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 Erd\H{o}s-R\'enyi 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.

翻译：本文提出了一种新的、归纳式的Vietoris-Rips复形构造方法。该方法利用了复形$k$维骨架中少量未被发掘的组合结构，从而在识别其$(k+1)$-单形时避免了不必要的比较运算。通过这种方式，与现有先进算法相比，我们在构建Vietoris-Rips复形时显著减少了所需的比较次数，这一优势可通过分析算法关键步骤的计算复杂度得到验证。在将我们算法的C/C++实现与GUDHI v3.9.0软件包进行对比的实验中，该方法在足够稀疏的Erdős-Rényi图上实现了5至10倍的加速比，且图越稀疏或Vietoris-Rips复形维数越高，优势越明显。我们进一步阐明，Boissonnat与Maria（https://doi.org/10.1007/978-3-642-33090-2_63）中描述的Vietoris-Rips复形构造算法本质上等同于Zomorodian（https://doi.org/10.1016/j.cag.2010.03.007）提出的增量算法，只是额外要求将结果存储于树结构中，并解释了这些技术与本文所提算法的区别。