Preclustering Algorithms for Imprecise Points

We study the problem of preclustering a set B of imprecise points in ℝ^d: we wish to cluster the regions specifying the potential locations of the points such that, no matter where the points are located within their regions, the resulting clustering approximates the optimal clustering for those locations. We consider k-center, k-median, and k-means clustering, and obtain the following results. Let B:={b₁,…,b_n} be a collection of disjoint balls in ℝ^d, where each ball b_i specifies the possible locations of an input point p_i. A partition 𝒞 of B into subsets is called an (f(k),α)-preclustering (with respect to the specific k-clustering variant under consideration) if (i) 𝒞 consists of f(k) preclusters, and (ii) for any realization P of the points p_i inside their respective balls, the cost of the clustering on P induced by 𝒞 is at most α times the cost of an optimal k-clustering on P. We call f(k) the size of the preclustering and we call α its approximation ratio. We prove that, even in ℝ^1, one may need at least 3k-3 preclusters to obtain a bounded approximation ratio - this holds for the k-center, the k-median, and the k-means problem - and we present a (3k,1) preclustering for the k-center problem in ℝ^1. We also present various preclusterings for balls in ℝ^d with d⩾2, including a (3k,α)-preclustering with α≈13.9 for the k-center and the k-median problem, and α≈254.7 for the k-means problem.