Deterministic O(1)-Approximation Algorithms to 1-Center Clustering with Outliers
share
The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant 0 < α < 1 and n points such that α n of them are in some (unknown) ball of radius r, the goal is to compute a ball of radius O(r) that also contains α n points. This problem can be formulated with the points in a normed vector space such as R^d or in a general metric space. The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any L^p vector space, we show an O(nd)-time solution with a ball of radius O(r) for a fixed α > 1/2, and for any normed vector space, we show an O(nd)-time solution with a ball of radius O(r) when α > 1/2 as well as an O (nd ^(k)(n))-time solution with a ball of radius O(r) for all α > 0, k ∈N, where ^(k)(n) represents the kth iterated logarithm, assuming distance computation and vector space operations take O(d) time. For an arbitrary metric space, we show for any C ∈N an O(n^1+1/C)-time solution that finds a ball of radius 2Cr, assuming distance computation between any pair of points takes O(1)-time, and show that for any α, C, an O(n^1+1/C)-time solution that finds a ball of radius ((2C-3)(1-α)-1)r cannot exist.
READ FULL TEXT