Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
An O(n log n) algorithm for the all-nearest-neighbors problem
Discrete & Computational Geometry
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Approximation algorithms for geometric problems
Approximation algorithms for NP-hard problems
Faster construction of planar two-centers
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Exact and approximation algorithms for clustering
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
ACM Computing Surveys (CSUR)
Reductions among high dimensional proximity problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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In this paper, we consider the problem of clustering a set of n finite point-sets in d-dimensional Euclidean space. Different from the traditional clustering problem (called points clustering problem where the to-be-clustered objects are points), the point-sets clustering problem requires that all points in a single point-set be clustered into the same cluster. This requirement disturbs the metric property of the underlying distance function among point-sets and complicates the clustering problem dramatically. In this paper, we use a number of interesting observations and techniques to overcome this difficulty. For the k-center clustering problem on point-sets, we give an O(m+nlogk)-time 3-approximation algorithm and an O(km)-time (1+3)-approximation algorithm, where m is the total number of input points and k is the number of clusters. When k is a small constant, the performance ratio of our algorithm reduces to (2+@e) for any @e0. For the k-median problem on point-sets, we present a polynomial time (3+@e)-approximation algorithm. Our approaches are rather general and can be easily implemented for practical purpose.