Stability of ε-Kernels

Given a set P of n points in R, 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 + log n) 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, κbd/2c logd−2(1/ε)}). Moreover, there exists a point set P in R 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(κbd/2c). 1 Research supported by subaward CIF-32 from NSF grant 0937060 to CRA, by NSF under grants CNS-05-40347, CFF-0635000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, by an NIH grant 1P50-GM-08183-01, by a DOE grant OEG-P200A070505, and by a grant from the U.S.–Israel Binational Science Foundation.