Abstract

Given a set of points P ⊂ Rd and value ɛ> 0, an ɛ-core-set S ⊂ P has the property that the smallest ball containing S is within ɛ of the smallest ball containing P. This paper shows that any point set has an ɛ-core-set of size ⌈1/ɛ⌉, and this bound is tight in the worst case. A faster algorithm given here finds an ɛ-core-set of size at most 2/ɛ. These results imply the existence of small core-sets for solving approximate k-center clustering and related problems. The sizes of these core-sets are considerably smaller than the previously known bounds, and imply faster algorithms; one such algorithm needs O(dn/ɛ + (1/ɛ) 5) time to compute an ɛ-approximate minimum enclosing ball (1-center) of n points in d dimensions. A simple gradient-descent algorithm is also given, for computing the minimum enclosing ball in O(dn/ɛ2) time. This algorithm also implies slightly faster algorithms for computing approximately the smallest radius k-flat fitting a set of points. 1

Citations