A Composable Coreset for k-Center in Doubling Metrics (original) (raw)

A set of points PPP in a metric space and a constant integer kkk are given. The kkk-center problem finds kkk points as centers among PPP, such that the maximum distance of any point of PPP to their closest centers (r)(r)(r) is minimized. Doubling metrics are metric spaces in which for any rrr, a ball of radius rrr can be covered using a constant number of balls of radius r/2r/2r/2. Fixed dimensional Euclidean spaces are doubling metrics. The lower bound on the approximation factor of kkk-center is 1.8221.8221.822 in Euclidean spaces, however, (1+epsilon)(1+\epsilon)(1+epsilon)-approximation algorithms with exponential dependency on frac1epsilon\frac{1}{\epsilon}frac1epsilon and kkk exist. For a given set of sets P1,ldots,PLP_1,\ldots,P_LP1,ldots,PL, a composable coreset independently computes subsets C1subsetP1,ldots,CLsubsetPLC_1\subset P_1, \ldots, C_L\subset P_LC1subsetP1,ldots,CLsubsetPL, such that cupi=1LCi\cup_{i=1}^L C_icupi=1LCi contains an approximation of a measure of the set cupi=1LPi\cup_{i=1}^L P_icupi=1LPi. We introduce a (1+epsilon)(1+\epsilon)(1+epsilon)-approximation composable coreset for kkk-center, which in doubling metrics has size sublinea...