The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes

Algorithmica volume 70, pages406–427(2014)Cite this article

Abstract

This paper introduces a new data structure, called simplex tree, to represent abstract simplicial complexes of any dimension. All faces of the simplicial complex are explicitly stored in a trie whose nodes are in bijection with the faces of the complex. This data structure allows to efficiently implement a large range of basic operations on simplicial complexes. We provide theoretical complexity analysis as well as detailed experimental results. We more specifically study Rips and witness complexes.

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References

  1. 1.

    Achtert, E., Böhm, C., Kröger, P., Kunath, P., Pryakhin, A., Renz, M.: Efficient reverse k-nearest neighbor search in arbitrary metric spaces. In: SIGMOD Conference, pp. 515–526 (2006)

  2. 2.

    Attali, D., Lieutier, A., Salinas, D.: Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Int. J. Comput. Geom. Appl. 22(4), 279–304 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Attali, D., Lieutier, A., Salinas, D.: Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes. Comput. Geom. 46(4), 448–465 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Bentley, J.L., Sedgewick, R.: Fast algorithms for sorting and searching strings. In: SODA, pp. 360–369 (1997)

  5. 5.

    Boissonnat, J.-D., Dey, T.K., Maria, C.: The compressed annotation matrix: an efficient data structure for computing persistent cohomology. In: ESA (2013)

  6. 6.

    Boissonnat, J.-D., Guibas, L.J., Oudot, S.: Manifold reconstruction in arbitrary dimensions using witness complexes. Discret. Comput. Geom. 42(1), 37–70 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Boissonnat, J.-D., Maria, C.: The simplex tree: an efficient data structure for general simplicial complexes. In: ESA, pp. 731–742 (2012)

  8. 8.

    Brisson, E.: Representing geometric structures in d dimensions: Topology and order. Discret. Comput. Geom. 9, 387–426 (1993)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008)

    Article  Google Scholar 

  10. 10.

    Chazal, F., Oudot, S.: Towards persistence-based reconstruction in euclidean spaces. In: SoCG, pp. 232–241 (2008)

  11. 11.

    de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Eurographics Symposium on Point-Based Graphics (2004)

  12. 12.

    Dey, T.K., Fan, F., Wang, Y.: Computing Topological Persistence for Simplicial Maps. CoRR, abs/1208.5018 (2012)

  13. 13.

    Edelsbrunner, H., Harer, J.: Computational Topology—An Introduction. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  14. 14.

    Jacobson, G.: Space-efficient static trees and graphs. In: FOCS, pp. 549–554 (1989)

  15. 15.

    Lee, A.B., Pedersen, K.S., Mumford, D.: The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vis. 54(1–3), 83–103 (2003)

    MATH  Article  Google Scholar 

  16. 16.

    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. J. Comput. Geometry Appl. 4(3), 275–324 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Martin, S., Thompson, A., Coutsias, E.A., Watson, J.-P.: Topology of cyclo-octane energy landscape. J. Chem. Phys. 132(23), 234115 (2010)

    Google Scholar 

  18. 18.

    Morozov, D.: Dionysus. http://www.mrzv.org/software/dionysus/

  19. 19.

    Mount, D.M., Arya, S.: ANN: Approximate Nearest Neighbors Library. http://www.cs.sunysb.edu/algorith/implement/ANN/implement.shtml

  20. 20.

    Sexton, H., Johansson, M.V.: JPlex, 2009. http://comptop.stanford.edu/programs/jplex/

  21. 21.

    SGI. Standard template library programmer’s guide. http://www.sgi.com/tech/stl/

  22. 22.

    The Stanford 3d scanning repository. http://graphics.stanford.edu/data/3Dscanrep/

  23. 23.

    Zomorodian, A.: The tidy set: a minimal simplicial set for computing homology of clique complexes. In: SoCG, pp. 257–266 (2010)

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Acknowledgments

The authors thanks A. Ghosh, S. Hornus, D. Morozov and P. Skraba for discussions that led to the idea of representing simplicial complexes by tries. They especially thank S. Hornus for sharing his notes with us. They also thank S. Martin and V. Coutsias for providing the cyclo-octane data set. This research has been partially supported by the 7th Framework Programme for Research of the European Commission, under FET-Open Grant Number 255827 (CGL Computational Geometry Learning).

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Affiliations

  1. INRIA Sophia Antipolis-Méditerranée, 2004, route des Lucioles, 93 06902, Sophia Antipolis Cedex, France

    Jean-Daniel Boissonnat & Clément Maria

Authors
  1. Jean-Daniel Boissonnat
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  2. Clément Maria
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Correspondence to Clément Maria.

Appendix: Additional Experiments

Appendix: Additional Experiments

In this section we provide more experiments on the running time of the algorithms for constructing Rips complexes and relaxed witness complexes on all datasets. The datasets used are described in Table 1.

See Tables 2 and 3.

Table 2 Timings \(T_{\text {Rips}}\) for the construction of the Rips complex on the data sets and size of the simplicial complexes \(|{\mathcal {K}}|\), for different values of the parameter \(r\)
Full size table
Table 3 Timings \(T_{{\mathrm{Wit}}^{\rho }}\) for the construction of the relaxed witness complex on the data sets and size of the simplicial complexes \(|{\mathcal {K}}|\), for different values of the parameter \(\rho \)
Full size table

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Boissonnat, J., Maria, C. The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes. Algorithmica 70, 406–427 (2014). https://doi.org/10.1007/s00453-014-9887-3

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Keywords

  • Simplicial complexes
  • Data structure
  • Computational topology
  • Flag complexes
  • Witness complexes