We study the design of efficient approximation algorithms for the
ℓ-center clustering and minimum-diameter
ℓ-clustering problems in high-dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the
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We study the design of efficient approximation algorithms for the
ℓ-center clustering and minimum-diameter
ℓ-clustering problems in high-dimensional Euclidean and Hamming spaces. Our main tool is randomized dimension reduction. First, we present a general method of reducing the dependency of the running time of a hypothetical algorithm for the
ℓ-center problem in a high-dimensional Euclidean space on the dimension. Utilizing this method in part, we provide
-approximation algorithms for the
ℓ-center clustering and minimum-diameter
ℓ-clustering problems in Euclidean and Hamming spaces that are substantially faster than the known 2-approximation algorithms when both
ℓ and the dimension are super-logarithmic. Next, we apply the general method to the recent fast approximation algorithms with higher approximation guarantees for the
ℓ-center clustering problem in a high-dimensional Euclidean space. Finally, we provide a speed-up of the known
-approximation method for the generalization of the
ℓ-center clustering problem that allows
z outliers (i.e.,
z input points may be ignored when computing the maximum distance from an input point to a center) in high-dimensional Euclidean and Hamming spaces.
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