Distributed and Provably Good Seedings for k-Means in Constant Rounds

The k-means++ algorithm is the state of the art algorithm to solve k-Means clustering problems as the computed clusterings are O(log k) competitive in expectation. However, its seeding step requires k inherently sequential passes through the full data set making it hard to scale to massive data sets. The standard remedy is to use the k-means‖ algorithm which reduces the number of sequential rounds and is thus suitable for a distributed setting. In this paper, we provide a novel analysis of the k-means‖ algorithm that bounds the expected solution quality for any number of rounds and oversampling factors greater than k, the two parameters one needs to choose in practice. In particular, we show that k-means‖ provides provably good clusterings even for a small, constant number of iterations. This theoretical finding explains the common observation that k-means‖ performs extremely well in practice even if the number of rounds is low. We further provide a hard instance that shows that an additive error term as encountered in our analysis is inevitable if less than k−1 rounds are employed.