Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

Harker, Shaun; Mischaikow, Konstantin; Mrożek, Marian; Nanda, Vidit

doi:10.1007/s10208-013-9145-0

Cited by 80 publications

(104 citation statements)

References 27 publications

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“…For example, two algorithms which are similar but slightly different to the one presented above are described in [17,18]. We plan to discuss and compare a range of algorithms in a subsequent article [2].…”

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confidence: 98%

Computing fundamental groups from point clouds

Brendel

Dłotko

Ellis

et al. 2015

AAECC

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We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the 3-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how the algorithm can be used to infer information about the fundamental group π 1 (K ) of a metric space K using only a finite point cloud X sampled from the space. In the special case where K is a d-dimensional compact manifold K ⊂ R d , we consider the closure of the complement of K in the d-sphere M K = S d \ K . For a base-point x in the boundary ∂ M K of the manifold M K one can attempt to determine, from the point cloud X , the induced homomorphism of fundamental groups φ : π 1 (∂ M K , x) → π 1 (M K , x) in the category of finitely presented groups. We illustrate a computer implementation for K a small closed tubular neigh-P.D. 123 28 P. Brendel et al.bourhood of a tame knot in R 3 . In this case the homomorphism φ is known to be a complete ambient isotopy invariant of the knot. We observe that low-index subgroups of finitely presented groups provide useful invariants of φ. In particular, the first integral homology of subgroups G < π 1 (M K ) of index at most six suffices to distinguish between all prime knots with 11 or fewer crossings (ignoring chirality). We plan to provide formal time estimates for our algorithm and characteristics of a high performance C++ implementation in a subsequent paper. The prototype computer implementation of the present paper has been written in the interpreted gap programming language for computational algebra.

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confidence: 98%

Computing fundamental groups from point clouds

Brendel

Dłotko

Ellis

et al. 2015

AAECC

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“…The first approach [18], that we call the coreduction-based algorithm, is based on the construction of an acyclic matching on a simplicial complex Σ by using coreduction pairs and removals of free simplices, where a free simplex is a simplex with an empty boundary. To obtain an acyclic matching (A, w : Q → K) of Σ, the approach in [18] initializes first sets A, Q, K and map w as null.…”

Section: A Using Coreduction Sequences and Reduction Sequencesmentioning

confidence: 99%

Efficient Computation of Simplicial Homology through Acyclic Matching

Fugacci

Iuricich

Floriani

2014

2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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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, coreductions 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.

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“…At the moment the best worst case complexity analysis of the SNF algorithm gives a lower bound that is super cubical in the dimensions of the chain complex. We developed a preprocessing technique based on discrete Morse theory that provides a linear time reduction in the size of the original complex after which the SNF algorithm is applied [5,6].…”

Section: Approximation Of Mapsmentioning

confidence: 99%

Databases for the Global Dynamics of Multiparameter Nonlinear Systems

Mischaikow¹

2014

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Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

Cited by 80 publications

References 27 publications

Computing fundamental groups from point clouds

Computing fundamental groups from point clouds

Efficient Computation of Simplicial Homology through Acyclic Matching

Databases for the Global Dynamics of Multiparameter Nonlinear Systems

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