Before revision:  We prove that if C is a convex closed differentiable curve in the plane such that the convex hull of C contains the evolute of C, then the homotopy type of a Vietoris--Rips complex of a subset of n points from C can be computed in time O(n  \log n) . Furthermore, we show how to compute the k-dimensional persistent homology of these n points in running time O(n^2(k+\log n)), which is nearly quadratic in the number of vertices n . This improves upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most k+1, and hence of running time O(n^{3(k+2)})
After revision:  We prove that if C is a convex closed differentiable curve in the plane such that the convex hull of C contains the evolute of C, then the homotopy type of a Vietoris--Rips complex of a subset of n points from C can be computed in time O(n \log n) ) algorithm for computing the k-dimensional persistent homology of these n points in running time O(n^2(k+\log n)), which is nearly quadratic in the number of vertices n . This improves upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most k+1, and hence of running time O(n^{3(k+2)})