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A New Construction of the Vietoris-Rips Complex

Antonio Rieser

TL;DR

The paper addresses the computational bottleneck of constructing Vietoris-Rips complexes by introducing New-VR, an inductive algorithm that exploits a combinatorial structure in the $k$-skeleton to identify $(k+1)$-simplices with a single additional edge check. It provides a rigorous complexity perspective, showing the critical step scales as $O\Big(n p^{(k-1)}\Big)$ compared to the $O(pn)$ behavior of the Incremental-VR, and reports 5–10x speedups on sparse Erdős–Rényi graphs and higher dimensions in experiments. By leveraging lexicographic minimal pair identification and a simplex-tree-based bookkeeping, New-VR achieves substantial practical gains while remaining embarrassingly parallelizable, offering a significant advance for efficient Vietoris-Rips construction in topological data analysis. The work also situates itself within the landscape of Zomorodian and Boissonnat–Maria algorithms and points to future work in multi-threaded, GPU-accelerated implementations and higher-dimensional persistence computations. Overall, the approach promises faster, scalable analysis of sparse data sets in TDA, enabling more widespread use of Vietoris-Rips complexes in practice.

Abstract

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.

A New Construction of the Vietoris-Rips Complex

TL;DR

The paper addresses the computational bottleneck of constructing Vietoris-Rips complexes by introducing New-VR, an inductive algorithm that exploits a combinatorial structure in the -skeleton to identify -simplices with a single additional edge check. It provides a rigorous complexity perspective, showing the critical step scales as compared to the behavior of the Incremental-VR, and reports 5–10x speedups on sparse Erdős–Rényi graphs and higher dimensions in experiments. By leveraging lexicographic minimal pair identification and a simplex-tree-based bookkeeping, New-VR achieves substantial practical gains while remaining embarrassingly parallelizable, offering a significant advance for efficient Vietoris-Rips construction in topological data analysis. The work also situates itself within the landscape of Zomorodian and Boissonnat–Maria algorithms and points to future work in multi-threaded, GPU-accelerated implementations and higher-dimensional persistence computations. Overall, the approach promises faster, scalable analysis of sparse data sets in TDA, enabling more widespread use of Vietoris-Rips complexes in practice.

Abstract

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 -skeleton of the complex in order to avoid unnecessary comparisons when identifying its -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 --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.
Paper Structure (7 sections, 5 theorems, 2 equations, 1 figure, 2 tables, 10 algorithms)

This paper contains 7 sections, 5 theorems, 2 equations, 1 figure, 2 tables, 10 algorithms.

Table of Contents

  1. Introduction
  2. Combinatorial preliminaries
  3. The New-VR Algorithm
  4. The Zomorodian Algorithms and the Simplex Tree
  5. The Complexity Difference Between the Incremental-VR and New-VR Algorithms
  6. Experiments
  7. Discussion and Future Work

Key Result

Lemma 2.1

Let $S$ be a set with $|S| = n+1$ for some natural number $n \geq 2$, and suppose that the subsets $S_0,S_1 \subset S$ satisfy $|S_i| = n$ and that $S_0 \neq S_1$. Let $t_i = S\backslash S_i$ for $i=0,1$. Then the only pair $\{a,b\} \subset S$ which is not contained in one of the $S_i$ is $\{t_0,t_1

Figures (1)

  • Figure 1: Graph which demonstrates the extra comparisons performed by Add-Coface. When adding node $10$ as a child to node $2$, the function merge-intersect will compare $10$ to all the nodes from $3$ to $10$. In New-Add-Cofaces, however, only a table lookup at the coordinates $(2,10)$ made, saving $7$ comparisons.

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 3.3
  • ...and 3 more