We consider the online hitting set problem for the range space $\Sigma=(\cal X,\cal R)$, where the point set $\cal X$ is known beforehand, but the set $\cal R$ of geometric objects is not known in advance. Here, geometric objects arrive one by one, the objective is to maintain a hitting set of minimum cardinality by taking irrevocable decisions. In this paper, we have considered the problem when the objects are unit balls or unit hypercubes in $\mathbb{R}^d$, and the points from $\mathbb{Z}^d$ are used for hitting them. First, we consider the problem for objects (unit balls and unit hypercubes) in lower dimensions. We obtain $4$ and $8$-competitive deterministic online algorithms for hitting unit hypercubes in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively. On the other hand, we present $4$ and $14$-competitive deterministic online algorithms for hitting unit balls in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively. Next, we consider the problem for objects (unit balls and unit hypercubes) in the higher dimension. For hitting unit hypercubes in $\mathbb{R}^d$, we present a $O(d^2)$-competitive randomized online algorithm for $d\geq 3$ and prove the competitive ratio of any deterministic algorithm for the problem is at least $d+1$ for any $d\in\mathbb{N}$. Then, for hitting unit balls in $\mathbb{R}^d$, we propose a $O(d^4)$-competitive deterministic algorithm, and for $d<4$, we establish that the competitive ratio of any deterministic algorithm is at least $d+1$.

翻译：我们考虑范围空间 $\Sigma=(\cal X,\cal R)$ 的在线击打集问题，其中点集 $\cal X$ 事先已知，但几何对象集 $\cal R$ 未知。在此问题中，几何对象逐一到达，目标是通过不可逆决策维护基数最小的击打集。本文考虑对象为 $\mathbb{R}^d$ 中单位球或单位超立方体，且使用 $\mathbb{Z}^d$ 中点集进行击打的情况。首先，我们研究低维空间中对象（单位球与单位超立方体）的问题。针对 $\mathbb{R}^2$ 和 $\mathbb{R}^3$ 中的单位超立方体击打，我们分别提出了竞争比为 $4$ 和 $8$ 的确定性在线算法；针对 $\mathbb{R}^2$ 和 $\mathbb{R}^3$ 中的单位球击打，则分别给出了竞争比为 $4$ 和 $14$ 的确定性在线算法。其次，考虑高维空间中对象的问题。对于 $\mathbb{R}^d$ 中单位超立方体的击打，我们为 $d\geq 3$ 提出了竞争比为 $O(d^2)$ 的随机化在线算法，并证明对任意 $d\in\mathbb{N}$，该问题任何确定性算法的竞争比至少为 $d+1$。对于 $\mathbb{R}^d$ 中单位球的击打，我们提出竞争比为 $O(d^4)$ 的确定性算法，且当 $d<4$ 时，证明任何确定性算法的竞争比至少为 $d+1$。