Before revision:  We then use this result, as well as the fact that \mathcal{M}^h can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with  rate of convergence O(h^{k-1}) .
After revision:  We then use this result, as well as the fact that \mathcal{M}^h can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence O(h^{k-1}) .