A Extension to k-Means and (k, p)-Clustering

The lower bound on opt( U) given in Lemma B.10 holds for ρ -metric spaces with no modifications. By making the appropriate modifications to the proof of Theorem C.1, we can extend this theorem to In particular, we can obtain a proof of Theorem A.5 by taking the proof of Theorem C.1 and adding extra ρ factors whenever the triangle inequality is applied. We first prove Lemma B.1, which shows that the sizes of the sets U By Lemma B.2, we get that Henceforth, we fix some positive ξ and sufficiently large α such that Lemma B.3 holds. By now applying Lemma B.4 it follows that The following lemma is proven in [25]. Lemma B.1, the third inequality follows from Lemma B.7, and the fourth inequality follows from the The second inequality follows from Lemma B.8, the third inequality from averaging and the choice Proof of Lemma 3.3: It follows that with probability at least 1 e Hence, by Theorem D.1, we must have that O (poly( k)) query time must have Ω( k) amortized update time.