On Vietoris-Rips complexes of planar curves
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A Vietoris--Rips complex is a way to thicken a (possibly discrete) metric space into a larger space containing more topological information. 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 n). Furthermore, we show how to compute the k-dimensional persistent homology of these n points in running time O(n^2(k+ 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)) 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.
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