Fast construction of the Vietoris-Rips complex

The Vietoris-Rips complex characterizes the topology of a point set. This complex is popular in topological data analysis as its construction extends easily to higher dimensions. We formulate a two-phase approach for its construction that separates geometry from topology. We survey methods for the first phase, give three algorithms for the second phase, implement all algorithms, and present experimental results. Our software can also be used for constructing any clique complex, such as the weak witness complex.