Computing discrete Morse complexes from simplicial complexes

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 extrema graphs for visualization of time-varying 3D scalar fields.

Betti splitting from a topological point of view

“…Apart its intrinsic interest, using Theorem 3.8, it is possible to write down an algorithm to test if a given standard decomposition of a simplicial complex is a Betti splitting over a given field k, see [8].…”

“…A version of this algorithm for 2-dimensional simplicial complexes has been developed and implemented in Python. The source code of this tool and of the other algorithms described in the paper can be found in [8]. In this algorithm, we take advantage of the fact that ∆ 1 ∩ ∆ 2 has dimension 1, i.e.…”

“…We provide obstructions to such kind of decomposition (Proposition 6.6). From this result, we can test algorithmically these obstructions, at least in dimension two, see [8].…”

Efficient Homology‐Preserving Simplification of High‐Dimensional Simplicial Shapes

“…The Simplex Tree has been defined for efficient extraction of boundary relations, as required in performing homology computations [ELZ02]. In [FIDF19], it has been shown that the Simplex Tree is generally five times more compact than the IG . Still, since it encodes all the simplices of the simplicial complex, the Simplex Tree also suffers from scalability issues.…”

“…Public‐domain implementations are available [Can14, Iur16]. In [FIDF19, FWD17], it has been shown that the data structure is more compact than the IG , requiring, on average, 80% less storage, and it is always more compact than the Simplex Tree, requiring from 30% of the storage on lower dimensional datasets, and a small fraction of the storage on higher dimensional datasets, where it can be observed a degenerate behaviour of the Simplex Tree as the complex dimension increases. A more compact representation for simplicial complexes embedded in the Euclidean space is provided by the Stellar tree [FWD17], as described in Section 6.…”