A bad 2-dimensional instance for k-means++

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: {quote} Pick the first center randomly from among the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the previously $(i-1)$ chosen centers. {quote} The k-means++ seeding algorithm is not only simple and fast but gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii \cite{av07}. There are datasets \cite{av07,adk09} on which this seeding algorithm gives an approximation factor $\Omega(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably large probability (say $1/poly(k)$). Brunsch and R\"{o}glin \cite{br11} gave a dataset where the k-means++ seeding algorithm achieves an approximation ratio of $(2/3 - \epsilon)\cdot \log{k}$ only with probability that is exponentially small in $k$. However, this and all other known {\em lower-bound examples} \cite{av07,adk09} are high dimensional. So, an open problem is 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 approximation ratio $c$ (for some universal constant $c$) only with probability exponentially small in $k$. This is the first step towards solving open problems posed by Mahajan et al \cite{mnv12} and by Brunsch and R\"{o}glin \cite{br11}.