorithm for computing a Forman gradient and for retrieving the discrete Morse complex defined by it, which is fundamental for computing, among others, homology and persistent homology.
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