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

Boissonnat, Jean‐Daniel; Maria, Clément

doi:10.1007/s00453-014-9887-3

Cited by 74 publications

(75 citation statements)

References 15 publications

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“…Before we can address the problem of considering chains on a simplicial complex we first need to have a model of a simplicial complex. For this, we decided to use a simplex tree model (Boissonnat & Maria, 2014) provided by the GUDHI library (The GUDHI Project, 2015) as it provides a compact representation of a simplicial complex (the number of nodes in the tree is in bijection with the number of simplices) which allows us to quickly get the enumeration of a given simplex σ . Indeed, the complexity of calculating…”

Section: Implementation Detailsmentioning

confidence: 99%

An algorithm for calculating top-dimensional bounding chains

Carvalho

Vejdemo-Johansson

Kragić

et al. 2018

PeerJ Computer Science

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We describe the Coefficient-Flow algorithm for calculating the bounding chain of an (n−1)-boundary on an n-manifold-like simplicial complex S. We prove its correctness and show that it has a computational time complexity of O(|S (n−1) |) (where S (n−1) is the set of (n − 1)-faces of S). We estimate the big-O coefficient which depends on the dimension of S and the implementation. We present an implementation, experimentally evaluate the complexity of our algorithm, and compare its performance with that of solving the underlying linear system.

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Section: Implementation Detailsmentioning

confidence: 99%

An algorithm for calculating top-dimensional bounding chains

Carvalho

Vejdemo-Johansson

Kragić

et al. 2018

PeerJ Computer Science

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“…This encompasses a large class of complexes encountered in practice and if the number of maximal simplices is small, CSD is a very efficient data structure as Γ 0 ≤ k. i+1 as given by π. First, we remark that this procedure is very expensive using ST, as vertex collapse is more expensive than the edge contraction operation, and Boissonnat and Maria [BM14b] provide an essentially optimal algorithm running in O(md log n) time for edge contraction.…”

Section: Performance Of Csd For Flag Complexesmentioning

confidence: 99%

“…We note here that typically P is a relatively small set. For example, in the experiments performed by Boissonnat and Maria (Table 1 of [BM14b]), we note that the cardinality of the witness set is about a few ten thousands while the number of simplices in the complex is over a hundred million. Therefore, this provides practical evidence of the compact representation of Del ρ (Q, P ) through CSD.…”

Section: Construction Of Relaxed Delaunay Complexesmentioning

confidence: 99%

“…Under the assumption that for any x, , D(x, ) could be computed in O(1) time (i.e., D is computed as part of the preprocessing), Boissonnat and Maria [BM14b] described an algorithm to construct the ST representation of the relaxed witness complex. Their algorithm can be easily adapted to construct Del ρ w (Q, P ) in time O(tmd log n).…”

Section: Construction Of Relaxed Delaunay Complexesmentioning

confidence: 99%

“…The most common representation of simplicial complexes uses the Hasse diagram of the complex that has one node per simplex and an edge between any pair of incident simplices whose dimensions differ by one. A more compact data structure, called Simplex Tree (ST), was proposed recently by Boissonnat and Maria [BM14b]. The nodes of both the Hasse diagram and ST are in bijection with the simplices (of all dimensions) of the simplicial complex.…”

Section: Introductionmentioning

confidence: 99%

See 2 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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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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Persistent 1-Cycles: Definition, Computation, and Its Application

Dey

Hou

Mandal

2018

Lecture Notes in Computer Science

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Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles, for H 1 -persistent homology with Z 2 coefficients. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.

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

Cited by 74 publications

References 15 publications

An algorithm for calculating top-dimensional bounding chains

An algorithm for calculating top-dimensional bounding chains

An Efficient Representation for Filtrations of Simplicial Complexes

Persistent 1-Cycles: Definition, Computation, and Its Application

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