Fast construction of the Vietoris-Rips complex

@article{Zomorodian2010FastCO,
  title={Fast construction of the Vietoris-Rips complex},
  author={Afra Zomorodian},
  journal={Comput. Graph.},
  year={2010},
  volume={34},
  pages={263-271}
}
  • A. Zomorodian
  • Published 1 June 2010
  • Computer Science, Mathematics
  • Comput. Graph.
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173 Citations
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Methods Citations
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Results Citations
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References

SHOWING 1-10 OF 49 REFERENCES

The tidy set: a minimal simplicial set for computing homology of clique complexes

The tidy set is introduced, a minimal simplicial set that captures the topology of a simplicial complex that is particularly effective for computing the homology of clique complexes.

Topological estimation using witness complexes

This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.

Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension

A new implementation of the well-known incremental algorithm for constructing Delaunay triangulations in any dimension is described and a modification of the algorithm that uses and stores only theDelaunay graph (the edges of the full triangulation) is proposed.

Topology of random clique complexes

Topological Persistence and Simplification

Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.

A New Algorithm for Generating All the Maximal Independent Sets

This paper presents a new efficient algorithm for generating all the maximal independent sets, for which processing time and memory space are bounded by $O(nm\mu)$ and $O (n+m)$, respectively, where n, m, and $\mu$ are the numbers of vertices, edges, and maximalIndependent sets of a graph.

Combinatorial Algebraic Topology

  • D. Kozlov
  • Mathematics
    Algorithms and computation in mathematics
  • 2008
Concepts of Algebraic Topology, Applications of Spectral Sequences to Hom Complexes, and Structural Theory of Morphism Complexes are presented.

Computing Persistent Homology

The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.

The flow complex: a data structure for geometric modeling

Reporting maximal cliques: new insights into an old problem

The bottom-line of this study is the investigation of the trade-off between output sensitivity and the overhead associated to the identification of dominated nodes.