A New Construction of the Vietoris-Rips Complex

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.