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We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is $α$-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most $α$. Our main contribution is in presenting efficient exact algorithms for $α$-stable clustering instances whose running times depend near-linearly on the size of the data set when $α\ge 2 + \sqrt{3}$. For $k$-center and $k$-means problems, our algorithms also achieve polynomial dependence on the number of clusters, $k$, when $α\geq 2 + \sqrt{3} + ε$ for any constant $ε> 0$ in any fixed dimension. For $k$-median, our algorithms have polynomial dependence on $k$ for $α> 5$ in any fixed dimension; and for $α\geq 2 + \sqrt{3}$ in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.