2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing 2014
DOI: 10.1109/synasc.2014.84
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Abstract: 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… Show more

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Cited by 10 publications
(11 citation statements)
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References 26 publications
(36 reference statements)
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“…An efficient implementation of [24], focused on regular grids, is discussed in [25] while, for simplicial complexes, the same algorithm as been extended to triangle [26] and tetrahedral meshes [27]. The first dimension independent implementation for simplicial complexes is presented in [28].…”
Section: Computing Persistent Homologymentioning
confidence: 99%
“…An efficient implementation of [24], focused on regular grids, is discussed in [25] while, for simplicial complexes, the same algorithm as been extended to triangle [26] and tetrahedral meshes [27]. The first dimension independent implementation for simplicial complexes is presented in [28].…”
Section: Computing Persistent Homologymentioning
confidence: 99%
“…An encoding for the Forman gradient on such data structures associates the gradient pairs to the top simplices, and is called a compact gradient. The use of such data structures with the compact gradient makes the computation of the Forman gradient and of the Morse and Morse-Smale complexes feasible on simplicial complexes on large size [WIFD13,FID14,FIDW14].…”
Section: Forman Gradient Encoding On Simplicial Complexesmentioning
confidence: 99%
“…Comparison The two approaches described above are dual to each other. The equivalence in the use of reductions or coreductions has been proven in [FID14] in the context of simplicial complexes. Any Forman gradient obtained through a sequence of reductions and removals of top cells can be obtained through a suitable sequence of coreductions and removals of free cells, and vice versa.…”
Section: Algorithms Based On Reduction and Coreductionmentioning
confidence: 99%
See 1 more Smart Citation
“…We describe a third method initially formulated in [16], obtained by interleaving reductions and coreductions, and we prove the equivalence of all three techniques. This equivalence will provide us the freedom to implement the method that best fits any given data structure.…”
Section: Introductionmentioning
confidence: 98%