Before revision:  in the number of vertices n  . We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.
After revision:  in the number of vertices n . Our algorithm applies, for example, to Vietoris--Rips complexes of points sampled from a curve in \mathbb{R . We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.