Improved Coresets for Euclidean k -Means

Given a set of n points in d dimensions, the Euclidean k -means problem (resp. Euclidean k -median) consists of finding k centers such that the sum of squared distances (resp. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a (1\pm \varepsilon) factor. The best known lower bound for both problems is \Omega(k\varepsilon {-2}) .