ε-核的稳定性
Stability of epsilon-Kernels
摘要：
给定n个点的集合P，|R^d，eps核K子集P在相对（1-eps）因子内在每个方向上近似P的方向宽度。本文研究了eps核在P上动态插入和删除点以及改变eps逼近因子时的稳定性。在第一种情况下，我们说动态维护eps-内核的算法是稳定的，如果在K中最多O（1）点的变化，因为一个点被插入或删除P。我们描述了一个算法，以维持大小为O（1/eps ^{（d-1）/2}）的eps-内核在O（1/eps^{（d-1）/2} + log n）时间每次更新。我们的算法不仅保持了一个稳定的eps内核，其更新时间比任何已知的算法，保持一个大小为O（1/eps^{（d-1）/2}）的eps内核更快。接下来，我们证明了如果存在大小为k的P的eps-核，其可能显著小于O（1/eps^{（d-1）/2}），则存在大小为O（min {1/eps^{（d-1）/2}，k^{floor（d/2）} log^{d-2}（1/eps）}）的P的（eps/2）-核。此外，存在点集P，|R^d和参数eps > 0，使得如果P的每个eps-核的大小至少为k，则P的任何（eps/2）-核的大小为Ω（k^{floor（d/2）}）。
英文摘要：
Given a set P of n points in |R^d, an eps-kernel K subset P approximates the directional width of P in every direction within a relative (1-eps) factor. In this paper we study the stability of eps-kernels under dynamic insertion and deletion of points to P and by changing the approximation factor eps. In the first case, we say an algorithm for dynamically maintaining a eps-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 eps-kernel of size O(1/eps^{(d-1)/2}) in O(1/eps^{(d-1)/2} + log n) time per update. Not only does our algorithm maintain a stable eps-kernel, its update time is faster than any known algorithm that maintains an eps-kernel of size O(1/eps^{(d-1)/2}). Next, we show that if there is an eps-kernel of P of size k, which may be dramatically less than O(1/eps^{(d-1)/2}), then there is an (eps/2)-kernel of P of size O(min {1/eps^{(d-1)/2}, k^{floor(d/2)} log^{d-2} (1/eps)}). Moreover, there exists a point set P in |R^d and a parameter eps > 0 such that if every eps-kernel of P has size at least k, then any (eps/2)-kernel of P has size Omega(k^{floor(d/2)}).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
