A New Construction of the Vietoris-Rips Complex
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909226549379072 |
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| author | Rieser, Antonio |
| author_facet | Rieser, Antonio |
| contents | We present a new, inductive construction of the Vietoris-Rips complex, in which we take advantage of a small amount of unexploited combinatorial structure in the $k$-skeleton of the complex in order to avoid unnecessary comparisons when identifying its $(k+1)$-simplices. In doing so, we achieve a significant reduction in the number of comparisons required to construct the Vietoris-Rips compared to state-of-the-art algorithms, which is seen here by examining the computational complexity of the critical step in the algorithms. In experiments comparing a C/C++ implementation of our algorithm to the GUDHI v3.9.0 software package, this results in an observed $5$-$10$-fold improvement in speed of on sufficiently sparse Erdős-Rényi graphs with the best advantages as the graphs become sparser, as well as for higher dimensional Vietoris-Rips complexes. We further clarify that the algorithm described in Boissonnat and Maria (https://doi.org/10.1007/978-3-642-33090-2_63) for the construction of the Vietoris-Rips complex is exactly the Incremental Algorithm from Zomorodian (https://doi.org/10.1016/j.cag.2010.03.007), albeit with the additional requirement that the result be stored in a tree structure, and we explain how these techniques are different from the algorithm presented here. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_07191 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A New Construction of the Vietoris-Rips Complex Rieser, Antonio Combinatorics Computational Geometry Data Structures and Algorithms Algebraic Topology Geometric Topology 05E45, 06A07, 68R05, 68R10, 05C85, 62R40 We present a new, inductive construction of the Vietoris-Rips complex, in which we take advantage of a small amount of unexploited combinatorial structure in the $k$-skeleton of the complex in order to avoid unnecessary comparisons when identifying its $(k+1)$-simplices. In doing so, we achieve a significant reduction in the number of comparisons required to construct the Vietoris-Rips compared to state-of-the-art algorithms, which is seen here by examining the computational complexity of the critical step in the algorithms. In experiments comparing a C/C++ implementation of our algorithm to the GUDHI v3.9.0 software package, this results in an observed $5$-$10$-fold improvement in speed of on sufficiently sparse Erdős-Rényi graphs with the best advantages as the graphs become sparser, as well as for higher dimensional Vietoris-Rips complexes. We further clarify that the algorithm described in Boissonnat and Maria (https://doi.org/10.1007/978-3-642-33090-2_63) for the construction of the Vietoris-Rips complex is exactly the Incremental Algorithm from Zomorodian (https://doi.org/10.1016/j.cag.2010.03.007), albeit with the additional requirement that the result be stored in a tree structure, and we explain how these techniques are different from the algorithm presented here. |
| title | A New Construction of the Vietoris-Rips Complex |
| topic | Combinatorics Computational Geometry Data Structures and Algorithms Algebraic Topology Geometric Topology 05E45, 06A07, 68R05, 68R10, 05C85, 62R40 |
| url | https://arxiv.org/abs/2301.07191 |