Abstract
Given a set of points P ⊂ ℝd, the k-means clustering problem is to find a set of k centers C = {c 1,...,c k }, c i ∈ ℝd, such that the objective function ∑ x ∈ P d(x,C)2, where d(x,C) denotes the distance between x and the closest center in C, is minimized. This is one of the most prominent objective functions that have been studied with respect to clustering.
D 2-sampling [1] is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points P ⊆ ℝd, the first point is chosen uniformly at random from P. Subsequently, a point from P is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled points.
D 2-sampling has been shown to have nice properties with respect to the k-means clustering problem. Arthur and Vassilvitskii [1] show that k points chosen as centers from P using D 2-sampling gives an O(logk) approximation in expectation. Ailon et. al. [2] and Aggarwal et. al. [3] extended results of [1] to show that O(k) points chosen as centers using D 2-sampling give O(1) approximation to the k-means objective function with high probability. In this paper, we further demonstrate the power of D 2-sampling by giving a simple randomized (1 + ε)-approximation algorithm that uses the D 2-sampling in its core.
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Jaiswal, R., Kumar, A., Sen, S. (2012). A Simple D 2-Sampling Based PTAS for k-Means and other Clustering Problems. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_2
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