Stability of ε-kernels

Abstract

Given a set P of n points in ℝd, an ε-kernel K ⊆ P approximates the directional width of P in every direction within a relative (1 - ε) factor. In this paper we study the stability of ε-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor ε. In the first case, we say an algorithm for dynamically maintaining a ε-kernel is stable if at most O(1) points change in K as one point is inserted or deleted from P. We describe an algorithm to maintain an ε-kernel of size O(1/ε(d - 1)/2) in O(1/ε(d - 1)/2 + logn) time per update. Not only does our algorithm maintain a stable ε-kernel, its update time is faster than any known algorithm that maintains an ε-kernel of size O(1/ε (d - 1)/2). Next, we show that if there is an ε-kernel of P of size κ, which may be dramatically less than O(1/ε (d - 1)/2), then there is an (ε/2)-kernel of P of size O(min{1/ε(d-1)/2, κ⌊d/2⌋ log d-2(1/ε)}).. Moreover, there exists a point set P in ℝd and a parameter ε > 0 such that if every ε-kernel of P has size at least κ, then any (ε/2)-kernel of P has size Ω(κ⌊d/2⌋). © 2010 Springer-Verlag.