Stability of ε-Kernels

Abstract

Given a set P of n points in ℝ^d, 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 $O(\min\{ 1/\varepsilon^{(d-1)/2}, \kappa^{\lfloor d/2 \rfloor} \log^{d-2} (1/\varepsilon)\})$. Moreover, there exists a point set P in ℝ^d 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 $\Omega(\kappa^{\lfloor d/2 \rfloor})$.