A Note on the Simplex-Tree Construction of the Vietoris-Rips Complex

We give an alternative presentation of the Simplex Tree construction of the Vietoris-Rips complex \cite{BoissonnatMaria2012}, which highlights how it takes advantage of a small amount of combinatorial structure in the $k$-skeleton of the complex in order to avoid unnecessary comparisons when identifying its $(k+1)$-simplices. We then show that it achieves an order-of-magnitude speedup over the Incremental-VR algorithm in Zomorodian \cite{Zomorodian_2010} when constructing the clique complexes of Erd\H{o}s-R\'enyi graphs.