Building Efficient and Compact Data Structures for Simplicial Complexes

Boissonnat, Jean‐Daniel; Karthik, C.; Tavenas, Sébastien

doi:10.1007/s00453-016-0207-y

Cited by 32 publications

(36 citation statements)

References 23 publications

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“…In this case, |M | = k and CSD requires O(kd log k) memory space, which matches the lower bound in Theorem 2.1, when k = n O(1) . In this case, CSD is very similar to SAL [BKT16]. Marc Glisse and Sivaprasad S. [GS] have performed experiments on SAL and concluded that it is not only smaller in size but also faster than the Simplex Tree in performing insertion, removal, and edge contraction.…”

Section: Summary We Summarize Inmentioning

confidence: 82%

“…It follows that determining if σ is in K reduces to computing A σ and checking whether it is non-empty. This procedure is very similar to the analogous procedure using SAL [BKT16]. Therefore, membership of a simplex can be determined in time O(d σ Γ 0 log k).…”

Section: Membership Of a Simplexmentioning

confidence: 99%

“…For example, the critical simplices in the example described in Figure 1 Theorem 2.1. [BKT16] Consider the class of all ddimensional simplicial complexes with n vertices containing k maximal simplices, where d ≥ 2 and k ≥ n+1, and consider any data structure that can represent the simplicial complexes of this class. Such a data structure requires log ( n/2 d+1 ) k−n bits to be stored.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section, we introduce the Critical Simplex Diagram, CSD(K), which is an evolved version of SAL [BKT16]. CSD is a collection of n arrays that correspond to the n vertices of K. The elements of an array, referred to as nodes in the rest of the paper, correspond to copies of the vertex of K associated to the array.…”

Section: Critical Simplex Diagrammentioning

confidence: 99%

“…A dual idea is to store only the simplices that are maximal for the inclusion. Following this last idea, Boissonnat et al [BKT16] introduced a new data structure, called Simplex Array List, which was the first data structure whose size and query time were sensitive to the geometry of the simplicial complex. SAL was shown to outperform ST for a large class of simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Representation for Filtrations of Simplicial Complexes

Boissonnat¹,

Karthik²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A filtration over a simplicial complex K is an ordering of the simplices of K such that all prefixes in the ordering are subcomplexes of K. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest.This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. [SoCG '15] considered storing the simplices that are maximal for the inclusion and Attali et al. [IJCGA '12] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the filtration of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex.In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [SoCG '15]. Our data structure allows to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Next, we show that the CSD representation admits the following construction algorithms.• A new edge-deletion algorithm for the fast construction of Flag complexes, which only depends on the number of critical simplices and the number of vertices.• A new matrix-parsing algorithm to quickly construct relaxed Delaunay complexes, depending only on the number of witnesses and the dimension of the complex.

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Section: Summary We Summarize Inmentioning

confidence: 82%

Section: Membership Of a Simplexmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Critical Simplex Diagrammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Representation for Filtrations of Simplicial Complexes

Boissonnat¹,

Karthik²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Topological Data Analysis

Boissonnat

Chazal²,

Michel

2022

Mathematics in Industry

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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