We present algorithms for the online minimum hitting set problem: Given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times. For disks of radii in the interval $[1,M]$, we present a $O(\log M \log n)$-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval $[1,M]$. As a main technical tool, we reduce the problem to the online hitting set problem for integer points and bottomless rectangles. Specifically, we present an $O(\log N)$-competitive algorithm for the variant where $P$ is a set of integer points in an $N\times N$ box, and the geometric objects are bottomless rectangles.

翻译：本文针对在线最小击中集问题提出算法：给定平面上包含$n$个点的集合$P$以及按序到达的几何对象序列，我们需要实时维护一个击中集。对于半径属于区间$[1,M]$的圆盘，我们提出了一个$O(\log M \log n)$竞争比的算法。该结果可从圆盘推广至平面上任意凸体在缩放因子属于区间$[1,M]$时的正位似变换对象。作为核心技术工具，我们将问题规约至整数点与无顶矩形的在线击中集问题。具体而言，针对$P$为$N\times N$网格中整数点集且几何对象为无顶矩形的变体问题，我们提出了$O(\log N)$竞争比的算法。