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

Algorithmica volume 70, pages 406–427 (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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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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Authors and 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, JD., 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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  • DOI: https://doi.org/10.1007/s00453-014-9887-3

Keywords

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