A Tight Lower Bound Instance for k-means++ in Constant Dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial k centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For i > 1, pick a point to be the i center with probability proportional to the square of the Euclidean distance of this point to the closest previously (i− 1) chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an O(log k) approximation in expectation as shown by Arthur and Vassilvitskii [7]. There are datasets [7, 3] on which this seeding algorithm gives an approximation factor of Ω(log k) in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say 1/poly(k)). Brunsch and Röglin [9] gave a dataset where the k-means++ seeding algorithm achieves an O(log k) approximation ratio with probability that is exponentially small in k. However, this and all other known lower-bound examples [7, 3] are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an O(log k) approximation ratio with probability exponentially small in k. This solves open problems posed by Mahajan et al. [13] and by Brunsch and Röglin [9].