Skip to Main content Skip to Navigation
Journal articles

Efficient data structure for representing and simplifying simplicial complexes in high dimensions

Abstract : We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the {\em link condition} can be contracted while preserving the homotopy type. Our primary interest is to simplify {\em flag complexes} such as {\em Rips complexes} for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called {\em clique complexes}) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex $K$ by the graph $G$ of its edges together with the inclusion-minimal simplices in the set difference $\Flag{G} \setminus K$. We call these minimal simplices {\em blockers}. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.
Document type :
Journal articles
Complete list of metadata

Cited literature [21 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00785082
Contributor : Dominique Attali <>
Submitted on : Tuesday, February 5, 2013 - 12:11:16 PM
Last modification on : Thursday, November 19, 2020 - 1:01:02 PM
Long-term archiving on: : Monday, June 17, 2013 - 7:21:28 PM

File

2012-ijcga-Blockers.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Dominique Attali, André Lieutier, David Salinas. Efficient data structure for representing and simplifying simplicial complexes in high dimensions. International Journal of Computational Geometry and Applications, World Scientific Publishing, 2012, 22 (4), pp.279-303. ⟨10.1142/S0218195912600060⟩. ⟨hal-00785082⟩

Share

Metrics

Record views

562

Files downloads

586